{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p> <br> <p> <br>  <p> <br>  <p>\n",
    "<hr>\n",
    "# Κεφάλαιο 7\n",
    "<hr>\n",
    "<p> <br> <p> <br>  <p> <br>  <p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7. Σειρές Fourier"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"imgContainer\">\n",
    "        <img src=\"fourier.jpg\" height=\"150\" width=\"150\"/>\n",
    "            </div> "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p> <font style=\"font-family:Dejavu Serif;FONT-SIZE:15px;LINE-HEIGHT:25px;\"> \n",
    "Λίγο πριν το 1800, ο Γάλλος μαθηματικός/φυσικός/μηχανικός Jean Baptiste Joseph Fourier έκανε μια εκπληκτική ανακάλυψη. Μέσω των ενδελεχών αναλυτικών ερευνών του στις Μερικές Διαφορικές Εξισώσεις που μοντελοποιούν την διάδοση της θερμότητας σε σώματα, ο Fourier οδηγήθηκε στον ισχυρισμό ότι κάθε συνάρτηση μπορεί να παρασταθεί ώς ένα άπειρο άθροισμα από στοιχειώδεις τριγωνομετρικές συναρτήσεις, ημιτόνων και συνημιτόνων. <br> <br>\n",
    "\n",
    "Για παράδειγμα,  αναλύοντας τον ήχο που παράγεται από ένα πιάνο, βιολί, τρομπέτα ή ένα τύμπανο στις τριγωνομετρικές συνιστώσες του αποκαλύπτονται οι βασικές συχνότητες οι οποίες συνδυάζονται για να παράγουν το ιδιαίτερο ηχόχρωμα του κάθε μουσικού οργάνου. <br><br>\n",
    "\n",
    "Η ανακάλυψη αυτή του Fourier συγκαταλέγεται πολύ εύκολα στις δέκα σημαντικότερες μαθηματικές εξελίξεις όλων των εποχών, συμπεριλαμβανομένων της ανάλυσης του Newton, και της διαφορικής γεωμετρίας των Gauss και Riemann. Η ανάλυση Fourier αποτελεί πλέον μια ουσιώδη συνιστώσα πολλών σύγχρονων κλάδων τόσο των εφαρμοσμένων όσο και των θεωρητικών μαθηματικών. Αποτελεί ένα ισχυρότατο αναλυτικό εργαλείο για να επιλύσουμε ένα ευρύ φάσμα γραμμικών διαφορικών εξισώσεων στην φυσική, μηχανική, βιολογία, οικονομικά, ή ακόμα και για να \"ακούσουμε\" την κατανομή των πρώτων αριθμών! Η ανάλυση Fourier βρίσκεται στην καρδιά της επεξεργασίας ήχου, φωνής, εικόνας, σεισμικών δεδομένων και ραδιοφωνικής μετάδοσης. Πολλές σύγχρονες τεχνολογίες όπως η τηλεόραση, μουσικά CD και DVD, κινητά τηλέφωνα, κινηματογραφικές ταινίες, γραφικά υπολογιστών, ανάλυση δακτυλογραφικών αποτυπωμάτων κ.α. έχουν τα θεμέλιά τους στην θεωρία του Fourier. Είναι ένα από τα ισχυρότερα όπλα στην φαρέτρα κάθε μαθηματικού, φυσικού, μηχανικού, όπως ακριβώς η ανάλυση και η γραμμική άλγεβρα.\n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.1 Τι είναι μια σειρά Fourier"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p> <font style=\"font-family:Dejavu Serif;FONT-SIZE:15px;LINE-HEIGHT:25px;\"> \n",
    "Ας θεωρήσουμε μια συνεχή συνάρτηση $f(x)$ που ορίζεται στο διάστημα $[-\\pi,\\pi]$. Η σειρά Fourier της $f(x)$ ορίζεται από την σχέση  <br><br>\n",
    "$$ f(x) \\sim \\frac{a_0}{2} + \\sum_{k=1}^\\infty \\left[\\,\\, a_k \\, \\cos k\\,x + b_k \\, \\sin k\\,x \\,\\,\\right] \\,, $$ <br>\n",
    "όπου το $\\sim$ δηλώνει ότι η συνάρτηση $f(x)$ έχει την αναπαράσταση με τον άπειρο γραμμικό συνδυασμό των ημιτόνων και συνημιτόνων στο δεξί μέλος της παραπάνω σχέσης. Τα $a_k$, $b_k$ ονομάζονται συντελεστές Fourier."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Όπως ήδη γνωρίζουμε ένα άπειρο άθροισμα, ή καλύτερα σειρά, είναι πολύ πιο ευαίσθητη από μια πεπερασμένη σειρά και συνεπώς μια τέτοια φορμαλιστική κατασκευή απαιτεί μια πολλή προσεχτική μαθηματική ανάλυση. \n",
    "Τα κύρια ερωτήματα που θα μας επιτρέψουν να ξεκλειδώσουμε το πρόβλημα από μαθηματική σκοπιά είναι:\n",
    "<ul> <li>\n",
    " Πότε μια άπειρη τριγωνομετρική σειρά συγκλίνει? </li> <li>\n",
    " Τι είδους συναρτήσεις $f(x)$ μπορούν να παρασταθούν από μια συγκλίνουσα σειρά Fourier? </li> <li>\n",
    " Δοσμένης μια τέτοιας συνάρτησης $f(x)$, πως μπορούμε να καθορίσουμε τους συντελεστές Fourier $a_k$, $b_k$; </li> <li>\n",
    " Επιτρέπεται να παραγωγίζουμε και να ολοκληρώνουμε σειρές Fourier όρο προς όρο?</li>\n",
    " </ul>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Από τα παραπάνω ερωτήματα η πρώτη δουλειά που έχουμε να κάνουμε είναι να δούμε αν μπορούμε να βρούμε τους συντελεστές $a_k$, $b_k$, και μετά οτιδήποτε άλλο όπως θέματα σύγκλισης κτλ.\n",
    "<br><br>\n",
    "Το κλειδί που ξεκλειδώνει το σεντούκι του θησαυρού Fourier είναι η **ορθογωνιότητα**. Δυο διανύσματα $\\mathbf{v}$, $\\mathbf{w}$ στον $\\mathbb{R}^n$ είναι ορθογώνια αν και μόνο αν το  εσωτερικό τους γινόμενο είναι μηδέν $\\mathbf{v}\\cdot\\mathbf{w}=0$. Όπως έχουμε ήδη διαπιστώσει η ορθογωνιότητα, και ειδικότερα οι ορθοκανονικές βάσεις, έχουν πολύ σημαντικές συνέπειες στην γραμμική άλγεβρα, όπως στη προσέγγιση συνεχών συναρτήσεων σε ένα διάστημα. <br><br> \n",
    "Η αφετηρία είναι να ορίσουμε ένα κατάλληλο εσωτερικό γινόμενο στον απειροδιάστατο χώρο συναρτήσεων σε ένα διάστημα, το οποίο θα παίζει τον ρόλο του εσωτερικού γινομένου στο πεπερασμένο χώρο των οικείων μας διανυσμάτων. Για τις κλασικές σειρές Fourier, χρησιμοποιούμε το $\\mathbf{L}^2$ εσωτερικό γινόμενο<br> <br>\n",
    "$$ < f , g > = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x)\\,g(x) \\, {\\mathrm{d}} x\\,,\\qquad\\qquad\\qquad (1)$$ <br>\n",
    "στο χώρο $C_{\\mathbb{R}}[-\\pi,\\pi]$ των συνεχών συναρτήσεων με πεδίο ορισμού στο διάστημα $[-\\pi,\\pi]$. Δεν είναι δύσκολο να δείξει κανείς ότι το παραπάνω εσωτερικό γινόμενο ικανοποιεί όλα τα αξιώματα (διγραμμικότητα, συμμετρία, θετικά ορισμένο) ενός εσωτερικού γινομένου στον (απειροδιάστατο) διανυσματικό χώρο  $C_{\\mathbb{R}}[-\\pi,\\pi]$, των πραγματικών συνεχών συναρτήσεων στο διάστημα $[-\\pi,\\pi]$. <br>\n",
    "Η αντίστοιχη νόρμα είναι \n",
    "$$ \\left\\lVert f \\right\\rVert = \\sqrt{< f,f > } = \\sqrt{\\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x)^2 \\, {\\mathrm{d}} x\\,}\\,.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Λήμμα 1.** Με το εσωτερικό γινόμενο (1), οι τριγωνομετρικές συναρτήσεις <br><br>\n",
    "$$1\\,,\\quad \\cos x\\,,\\quad \\sin x \\,,\\quad \\cos 2\\,x \\,, \\quad \\sin 2\\,x \\,,\\ldots $$<br>\n",
    "ικανοποιούν τις ακόλουθες σχέσεις ορθογωνιότητας <br><br>\n",
    "$$\\begin{array}{ccc} < \\cos k \\,x , \\cos \\ell\\,x > \\,=\\, < \\sin k \\,x , \\sin \\ell\\,x > = 0& \\mbox{για}  & k \\neq \\ell \\,,\\\\ \\\\ < \\cos k \\,x , \\sin \\ell\\,x > = 0\\,, & \\mbox{για κάθε} & k\\,,\\ell \\,,\n",
    "\\\\  \\\\ \\left\\lVert 1 \\right\\rVert = \\sqrt{2}\\,,\\quad \\left\\lVert \\cos k\\,x \\right\\rVert = \\left\\lVert \\sin k\\,x \\right\\rVert = 1\\,, & \\mbox{για} & k\\neq 0\\,,\n",
    "\\end{array} $$<br>\n",
    "όπου $k,\\ell$ μη αρνητικοί ακέραιοι. <br>\n",
    "Η απόδειξη είναι καθαρά θέμα πράξεων, οπότε χρησιμοποιούμε το Sage! <br><br>\n",
    "***Απόδειξη***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def inner(f,g):\n",
    "    return 1/pi * integrate(f*g,x,-pi,pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "0\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "var('k l')\n",
    "assume(k,'integer')\n",
    "assume(k>0)\n",
    "assume(l,'integer')\n",
    "assume(l>0)\n",
    "print inner(cos(k*x),cos(l*x))\n",
    "print inner(sin(k*x),sin(l*x))\n",
    "print inner(cos(k*x),sin(l*x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sqrt(2)\n"
     ]
    }
   ],
   "source": [
    "norm_1 = sqrt(inner(1,1)) ; print norm_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "norm_cos = sqrt(inner(cos(k*x),cos(k*x))) ; print norm_cos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "norm_sin = sqrt(inner(sin(k*x),sin(k*x))) ; print norm_sin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "για κάθε μη-αρνητικούς ακέραιους $k\\,,\\ell \\geq 0$. <br><br> Το προηγούμενο λήμμα συνεπάγεται ότι οι στοιχειώδεις τριγωνομετρικές συναρτήσεις φτιάχνουν ένα **ορθογώνιο σύστημα**, που σημαίνει ότι κάθε ζευγάρι συναρτήσεων είναι ορθογώνιο μεταξύ τους, ως προς το εσωτερικό γινόμενο (1). Αν αντικαθιστούσαμε το $1$ με το $\\frac{1}{\\sqrt{2}}$, τότε θα είχαμε ένα **ορθοκανονικό σύστημα**, που σημαίνει ότι επιπλέον όλα τα στοιχεία θα είχαν νόρμα $1$. Όμως, το επιπλέον $\\sqrt{2}$ είναι  εντελώς ενοχλητικό, και είναι καλύτερα να μην το λαμβάνουμε υπόψη και να χρησιμοποιούμε το $1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Παρατήρηση:*** Δεν είναι τυχαίο το γεγονός ότι οι τριγωνομετρικές συναρτήσεις είναι ορθογώνιες μεταξύ τους. Είναι απόρροια του γεγονότος ότι οι τριγωνομετρικές συναρτήσεις είναι οι ιδιοσυναρτήσεις ενός αυτοσυζυγούς προβλήματος συνοριακών τιμών, το οποίο είναι το \"συναρτησιακό\" ανάλογο της ορθογωνιότητας των ιδιοδιανυσμάτων για συμμετρικούς πίνακες στην γραμμική άλγεβρα."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Αν αφήσουμε προς το παρόν θέματα σύγκλισης τριγωνομετρικών σειρών, τότε οι σχέσεις ορθογωνιότητας χρησιμεύουν στο να καθορίσουμε τους συντελεστές Fourier $a_k$ , $b_k$. παίρνοντας το εσωτερικό γινόμενο στα δυο μέλη της εξίσωσης <br><br>\n",
    "$$ f(x) = \\frac{a_0}{2} + \\sum_{k=1}^\\infty \\left[\\,\\, a_k \\, \\cos k\\,x + b_k \\, \\sin k\\,x \\,\\,\\right] \\,,$$ <br>\n",
    "έχουμε \n",
    "$$ \\begin{array}{rl} \n",
    "< f \\,,\\, \\cos \\ell\\,x > & = & \\frac{a_0}{2} < 1 \\,,\\, \\cos \\ell\\,x > + \n",
    "\\sum_{k=1}^\\infty \\left[\\,\\, a_k \\, < \\cos k\\,x \\, ,\\, \\cos \\ell \\, x >  + \n",
    "b_k \\, < \\sin k\\,x \\, ,\\, \\cos \\ell \\, x >\\,\\,\\right]  \\\\ \\\\\n",
    "& = & a_\\ell < \\cos \\ell \\,x \\, , \\,  \\cos \\ell \\, x > = a_\\ell \\,,\\end{array} \n",
    "$$ <br>\n",
    "δηλαδή  <br><br>\n",
    "$$ a_k \\,=\\, \\, < f \\, , \\, \\cos k\\,x > \\,,$$<br>\n",
    "Με παρόμοιο τρόπο παίρνουμε ότι <br><br>\n",
    "$$ b_k \\,=\\, \\, < f \\, , \\, \\sin k\\,x > \\,,$$ <br>\n",
    "και το εσωτερικό γινόμενο της $f(x)$ με την μονάδα δίνει <br><br>\n",
    "$$< f \\, , \\, 1 > =  \\frac{a_0}{2} < 1 \\,, 1 > + \n",
    "\\sum_{k=1}^\\infty \\left[\\,\\, a_k \\, < \\cos k\\,x \\, , \\, 1 >  +  b_k \\, < \\sin k\\,x \\, ,\\, 1 > \\,\\,\\right]   = a_0 \\,,$$ <br>\n",
    "που εξηγεί τον λόγο που πήραμε ως νόρμα του $1$ το $\\frac{1}{\\sqrt{2}}$. Οπότε καταλήγουμε στο συμπέρασμα ότι *αν η σειρά Fourier συγκλίνει στην συνάρτηση $f(x)$ , τότε οι συντελεστές Fourier της σειράς καθορίζονται παίρνοντας τα εσωτερικά γινόμενα της συνάρτησης με τις βασικές τριγωνομετρικές συναρτήσεις.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Ορισμός 2.** Η σειρά Fourier μιας συνάρτησης $f(x)$ που ορίζεται στο διάστημα $[-\\pi,\\pi]$ είναι η <br><br>\n",
    "$$ f(x) \\sim \\frac{a_0}{2} + \\sum_{k=1}^\\infty \\left[\\,\\, a_k \\, \\cos k\\,x + b_k \\, \\sin k\\,x \\,\\,\\right]\\,, \\qquad\\qquad\\qquad\\qquad\\quad (2.a)$$ <br>\n",
    "της οποίας οι συντελεστές Fourier δίνονται από τις σχέσεις των εσωτερικών γινομένων <br><br>\n",
    "$$ \n",
    "a_k \\,=\\, \\, < f \\, , \\, \\cos k\\,x > = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x)\\,\\cos k\\,x \\, {\\mathrm{d}} x\\,,\\qquad k=0,1,2,\\ldots \\qquad (2.b) \n",
    "$$<br>\n",
    "$$ \n",
    "b_k \\,=\\, \\, < f \\, , \\, \\sin k\\,x > = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x)\\,\\sin k\\,x \\, {\\mathrm{d}} x\\,,\\qquad k=1,2,\\ldots \\qquad\\quad (2.c) \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Η συνάρτηση $f(x)$ δεν μπορεί να είναι τελείως αυθαίρετη, αφού τουλάχιστον τα ολοκληρώματα θα πρέπει να υπάρχουν και να είναι πεπερασμένοι πραγματικοί αριθμοί. Έστω κι αν οι συντελεστές είναι πεπερασμένοι αριθμοί δεν υπάρχει εγγύηση ότι οι σειρές που προκύπτουν θα συγκλίνουν, και έστω κι αν συγκλίνουν δεν υπάρχει εγγύηση ότι θα συγκλίνουν στην συνάρτηση $f(x)$, έστω και κατά σημείο. Γι' αυτούς τους λόγους τείνουμε να χρησιμοποιούμε το σύμβολο $\\sim$ αντί αυτού της ισότητας όταν γράφουμε μια σειρά Fourier.<br><br>\n",
    "\n",
    "**Παράδειγμα 3.** Θεωρούμε την συνάρτηση $f(x) = x$. Οι συντελεστές Fourier της $f(x)$ είναι: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x |--> x\n"
     ]
    }
   ],
   "source": [
    "f(x)=x ; print f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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oIq6mJSA/2rzZLPns32/O7R80SOWdtmRkZNCuXTv27NlDvXr1LvgaLQGJ12kG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\nHqUBQETEozQAiIh41P8Dk0jWXtLOUQcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pf = plot(f,(x,-pi,pi)); pf.show(figsize=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def inner(f,g):\n",
    "    return 1/pi * integrate(f*g,x,-pi,pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a0 =  0\n",
      "ak =  0\n",
      "bk =  -2*(-1)^k/k\n"
     ]
    }
   ],
   "source": [
    "var('k')\n",
    "assume(k,'integer')\n",
    "a0 = inner(f(x),1); print str('a0 = '), a0; \n",
    "a(k) = inner(f(x),cos(k*x)) ; print str('ak = '), a(k);\n",
    "b(k) = inner(f(x),sin(k*x)); print str('bk = '), b(k);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, \\sin\\left(4 \\, x\\right) + \\frac{2}{3} \\, \\sin\\left(3 \\, x\\right) - \\sin\\left(2 \\, x\\right) + 2 \\, \\sin\\left(x\\right)</script></html>"
      ],
      "text/plain": [
       "-1/2*sin(4*x) + 2/3*sin(3*x) - sin(2*x) + 2*sin(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fourier4 = a0/2 + sum( a(k)*cos(k*x) + b(k)*sin(k*x) for k in [1..4]); \n",
    "fourier4.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "fourier6 = a0/2 + sum( a(k)*cos(k*x) + b(k)*sin(k*x) for k in [1..6]); \n",
    "fourier10 = a0/2 + sum( a(k)*cos(k*x) + b(k)*sin(k*x) for k in [1..10]); "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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EVXAwACD/xg2Dj8EwpmoxiImk7d4efnNsFi6sWIGY55+Hqy/voDYF5Z3pTfnXrhl8DIYx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\na6tlZ/JtGAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics Array of size 2 x 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pfr4 = plot(fourier4,(x,-pi,pi),color='brown'); \n",
    "pfr6 = plot(fourier6,(x,-pi,pi),color='green');\n",
    "pfr10 = plot(fourier10,(x,-pi,pi),color='red'); \n",
    "myGraphicsArray = graphics_array([[pf,pf+pfr4],[pf+pfr6,pf+pfr10]])\n",
    "myGraphicsArray.show(figsize=5,ticks=[[],[]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Παρατηρούμε ότι όσο περισσότερους όρους περιλαμβάνουμε στην σειρά Fourier τόσο καλύτερα η τελευταία προσεγγίζει την συνάρτησή μας $f(x)$, όχι όμως σε όλα τα σημεία στο διάστημα $[-\\pi,\\pi]$. Όσους όρους και να πάρουμε στην σειρά, στα άκρα του διαστήματος η σειρά εξακολουθεί να  αστοχεί να βρει την πραγματική τιμή της $f(x)$. Το φαινόμενο αυτό είναι γνωστό ως **φαινόμενο Gibbs**. Για παράδειγμα, αν πάρουμε 50 όρους στην σειρά Fourier, τότε προκύπτει η παρακάτω προσέγγιση"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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FTy9ba3NrCmIRkdwtIwMmTYJ5A7axJnE4v7cdzOh5dbUUOKtoHbGISC60axf88gv7vKvy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\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fourier50 = a0/2 + sum( a(k)*cos(k*x) + b(k)*sin(k*x) for k in [1..50]); \n",
    "pfr50 = plot(fourier50,(x,-pi,pi),color='red',ticks=[pi,pi], tick_formatter=[pi,pi] ); \n",
    "(pf+pfr50).show(figsize=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Τι γίνεται όμως πέρα από το διάστημα $[-\\pi,\\pi]$ ;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Wllbpz83JyXHv+wxDLrJZs8hZvpy0mTPh/vsla6eCn+Hq7+cjN5ccIO3YMQgJ\nsb69aryv0u9NSSHn3xO45MO1AJw9uJu0T+ZDWddyddtz6n0hId6wsyrvmzcPvvqKnIQE0mbPhrvu\nkvx8fxyDx4+X+ZLatWsTVME9A4IMwzAqZ0FJ0tLSqFOnTnU/RlEURakCp06dIjo6ukKvNUX0L+jp\nX3edxLnHjatuM+5h/3645hoYMQL+/nenrbGWZcskG2ntWu9uLF0ao0djfPYZ/euupMG1DZg1qyUH\nDhyo8MDxFPfeKyGeL75w2hLzOH5cyizcdx+88463d8Yygcp4+qaEd4KCgsoeLL5Vjf40mKZOle81\ndqwsxPJn6teXY1iY//RhWhpMmsSG20eycsZVrBiTxqxZEB0d7Z+iHxUlczL+9N3efluE/q23QKMM\nlcL6iVx/WwKemytZOsOG+b/gQ7Wyd1zLl19iZGby5+RHuesucRj9Gn9bkWsYMGmSpGVedJHT1ngO\n65OR/e2CW7sWjhyR28pAwB9L886cydH2PVmyuQWrpjptjA34m+O1bh3s2YO3tzRzDvX0K0t8vHj4\nXbs6bYk9+Ftp3pwcjCVL+Or4bdx2m2Ro+j3+5njFx0NkpH+tmrYR9fQry6JF/lVbpzz8Lbyzdi1B\naWlMTevDa/5dHaQAfxuD8fEi+J7f4cYZrPf0/WmP1awsWL7cf2rlVwQ/C+8YPyzidHAUYd2v85si\njOXiT6J/7pxklPlTQTybsczTX7JkCampqZz+5Rfm7d/P6HXr6NKli1XN2cOqVZCZCX36kJSUxPLl\ny0lPT2flypW88MIL9OzZ02kLzcfPwjsnv40nIe8WxrwcFjhZfn4SYk1KSmL5lCmkZ2SwcsYMXrjx\nRr8ac+c18/Rp5s2bx+jRoy3RTMs8/XvvvZesrCyGt2tHt1q1GDRokFVN2ceiRdCgAZlt2zJ9+nSe\ne+45XnnlFR599FH69+/P4cOHnbbQfPzJ08/Kotb65fzcvA/9+jltjI34gaefmZkpY65xY16pXZtH\nn33W78bcec0cPpxu3bpZppmWif6SJUsYMmSIeBk5OeTm5lrVlH0sWgS9e7Nz927eeOMNdu/eDcAd\nd9xBZmYmK1ascNhACygU03/llVe48sorueSSS+jQocP548MPP+yoiRVl8yeriMg7S+dn+gSEl3++\nv957jw6pqZ7rr8Ls3LlTxtycOdCzJ3cMGOB3Y+68ZuZjmWYaVjNihNErKsqYPHlyiadefvllo1On\nTkbr1q32cK6xAAAckklEQVSN9u3bnz8OGzbMcrMqTVqaYYSGGsYHHxiGYRgrV648/1RycrIRHBxs\nrF+/vshbPPX9LkRwsDFl+HBj2bJlRl5envHOO+8YhmEYr7/+usOGVY4v2vzVOBHSwMg9l1vk/KlT\npwzAOHXqlEOWmc+UKVMK+uvuuw0jKspz/VWclUuWGEaNGoYxfnypY85vxpthGL169SqhmWZ9P0tF\nPykpyXi5WzdjVIMGRlZWVpHnilyUXhCR2bMNAwxj+/YSTz300EPGX/7ylyLnvPT9/v3vfxvPP/+8\nMXLkyCL/fOdmhocbRv53mD9/vhEfH29s2rTJ+Ne//uWw5RVn3TrDWMZNxt6uQ0o854+i72P+/PlG\n/J/+ZGwKDfVUf5XK4sUyBteuLTHmvDTeLkRSUpLx8ssvG6NGjSqimWZ+v0pP5H7wwQfs2bOnRJ0H\nwzAICgqiZ8+eDMzfOapr16507dWLCbt2cfPNN7N48WJq5pdleOCBBwBYsGABnTt3Jjk5maioqOrd\ntphAmd8vPp6gqCh6bt/OwLZtz5+fOHEiTZs25Y033ijyerd+v9IYMWLEhV/w8cfnJwInTpzIpEmT\nmDNnjg2Wmcf4v2fwKYkE//49p02xlYkTJzKpZ0/m5OTISlYvEx8P9eoxce3aEmPOS+PtQnTt2pWu\nXbsyYcKEIppp5vertOiXKxBAYmIid999N4mJiVwcHk6v0FCeXLOGefPmMXjw4CKvdZuIlPn9unSB\nwYOldGs+c+fOJS8vjzfeeIOsrCxSUlK4+OKLi7zNbd+vSuRvzLxt2zYyMzMJCwsjKyuLDRs2OG1Z\nhdi6FY5/t4wwcqBv4KT6ne+vWrXIAjasW+e0SdUjPp657dqRB2WOOa+OtyKaefHF9OrViyeffLKE\nZprx/SyZyA0NDaVTp040b94cIiLYlZlJeHg4V111VZHXeUZEUlNh/Xq47bbzp5YuXcqhQ4cYOHAg\nKSkpzJs3j5SUlCJv88z3K4/87I+ZM2cybNgwADp37syxY8ccNqxivP463FtrHkaLllDoLs3fOd9f\nERF0Bo4dOeK0SVXn5EmWJiRwqE2bMsecl8dbEc0Edu3aVUIzTft+ZsShSmPq1KnG22+/bfzzrruM\nIWFhxg8//FDiNW+++aYxbdo0wzAMY+vWrcZvfvMbq8ypHp9/LrHE1FTDMAxj9+7dRnR0tBEcHGwE\nBwcbQUFBRnBwsJGenl7kbZ75fuXRpo1hjBrltBVVYvduwwgJMYwTDS83jMceK/U1/hzTNwzDMKZN\nk+v36FGnLakyu99/34iGC445r4+385r5z38aQ4YMKaGZZn0/U+rpX5D33oPRo+HMGUubsZSYGNi1\nC1avdtoSZ+jUSe5y3nnHaUsqzRNPwNqvd5F0og189x3cc0+J1/g2AarMRhSeYs4cuPNOOHgQ8j1J\nzzF8uIy/5GSnLfE8WnCtPNLTYebMUsUiYMiP6XuNX36BiRPhza5fSYGuQCqfURivL7DLzoZZs2DA\nAKct8Qvsqb2Tmyv/vMg338hdykMPOW2Jc3i0ftL48VAz0qDHjk9hyBCoXdtpk5zB66L/3Xdw7Bh4\ncFGZG7HH0wdPigaZmTILOGAAtGrltDXO4cFl/EePwkcfwX96xxKyeyc8/rjTJjmHl8dgTg688YZU\ntu3QwWlr/AJ7SiuDiEZkpOXNVZtduyA2Fk6fhoQEOHBAvP1AxkvhnZwciItj+/tr+TT7CEMWfAe/\n/S3cdJPTljmH1zz9PXvg66+loubixbBpk1S3VUzBetH3kpexZo14FGFhsg1b/frw/fdw5ZVOW+Ys\nXgnvGAY88gh8/jmNg9vStFkTgu75g3iKgYyXRH/zZrjlFrnLjoqCxo1h+vQA2e3GHuz19N2MYcCT\nT0L79rB0aWDsf1tRIiIgI8NpK8pn4UL4/HNmDP6c+2f/nj1JQFOnjXIBXhmDIGOwcWPx7OvVc9oa\nv8Q+T9/tF9yiReLpL1iggl8cr4R33nyT3C7X8sfFD/HII9BUBV/wyt326tWyQcr06Sr4FmJP9g64\n/4KbMgWuuKLIqlslHy9M5B48CIsWsaDd05xKC2LUKKcNchFe8fQ/+wxatoT82l2KNdiXvePmCy47\nG2bMgPvuIyAKrVcWL8T0v/sOIyyMZxYN4qGHAjvZqgReEP28PEnNvPdeCAlx2hq/xr6YvptFIykJ\nfv0V7r7baUvciRfCO7NmcaBNb3Zur8vM0U4b4zK8IPrr18PhwzoGbUA9fZCJ29q14eqrnbbEnbg9\nvJOTg5GQQOzh3tx3X0DVVKsYYWFydLPjtXSpXGc33OC0JX6PxvRBLribb9bbyrJwe3hn/XqCMjKY\n8WtPXnih6h8TExPDoEGDiI2NNc82NxAU5P67tWXLJC3TpxeKZWj2Tk4OrFgBL77otCXuxeWeft7i\npWQH1aDZwOuqtaQiLi7OPwuugbv70DDE8XriCactCQg0pr9hg+Sg9+zptCXuxeVe4qGvlrHD6M7/\n/jXcaVPci5tFf9s2qa2jY9AW7AvvuPWCW7oUatSA665z2hL34uLwjpGbR621yzh4aU+6dnXaGhfj\n5mq3y5ZJaFXj+bagBdeWLpWLLVy9xDJxsZe4+KNt1Ms9TucRPZw2xd24uA9ZuhSuuUbKLiiWY73o\n+zIH3HjB5eWJl6G3lRcmPFzmPvLynLakCIYBSf9vKTlBoXR+XL3EC+J20dcxaBvWi74vc8CNnv7W\nrXD8uF5w5eHSeZn4eGixdxkZba8hKEpLZ1wQt4r+3r1SybaH3qnZhfWiD+694JYskTsRreB3YVw6\nL/PqPwxuDVtKnYH6o10ubnW8Fi8Wx1AdL9uwPnsH3HvBLVoE3btrgbXycGHabUIC7FuyhyYchJ7q\nJZaLWx2v+HhZFKkF1mwjcD393Fz48Ufo08dpS9yPC8M7Y8fCsKYLMUJCoFcvp81xP24cg4YhY7B3\nb6ctCSjsEX035nmvXw8nTwbuZtmVwWXhnbVrYc4cGN5iAUHdu4O/LqgyEzeK/q5dUh1VRd9W7PP0\nXeQlAjB7tqSIaTy/fFwW3hk3DtpfmkXLnxdB375Om+MN3BhinTFDtEHj+bYSmJ6+YcDUqTB4sObn\nVwQXhXe2bIFp0+C922cSdOqUlMNWyseNnv6XX8KAAXqnZjOB6emvWAE//wy/+53TlngDF4V3XnsN\nmjeHPns+lbu09u2dNskbuE30N2yQnbJiYpy2JOCwL3vHyQvuhx/Es2/fXlbfPvssXHWV7pJVUZwW\n/cxMGDuWU/t/5ewXvfnvg1kET5oLn3/ujD1exGnRX7ECPv0ULr8cbr8dnnlGamAPHuycTQGKPaLv\npKefkAD9+sFll8FXX8GZM3I7uXChllKuKE7H9IcPh+nTyYpowdd5/4ZJwF13wYMPOmOPF3Eypr9h\ngyRMNG8OcXEwZgxERsLcuRBqjwQpBfi/p//SS+LVJyVJmuaaNbKXXosWztjjRZyM6a9eDV9+yYl3\nJtFi5EO8M/oAI/rvgZtugmB7opN+gZOe/osvQuvWIv45ObBypdx1N2/ujD0Bjn97+tu2SR5wbKx4\n9SEhcOON9tvhdZwM73zwAVx6Kf/Y9TuiouDBMS0huqX9dngdp0T/l1/Eo58wQWyIiNCwqsP4d/bO\nzJlyG3nPPfa37U84Fd7JzYVZszg98H4m/DeEp5/WRI8q45Toz5ghztZvf2t/20qp+Hf2zty5suK2\nRg372/YnnArvrFkDx44x+cSdhITA00/b27xf4VRMf8UKKZtcp479bSul4r+efnq6lE3u39/edv2R\n0FApimV3H86dS17deoyZ3o0RI6B+fXub9yuc8vRXrtSQqsvwX08/MVEmjbTMQvVxamPt5cvZ0aQn\nZ3NCee45e5v2O5wQ/ZQU2LNHd8RyGf7r6Scmyi1l27b2tuuv2P3DnZeHsXo1X+3rxiOPQOPG9jXt\nl0REwLlz9m6Es3KlHFX0XYX/Zu8kJsL112tan1nY7Slu305QWhrLQ67n45H2Neu3+Cbjz50rmKOx\nmoQETY92If7p6RtGgegr5mBzH2YvSySPINr+7jpatbKtWf/FibTbhAT18l2If9bT37cPjhzRCppm\nYvPd2s+TE9lGO/78smZ9mILdop+VBT/9pJO4LsQ+T9/O8E5iohzV0zcPG3+4z50DIzGJo5d0o00b\nW5r0f+wW/XXrpC0Vfdfhn55+UhJcfLHO/pmJjX0Y+2km7c5tpM2DeqdmGr6Yvl3OV0KCLIy86ip7\n2lMqjP96+hraMReb+jA3F2a/uo4wcmh+j96pmYbdnv7KldC1K4SF2dOeUmHs8/RzcuxJF8vOlpWc\nOoFkLjZ5+l9/Dc0OJpIXUQOuvNLy9goTExPDoEGDiI2NtbVdW7BT9A1DPH0N7bgS+1I2QQTZ6pII\nGzbIhd29u7XtBBo2iH5enmyF+H6TVQRfdq3tXmJcXBzR/lrcx07RP3AADh1Sx8ul2BfeAXsuuFWr\npL0uXaxvK5CwIWVz1izYtMmge85yKZ2smIedMf2EBDmq6LsS+8I7YM8Ft2qVFHiyawFKoGBxyqZh\nwKuvwm+77iPi2CEVfbOx09Nfvlx2yGrY0Pq2lErjX56+YcgFp6Ed87E4vPPDD7Jfykt9VsgJjQeb\ni52iv2iRVLdVXIl/efrbtsH+/bIHp2IuFod3xo6F666DK4/EQ8eOcNFFlrUVkNgl+r/8IuNQCx26\nFu96+t9+C489Bps3F5ybM0cminv1Mq8dRTA7vJOWBn/+M4wfz/LFOSxZAi+OziVo7hzZ01gxFyti\n+vPmwZNPysrbwueCgqB3b/PaUUzF/uwdM1i9WnbiqVVLcvxWroQrroCpU8XLj4w0px2lALPDO489\nJj/ceXmkN9tOp44TGBQxX8rxxsSY144imO3pb9kCgwbJrliTJ8P8+TIPM2WKhHb0Ts212Cv6Z8+a\n83nvvw9t2sikbc+eMGAAjBghS79ffdWcNpSimCn6e/fCV1/BRx+x91A4/f82nIu71CN47FK4+mq4\n9lpz2lEK8KW/mtWH77wDjRrBpk3wm9/IZkVPPw2LF8OXX5rThmIJ9oi+LzffDNHPzITvvoORI2Ur\npdmzoUcPeTx4sO6UZRVmrsj96iu5G3vgAZ57qBY96h/k2Zl/lR+WhQslPKCYS1CQeT/ceXkyBh95\nBOrVk1zb/v1lYqZfPxgypPptKJZhj+j7wi1miP7KlZCRId4FSI2drVth1y6ZAFTBsAYzPf1Fi6Bn\nTzbvrcV338HAT16CW4ZKuK5JE3PaUEpi1g93cjIcOwZ9+8rjqCjp0+Rk6NxZ97BwOfb0js/Tz8ys\n/metWAF164rA+6hVSy62kJDqf75SOmaJfna27F1866289hq0bAkPPghcdpkKvtWY1YeLFsmYLpxW\nGx4u62NC7fEjlapjr+ib4ekvXy4Xm3oT9mKWl7h1K2RmcrDVjcTGwqhRBYklisWYJforVkhBQ6tL\nqiiWYI9ymhXeMQwpm6yLr+zHLMFITgbgjdmdaNgQ/vjH6n+kUkHM7EMtmexZ7BH90FAJvVQ3vHPo\nEPz6q15wTuDL0zeM6n3Opk3kNG/FhNho/vIXza61FTPu1s6ehR07oFMnc2xSbMe+GElkZPU9/Xwv\nUS84Byi8sXZ1SE5mW9iVREXBE09U3yylEpjh6W/bJtk7OgY9i32iX6NG9T39TZtk0rZ1a1NMUiqB\nSYt7cjckM+9AJ555BmrXNsEupeKYIfo+x6twIoXiKbzn6XfsqJO4TmCG6KelEXJwH9vDOvGnP5lj\nllIJzBL9Vq3AX/cdCADs9fTNEH29rXQGE+onpa2UOkmdYjpRr54ZRimVwoyYvo5Bz+Od8E5urtT7\nsHkLPSUfE+onLfl3MjmEEPN/7UwySqkUZnn6KvqexjvhnT175EdDY4nOUM3wTno6HFqQzLF6l9P4\nYs3vdoTqin5aGuzbp6Lvcbzj6WvmjrNUU/Q/+gguz04m+kbtP8eoruhv2SJHHYOexjue/ubNUtxJ\nl+o7QzXqsWdmwvjxcF3EJmp2VcFwjOrG9JOTJYminYbnvIy7JnJPnZJCTqWxebMWVHOSinr6Bw7I\n/EshPvkEOHKE6LNH1Ut0kop4+qdPlz0Gk5OlpLmuqPM09nr6FwrvnD4NXbpI1czt20s+7xN9xRkq\nIvoTJ0o636OPnj+VnQ1vvglP99HwnOOUJ/pZWbJnZcuWsH59yed1EtcvcI+nP22aTNaeOSMbNBQm\nJ0dWAqroO0dFwjtvvinHiROlL5FNlQ4cgOFdk0V02rSx2FClTMoT/YULZZydPQuvv170OcOAjRtV\n9P0A90zkzpsnXsZf/wpffFFUXHbulMd6wTlHeZ6+b0PsyZNlqe3kyeTkiHYMHgxNj22CDh20/LWT\nlBfTnz0bLr8cxo2DmTPFAfORkgJHj8rOZoqncc9E7tq1Uj1zyBBJDVu8uOA53+bn6uk7R3min5go\nxz59YOBA+O47vv5afq9feAHYsEHXWDhNeZ7+Tz9J2fL77hPBnzev4LkNG+SoxQ49jzvCO2fOwM8/\nywV15ZVwySUwfXrB8+vXy36cjRrZY6tSkvLCO+vXS2ZVs2ayq9n69Xz2yh769YNrO5+TukkqGM5y\nIdE/d07CN9dcIxvadOggnr+PDRvkDk7rXnked0zkbtkilfs6d5bsnLvvhhkz5ByIF9mtm22mKqUQ\nHi59U1Yf7twJbdvK3/37kxsWQccd3/HSS4iYnD2rfeg0FxL9/fvluQ4d5PGAATBnTkEp7XXrdCtE\nP8Ednv7u3XK8/HI53nOP1M5fs0aEPzFRN05xmqAgqFmzaJy3MDt3np+kNWpFkVCzLw9Hf8dNNyH7\nGoeFwbXX2mdvFYiJiWHQoEHExsY6bYo1XCimnz/xziWXyHHAAInjr18vwr9sWdHtERXPYt+Glhfy\n9Pftk1vHunXl8U03QYMG4u2Hh0uM/4YbbDNVKYMLif6uXfJjDSxYAHGnBvNp0B8hNRWWLpWwgcu3\n14uLiyPan6tHRkRIGCcvr6THvmePnGvVSh7fdJOMyTlzZOPzQ4egVy/bTVbMxz7Rr1FDbh8No+QC\nq717JVboOx8aKpOB06fL39HRchEqzlKW6J84If/yPf2xY6FGl7tgQxB8+aUIxwsv2GysUoLCRfOK\n/wDv2QMtWsgdGYiz1b8/TJ0qYzAiAm6+2V57FUuwN7wDpccU9+0rOUF0330S6//732HoUN092w2U\nJfq+0MBll7F0qUQCnvq/iwjq1Qv+/GcJ6913n62mKqXgW0lb2h337t0FoR0fI0bIRvajR8O992oN\nfT/B3olcKP2C83n6henfH556SryL//s/i41TKkStWqWL/uHDcmzWjLFjJQFr4EDgvfdk7cVbb+mi\nLDdQq5YcT58u+dyePSVFv2dPEfwbbii5WEvxLPaGd6D0ydx9+6T8QmGCguD99623S6k4ZXn6KSkQ\nFMTqvQ0lnh+XHzLu2BFWr7bdTKUMLiT6Bw5Av35FzwUFwWuvWW+XYivOe/oZGfKvaVPbTFGqyIVE\nv2FDxr0ZStu2sr5OcSE1a8qxuOjn5cGRI1rBNkBw3tNPTZWjXnDup2bN0r3E1FQy6zRm+nT49FOt\ntOBafJ5+8R/uEyekMmrjxvbbpNiO/Z5+8QvOJ/p6wbmfC3j6O9Kb0KoVPPig/WYpFaSs8E5KihzV\n8QoI7PP0o6LkWPyCU9H3DmWIfuaeFDamtOZ//12Q8ae4kLJEX8dgQGGfp38hLyMkBOrXt80UpYqU\nIfqndqSSVrMJf/iDAzYpFccX0y/ehz5PX0U/ILBP9H2efkZG0fOpqXKxaU0P91OK6O/fDzXTUujQ\np4nbF9wqNWpIRk5pnn6tWgVjVPFrnPf0faKvuJ9S8vTfGXeGaNLpfrf2oesJCpI+1DEY0Ngn+iEh\n4mkU9/RTUvSC8wrFPP2UFJgzUeLBNVrrJKAnKG2BXUqKTuIGEPbGVKKiSg/v6AXnDXwpm/nldt9+\nG1qEauaHpygt7VY9/YDCXtHXW0tvU7OmCH5WFidOwIcfwsP9dBLQU5Q1BvVHO2Bw3tPX8I53KJT9\n8d57sp7n7u6pErpr0MBZ25SKUVZ4R8dgwOCsp5+RIRegXnDeIF/001PP8O678NhjUPt0imxjqdlX\n3qD4GMzNlQ3P1dMPGJz19LUEg7fIz8CK+28GZ87A88+jk4Beo3hM//hxLcEQYNjv6Zcm+nrBeYP8\neurfTEzn4Ydlzw2NB3uM4p6+jsGAw35Pv/AFpysBvUW+6Bun0vjf/80/p/Fgb1GW6OsPd8DgfHhH\nSzB4huwaIvoDe5zi0kvzT6qn7y2io2XPaR/qeAUczk7kagkGTzF5hoj+kNvzRcMwNKbvNerUgVOn\nCh6npooz5svMUvwee9W2du2SXoZ6GJ4gJwfGvRVGVkgkzaLy+zA9XTbF0T70DnXrwq+/FjxOSdEN\njAIMe0VfLzjP8uWXsnd2cN1C4QGNB3u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      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pfr10 = plot(fourier10,(x,-3*pi,3*pi),color='red' , \\\n",
    "            ticks=[pi,pi], tick_formatter=[pi,pi]); \n",
    "(pf+pfr10).show(figsize=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Παρατηρούμε ότι η σειρά Fourier πέρα από το διάστημα $[-\\pi,\\pi]$ δεν ακολουθεί πλέον την συνάρτησή μας $f(x)=x$, αλλά τραβάει τον δικό της δρόμο, ο οποίος μάλιστα δείχνει να επαναλαμβάνεται περιοδικά με περίοδο $2\\,\\pi$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.2 Περιοδικές επεκτάσεις συναρτήσεων <br><br>\n",
    "\n",
    "Η τελευταία παρατήρηση στο προηγούμενο παράδειγμα είναι ένα γεγονός που έπρεπε να το περιμέναμε αφού οι βασικές τριγωνομετρικές συναρτήσεις είναι όλες μαζί περιοδικές συναρτήσεις με περίοδο $2\\,\\pi$. Οπότε αν η σειρά Fourier συγκλίνει, η οριακή συνάρτηση ${\\widetilde{f}}(x)$ οφείλει και η ίδια να είναι περιοδική συνάρτηση, και μάλιστα με την ίδια περίοδο $2\\,\\pi$: <br>\n",
    "$${\\widetilde{f}}(x +2\\,\\pi) = {\\widetilde{f}}(x) \\qquad \\mbox{για κάθε}\\qquad x\\in\\mathbb{R}\\,. $$<br>\n",
    "Μια σειρά Fourier μπορεί να συγκλίνει μόνο σε μια $2\\,\\pi$-περιοδική συνάρτηση. Οπότε είναι παράλογο να περιμένουμε η σειρά Fourier να συγκλίνει στην **α**περιοδική συνάρτηση $f(x)=x$, του προηγούμενου παραδείγματος παντού στο $\\mathbb{R}$. Μαλλον θα έπρεπε να συγκλίνει στην $2\\,\\pi$- περιοδική επέκταση της $f(x)$, η οποία ικανοποιεί ${\\widetilde{f}}(x) = f(x)$ για κάθε $-\\pi < x \\leq \\pi$.<br><br>\n",
    "**Λήμμα 4.** Αν $f(x)$ είναι μια οποιαδήποτε συνάρτηση ορισμένη στο $[-\\pi,\\pi]$, τότε υπάρχει μοναδική $2\\,\\pi$-περιοδική συνάρτηση ${\\widetilde{f}}(x)$, που ονομάζεται η $2\\,\\pi$-περιοδική επέκταση της $f(x)$, η οποία ικανοποιεί ${\\widetilde{f}}(x) = f(x)$ για κάθε $-\\pi < x \\leq \\pi$.<br> \n",
    "***Απόδειξη*** <br> \n",
    "Δοσμένου $x\\in \\mathbb{R}$, υπάρχει μοναδικός ακέραιος $m \\in \\mathbb{Z}$, τέτοιος ώστε \n",
    "$(2\\,m-1)\\,\\pi < x \\leq  (2\\,m+1)\\,\\pi$. Η περιοδικότητα της ${\\widetilde{f}}(x)$ μας οδηγεί στο να ορίσουμε <br><br>\n",
    "$$ {\\widetilde{f}}(x)  = {\\widetilde{f}}(x-2\\,m\\,\\pi) = f(x-2\\,m\\,\\pi)\\,. $$<br>\n",
    "Ειδικότερα, αν $-\\pi < x \\leq \\pi$, τότε $m=0$ και συνεπώς ${\\widetilde{f}}(x) = f(x)$ για ένα τέτοιο $x$. Ο αναγνώστης καλείται να γεμίσει την απόδειξη ότι πραγματικά η συνάρτηση ${\\widetilde{f}}(x)$ που προκύπτει από τον παραπάνω τρόπο είναι $2\\,\\pi$-περιοδική. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Παρατήρηση*** <br>\n",
    "Η περιοδική επέκταση ${\\widetilde{f}}(x)$, από την κατασκευή της, χρησιμοποιεί την τιμή ${\\widetilde{f}}(-\\pi) = {\\widetilde{f}}(\\pi)= f(\\pi)$ στο δεξί άκρο της $f$. Εναλλακτικά, θα μπορούσαμε να ορίσουμε  ${\\widetilde{f}}(-\\pi) = {\\widetilde{f}}(\\pi)= f(-\\pi)$, όπου γενικά $f(-\\pi)\\neq f(\\pi)$. Δεν υπάρχει κανένας λόγος να είναι προτιμητέα μια επιλογή από την άλλη. Μάλιστα η σειρά Fourier δεν προτιμά καμιά από τις δύο τιμές, γιατί η σειρά Fourier συγκλίνει στην μέση τιμή των τιμών στα άκρα:<br><br>\n",
    "$$ {\\widetilde{f}}(-\\pi) = {\\widetilde{f}}(\\pi) = \\frac{1}{2} \\left(\\, f(\\pi)+f(-\\pi) \\,\\right) \\,, $$<br>\n",
    "η οποία σταθεροποιεί τις τιμές της ${\\widetilde{f}}(x)$ στα περιττά πολλαπλάσια του $\\pi$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Παράδειγμα 5.** H $2\\,\\pi$-περιοδική επέκταση της $f(x)=x$ είναι η πριονωτή συνάρτηση του παρακάτω σχήματος. Αναλυτικά η $2\\,\\pi$-περιοδική επέκταση ${\\widetilde{f}}(x)$ είναι <br><br>\n",
    "$$ {\\widetilde{f}}(x) =  \\left\\lbrace \\begin{array}{lc} \n",
    "x-2\\,\\,m\\,\\pi \\,, & (2\\,m-1)\\,\\pi <  x < (2\\,m+1)\\,\\pi \\,, \\\\  \n",
    "0\\,,  &  x = (2\\,m-1)\\,\\pi\\,. \\end{array} \\right.$$ <br>\n",
    "Στο Sage υλοποιούμε ως εξής:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x |--> x\n"
     ]
    }
   ],
   "source": [
    "f(x) = x ; print f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def f_ext(x):\n",
    "    if   x % pi.n() == 0.0 :\n",
    "        y = 0\n",
    "    else:\n",
    "        y = f( ( x ) % (2*pi.n()) )\n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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a7cc2DDF9vCGWlma2F9tcueK+LdW3L/D882b7sU1rfPTgvWgbwCUlJTh9+jQa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\nsQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 5 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "set_verbose(-1)\n",
    "pf_ext = plot(f_ext,(x,-5*pi,5*pi),ymin=-4,ymax=4, exclude=[(2*i-1)*pi for i in [-2..3] ], \\\n",
    "            thickness=2 , ticks=[pi,pi], tick_formatter=[pi,pi]) ;  pf_ext.show(figsize=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Με αυτές τις παραδοχές μπορεί να αποδειχθεί ότι η σειρά Fourier που βρήκαμε παραπάνω συγκλίνει παντού στην  $2\\,\\pi$ - περιοδική επέκταση ${\\widetilde{f}}(x)$. Ειδικότερα, <br><br>\n",
    "$$ 2\\, \\sum_{k=1}^\\infty \\frac{(-1)^{k+1}}{k} \\, \\sin k\\,x  = \n",
    "\\left\\lbrace \\begin{array}{lc} \n",
    "x \\,, & -\\pi <  x < \\pi \\,, \\\\  \n",
    "0\\,,  &  x = \\pm \\pi\\,. \\end{array} \\right. $$ <br>\n",
    "Ο τύπος αυτός, όσο απλοϊκός κι αν δείχνει, έχει εκπληκτικές και μη τετριμένες συνέπειες. Αν θέσουμε $x=\\frac{\\pi}{2}$ και διαιρέσουμε με $2$, παίρνουμε την σειρά Gregory  <br> <br>\n",
    "$$ \\frac{\\pi}{4}  = 1 - \\frac{1}{3} +\\frac{1}{5} - \\frac{1}{7} + \\frac{1}{9}- \\cdots \\,.$$ <br>\n",
    "η οποία αν και προϋπήρχε της θεωρίας του Fourier, είναι πολύ δύσκολο να αποδειχθεί άμεσα. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1/4*pi\n"
     ]
    }
   ],
   "source": [
    "print sum((-1)^(k+1)/k * sin (k*pi/2),k,1,oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.00999600796130951\n"
     ]
    }
   ],
   "source": [
    "print 1/2*fourier50(x=pi/2) - pi.n()/4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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H/8u3GcQNPvyPUttTO1a0SF9ehLPn8GmsIJGnsLh7sHK9RA27uUmem1OC0k2H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\n5XnT6zUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "Graphics object consisting of 6 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pfr10_b = plot(fourier10,(x,-5*pi,5*pi),color='red');\n",
    "(pf_ext+pfr10_b).show(figsize=4, ticks=[pi,pi], tick_formatter=[pi,pi])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.3 Τμηματικά συνεχείς συναρτήσεις <br><br>\n",
    "\n",
    "**Ορισμός 6.** Μια συνάρτηση $f(x)$ λέγεται *τμηματικά συνεχής* στο διάστημα $[a,b]$, αν ορίζεται και είναι συνεχής εκτός πιθανά από ένα πεπερασμένο πλήθος σημείων \n",
    "$a \\leq x_1 < x_2 < \\cdots < x_n \\leq b $. Επιπλέον, σε κάθε σημείο ασυνέχειας, θα πρέπει τα πλευρικά όρια <br><br>\n",
    "$$ f(x_k^-) = \\lim_{x\\rightarrow x_k^-}\\,f(x)\\,,\\qquad  f(x_k^+) = \\lim_{x\\rightarrow x_k^+}\\,f(x) \\,,$$<br>\n",
    "να υπάρχουν. Στα άκρα του διαστήματος $[a,b]$ αρκεί να υπάρχει μόνο το από δεξιά όριο $f(a^+)$, και αντίστοιχα μόνο το από αριστερά όριο $f(b^-)$. \n",
    "- Δεν απαιτείται η $f(x)$ να ορίζεται στα σημεία $x_k$. \n",
    "- Αν η $f(x)$ ορίζεται στα $x_k$, δεν απαιτείται η τιμή της στα σημεία αυτά να είναι ίση με τα πλευρικά όρια. <br>\n",
    "\n",
    "Παρόμοια ορίζεται και μια τμηματικά διαφορίσιμη συνάρτηση $f(x)$ και και γενικότερα μια τμηματικά $n$-φορές διαφορίσιμη σε ένα διάστημα $[a,b]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Η πιο απλή τμηματικά συνεχής συνάρτηση είναι η αλματική συνάρτηση $ \\mathrm{u}(x) = \\left\\lbrace \\begin{array}{cc} 1\\,, & x > 0 \\,, \\\\ 0\\,  &  x < 0\\,. \\end{array} \\right.$. Η συνάρτηση αυτή στο Sage είναι η unit_step."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARwAAADRCAYAAAAe9Q0+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nBAeIERwgRnCAGMEBYgQHiPkTzv6duiCBKfYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = plot(unit_step, -1, 1,ymin=-0.5,ymax=1.5 , exclude=[0],thickness=2 , ticks=[[],[1]])\n",
    "p.show(figsize=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.4 Το θεώρημα σύγκλισης των σειρών Fourier <br>\n",
    "\n",
    "**Θεώρημα 7.** Έστω $ {\\widetilde{f}}(x)$ η $2\\,\\pi$-περιοδική, τμηματικά διαφορίσιμη συνάρτηση, τότε για κάθε $x\\in\\mathbb{R}$, η σειρά Fourier συγκλίνει (κατά σημείο) στα\n",
    "- $\\qquad{\\widetilde{f}}(x)\\,,\\qquad\\qquad\\qquad\\,\\,$ αν η ${\\widetilde{f}}(x)$ είναι συνεχής στο $x$,<br><br>\n",
    "- $\\displaystyle \\frac{1}{2} \\big( {\\widetilde{f}}(x^+) + {\\widetilde{f}}(x^-) \\big)\\,,\\qquad\\quad$ αν το $x$ είναι σημείο ασυνέχειας. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Παράδειγμα 8.** Θα βρούμε την σειρά Fourier της συνάρτησης $ \\mathrm{u}(x) = \\left\\lbrace \\begin{array}{cc} 1\\,, & x > 0 \\,, \\\\ 0\\,,  &  x < 0\\,. \\end{array} \\right.$. Η γραφική παράσταση της αλματικής συνάρτησης είναι"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARwAAADRCAYAAAAe9Q0+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAAPYQAAD2EBqD+naQAABO5JREFUeJzt2rFqFGsYx+F3NrbZARWMJMHWRvAChGCXyvqcgHgFXoFX\noH1aK5FTewfRlIqksxARjhICWuwsKc1Y5HiIBDy74vmPmzwPLJtdvg1vMfz2m5lt+r7vCyBgNPQA\nwPkhOECM4AAxggPECA4QIzhAjOAAMYIDxAgOc+v7vrquK78ZZV6Cw9ym02m1bVvT6XToUVgwggPE\nCA4QIzhAjOAAMYIDxAgOECM4QIzgADGCA8QIDhAjOECM4AAxggPECA4QIzhAjOAAMYIDxAgOECM4\nQIzgADGCA8QIDhAjOECM4AAxggPECA4QIzhAjOAAMYIDxAgOECM4QIzgADGCA8QIDhAjOECM4AAx\nggPECA4QIzhAjOAAMYIDxAgOECM41IsXL+rOnTu1urpao9Gonj17NvRInFGCQx0eHtbNmzdre3u7\nmqYZehzOsAtDD8DwNjc3a3Nzs6qq+r4feBrOMjscIEZwgBinVMxtY+P4+fr1qpGvrHNpZaXq5cv5\nPyc4zO3g4Ph5f/+POn0I/fnPA04THOZ25UrV/n7V1at/1Wg0HnocBrCy8nOfExzq8PCw3r59++8d\nqnfv3tXe3l5dvHix1tfXT63f2alq26o3b6rGesMcmt590HNvZ2enbt++feo3OPfu3avHjx+fWt91\nXbVtW5PJpMaKwxwEh7kJDj/LPQYgRnCAGMEBYgQHiBEcIEZwgBjBAWIEB4gRHCBGcIAYwQFiBAeI\nERwgRnCAGMEBYgQHiBEcIEZwgBjBAWIEB4gRHCBGcIAYwQFiBAeIERwgRnCAGMEBYgQHiBEcIEZw\ngBjBAWIEB4gRHCBGcIAYwQFiBAeIERwgRnCYy+vXVQ8eHP+9tzfsLCyepu/7fughWAxPn1bdvVt1\ndNRVVVtNM6knT8a1tTX0ZCwKwWEmX75UXbtW9fFjVdVxcKomtbY2rvfvq5aWBh2PBeGUipl8+vQt\nNt/78KHq8+f8PCwmwWEmly9Xra6efn9trerSpfw8LCbBYSZLS1WPHlWNThwxTVP18KHTKWYnOMxs\na6vq1auq+/ePXz9/Xi4YMxcXjZlb13XVtm1NJpMaj8dDj8MCscMBYgQHiBEcIEZwgBjBAWIEB4gR\nHCBGcIAYwQFiBAeIERwgRnCAGMEBYgQHiBEcIEZwgBjBAWIEB4gRHCBGcIAYwQFiBAeIERwgRnCA\nGMEBYgQHiBEcIEZwgBjBAWIEB4gRHCBGcIAYwQFiBAeIERwgRnCAGMEBYgQHiBEcIEZwgJgLsyzq\n+76m0+l3721sVB0c/C8z8Zs7OuqqqurWra52dwceht/G8vJyNU3zwzVN3/f9f/2jruuqbdtfNhhw\n9kwmkxqPxz9cM1Nw7HA46eioq4OD9bpx4+/a3f3xAcb58ct2OHDStx3vLN9ocJKLxkCM4AAxggPE\nCA4QIzhAjOAAMYIDxAgOECM4QIzgADGCA8QIDhAjOECM4AAxggPECA4QIzhAjOAAMYIDxAgOECM4\nQIzgADGCA8QIDhAjOECM4AAxggPECA4QIzhAjOAAMYIDxAgOECM4QIzgADGCA8QIDhAjOECM4AAx\nggPECA4QIzhAjOAAMYIDxAgOECM4QIzgADGCA8Q0fd/3Qw/BYun7vqbTaS0vL1fTNEOPwwIRHCDG\nKRUQIzhAjOAAMYIDxAgOECM4QIzgADFfAd8c03ngeLlCAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "Graphics object consisting of 3 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "punit = plot(unit_step,-2,2,ymin=-0.5,ymax=1.5,exclude=[0], thickness=2);\n",
    "p0 = point((0,1/2),size=20); (punit+p0).show(figsize=3,ticks=[[],[1]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Η $2\\,\\pi$-περιοδική επέκταση της $\\mathrm{u}(x)$ είναι η συνάρτηση \n",
    "<br><br>\n",
    "$$ \\widetilde{\\mathrm{u}}(x) = \\left\\lbrace \\begin{array}{cc} \n",
    "0 \\,,        & (2\\,m-1)\\,\\pi < x < 2\\,m\\,\\pi  \\,, \\\\ \n",
    "1 \\,,        & 2\\,m\\,\\pi < x < (2\\,m\\,+1)\\,\\pi  \\,,  \\\\\n",
    "\\frac{1}{2}\\,,  & x = m\\,\\pi\\,, \\end{array} \\right.$$ <br>\n",
    "για οποιοδήποτε ακέραιο $m$. Με τις επόμενες δυο εντολές υλοποιούμε στο Sage την $2\\,\\pi$-περιοδική επέκταση της $\\mathrm{u}(x)$ και απεικονίζουμε την γραφική της παράσταση."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def u_ext(x):\n",
    "    if   x % pi.n() == 0.0 :\n",
    "         y = 0\n",
    "    else:\n",
    "        y = unit_step( ( x ) % (2*pi.n()) )\n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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PP/98tLW1RX9/f5RKpWhrazti287Ozvj5z38e69evf72jFFY9Ga1evTp6e3tj+fLlkWVZ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\nAElFTkSuQmCC\n",
      "text/plain": [
       "Graphics object consisting of 11 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pu_ext = plot(u_ext,-5*pi,5*pi, ymin=-0.3 , ymax=1.3, \\\n",
    "              exclude=[i*pi for i in [-4..5] ],  ticks=[pi,[1]], tick_formatter=[pi,1/2], \\\n",
    "              thickness=2 , aspect_ratio=5 )\n",
    "pu_points = point([(i*pi,1/2) for i in [-5..5]],size=15)\n",
    "(pu_ext+pu_points).show(figsize=8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Οι συντελεστές Fourier της σειράς Fourier της $\\mathrm{u}(x)$ είναι"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a0 =  1\n",
      "ak =  0\n",
      "bk =  -((-1)^k - 1)/(pi*k)\n"
     ]
    }
   ],
   "source": [
    "var('k')\n",
    "assume(k,'integer')\n",
    "a0 = inner(unit_step(x),1); print str('a0 = '), a0; \n",
    "a(k) = inner(unit_step(x),cos(k*x)) ; print str('ak = '), a(k);\n",
    "b(k) = inner(unit_step(x),sin(k*x)); print str('bk = '), b(k);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Παρατηρούμε ότι οι συντελεστές $b_k$ μπορούν να απλοποιηθούν περαιτέρω, ανάλογα αν ο ακέραιος $k$ είναι περιττός ή άρτιος"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "if k is odd : bk =  2/(pi*k)\n",
      "if k is even: bk =  0\n",
      "[k is integer, k > 0, l is integer, l > 0]\n"
     ]
    }
   ],
   "source": [
    "assume(k,'odd')\n",
    "print str('if k is odd : bk = '), b(k).simplify();\n",
    "forget(k,'odd')\n",
    "assume(k,'even')\n",
    "print str('if k is even: bk = '), b(k).simplify();\n",
    "forget(k,'even'); print assumptions()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Συνεπώς οι συντελεστές Fourier είναι $a_0=0$, $a_k=0$, και\n",
    "<br><br>\n",
    "$$ b_k = \\left\\lbrace \\begin{array}{cc} \n",
    "\\frac{2}{k\\,\\pi} \\,,  & k \\,\\,\\mbox{περιττός}  \\,, \\\\ \n",
    "0  \\,,               & k \\,\\,\\, \\mbox{άρτιος}\\,. \\end{array} \\right.$$ <br>\n",
    "Οι πρώτοι δέκα όροι στη σειρά Fourier της $\\mathrm{u}(x)$ είναι"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, \\sin\\left(9 \\, x\\right)}{9 \\, \\pi} + \\frac{2 \\, \\sin\\left(7 \\, x\\right)}{7 \\, \\pi} + \\frac{2 \\, \\sin\\left(5 \\, x\\right)}{5 \\, \\pi} + \\frac{2 \\, \\sin\\left(3 \\, x\\right)}{3 \\, \\pi} + \\frac{2 \\, \\sin\\left(x\\right)}{\\pi} + \\frac{1}{2}</script></html>"
      ],
      "text/plain": [
       "2/9*sin(9*x)/pi + 2/7*sin(7*x)/pi + 2/5*sin(5*x)/pi + 2/3*sin(3*x)/pi + 2*sin(x)/pi + 1/2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fourier_u10 = a0/2 + sum( a(k)*cos(k*x) + b(k)*sin(k*x) for k in [1..10]); \n",
    "fourier_u10.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Στο παρακάτω γραφικό αναπαριστούμε το μερικό άθροισμα των δέκα πρώτων όρων της σειράς Fourier της $\\mathrm{u}(x)$ μαζί με την $2\\,\\pi$-περιοδική επέκταση της $\\mathrm{u}(x)$ στην οποία συγκλίνει η σειρά Fourier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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V7qoquY9q7te4aJG81rp19V+r/f56g04KjdZIEGhFxttuk15abU/KPXvkGf7I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\nuKqQ9VJSZM5XQYHdKXEOrTGGDQzWYSB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8NP9q/DfpDXzR9NJ/zuTmVqexT+kKvOl+HY8k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\noUMHREdH44ILLkBycrL3D/Tu7R3ILVwInHUWEGDfhAYlJgY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      "text/plain": [
       "Graphics object consisting of 12 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pfu10 = plot(fourier_u10,(x,-5*pi,5*pi) , ymin=-0.3 , ymax=1.3,color='red' , aspect_ratio=5); \n",
    "(pu_ext+pu_points+ pfu10).show(figsize=9, ticks=[pi,[1]], tick_formatter=[pi,1/2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.5 Άρτιες και περιττές συναρτήσεις <br><br>\n",
    "\n",
    "**Ορισμός 9.** Μια συνάρτηση λέγεται **άρτια** αν $f(-x)=f(x)$, σε κάθε σημείο του πεδίου ορισμού της. Μια συνάρτηση λέγεται **περιττή** αν $f(-x)=-f(x)$, σε κάθε σημείο του πεδίου ορισμού της. <br><br>\n",
    "- Το άθροισμα $f(x)+g(x)$ δυο άρτιων συναρτήσεων είναι μια άρτια συνάρτηση, και ομοίως το άθροισμα $f(x)+g(x)$ δυο περιττών συναρτήσεων είναι μια περιττή συνάρτηση.\n",
    "- Το γινόμενο δυο άρτιων ή περιττών συναρτήσεων είναι μια άρτια συνάρτηση, ενώ το γινόμενο μιας άρτιας με μια περιττή συνάρτηση είναι περιττή συνάρτηση.\n",
    "- Όπως έχετε ήδη μάθει στον Απειροστικό Λογισμό αν η $f(x)$ είναι περιττή στο **συμμετρικό** διάστημα $[-a,a]$, τότε το ολοκλήρωμά της στο διάστημα αυτό είναι μηδέν \n",
    "$$ \\int_{-a}^a f(x)\\,{\\mathrm d} \\,x =0 \\,.$$\n",
    "- Αν η $f(x)$ είναι άρτια στο **συμμετρικό** διάστημα $[-a,a]$, τότε το ολοκλήρωμά της στο διάστημα αυτό είναι<br><br>\n",
    "$$ \\int_{-a}^a f(x)\\,{\\mathrm d} \\,x = 2\\, \\int_{0}^a f(x)\\,{\\mathrm d} \\,x \\,.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Θεώρημα 10.** \n",
    "- Αν η $f(x)$ είναι άρτια συνάρτηση τότε οι συντελεστές Fourier των ημιτόνων $b_k$ είναι μηδέν, οπότε η $f(x)$ έχει την αναπαράσταση σε σειρά Fourier <br><br>\n",
    "$$f(x) \\sim \\frac{a_0}{2} + \\sum_{k=1}^\\infty a_k\\,\\cos k\\,x \\,,\n",
    "\\qquad a_k = \\frac{2}{\\pi}\\, \\int_0^\\pi f(x) \\, \\cos k\\,x \\, {\\mathrm d}\\,x\\,,\\qquad k=0,1,2,\\ldots\\,.$$<br>\n",
    "- Αν η $f(x)$ είναι περιττή συνάρτηση τότε οι συντελεστές Fourier των συνημιτόνων $a_k$ είναι μηδέν, οπότε η $f(x)$ έχει την αναπαράσταση σε σειρά Fourier <br><br>\n",
    "$$f(x) \\sim \\sum_{k=1}^\\infty a_k\\,\\sin k\\,x \\,,\n",
    "\\qquad b_k = \\frac{2}{\\pi}\\, \\int_0^\\pi f(x) \\, \\sin k\\,x \\, {\\mathrm d}\\,x\\,,\\qquad k=1,2,\\ldots\\,.$$<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Παράδειγμα 11.** Η $f(x)=|x|$ είναι άρτια συνάρτηση, οπότε έχει σειρά Fourier μόνο κατά συνημίτονα. Στο Sage βρίσκουμε"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def inner_even_odd(f,g,a,b):\n",
    "    if a>=0:\n",
    "        return 2/pi * integrate(f*g,x,a,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x |--> abs(x)\n"
     ]
    }
   ],
   "source": [
    "f(x) = abs(x); print f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a0 =  pi\n",
      "ak =  2*((-1)^k/k^2 - 1/k^2)/pi\n",
      "bk =  0\n"
     ]
    }
   ],
   "source": [
    "var('k')\n",
    "assume(k,'integer')\n",
    "a0 = inner(f(x),1); print str('a0 = '), a0; \n",
    "a(k) = inner_even_odd(f(x),cos(k*x),0,pi) ; print str('ak = '), a(k);\n",
    "b(k) = inner(f(x),sin(k*x)); print str('bk = '), b(k);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "if k is odd : ak =  -4/(pi*k^2)\n",
      "if k is even: ak =  0\n",
      "[k is integer, k > 0, l is integer, l > 0]\n"
     ]
    }
   ],
   "source": [
    "assume(k,'odd')\n",
    "print str('if k is odd : ak = '), a(k).simplify();\n",
    "forget(k,'odd')\n",
    "assume(k,'even')\n",
    "a.simplify(); print str('if k is even: ak = '), a(k).simplify();\n",
    "forget(k,'even'); print assumptions()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Συνεπώς οι συντελεστές Fourier είναι $b_k=0$, $a_0=\\pi$, και\n",
    "<br><br>\n",
    "$$ a_k = \\left\\lbrace \\begin{array}{cc} \n",
    "-\\frac{4}{k^2\\,\\pi} \\,, & k \\,\\, \\mbox{περιττός}  \\,, \\\\ \n",
    "0  \\,,               & k \\neq 0 \\,\\,\\, \\mbox{άρτιος}\\,. \\end{array} \\right.$$ <br>\n",
    "Οι πρώτοι οχτώ όροι στη σειρά Fourier της $f(x)=|x|$ είναι"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\pi - \\frac{4 \\, \\cos\\left(7 \\, x\\right)}{49 \\, \\pi} - \\frac{4 \\, \\cos\\left(5 \\, x\\right)}{25 \\, \\pi} - \\frac{4 \\, \\cos\\left(3 \\, x\\right)}{9 \\, \\pi} - \\frac{4 \\, \\cos\\left(x\\right)}{\\pi}</script></html>"
      ],
      "text/plain": [
       "1/2*pi - 4/49*cos(7*x)/pi - 4/25*cos(5*x)/pi - 4/9*cos(3*x)/pi - 4*cos(x)/pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fourier_abs = a0/2 + sum( a(k)*cos(k*x) + b(k)*sin(k*x) for k in [1..8]); \n",
    "fourier_abs.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Αντικαταθιστώντας στην σειρά $x=0$, τότε παίρνουμε το εξής αποτέλεσμα <br><br>\n",
    "$$ \\frac{\\pi^2}{8} = 1 +\\frac{1}{9} + \\frac{1}{25} + \\frac{1}{49} +\\cdots = \\sum_{j=0}^\\infty \\frac{1}{(2\\,j+1)^2}\\,, $$<br>\n",
    "το οποίο το γνωρίζει και το Sage!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1/8*pi^2\n"
     ]
    }
   ],
   "source": [
    "var('j')\n",
    "print sum(1/(2*j+1)^2,j,0,oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Αλλάζοντας μεταβλητή στην άθροιση $k=2\\,j+1$, με $k=1,2,\\ldots$, τότε το Sage μας δίνει"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{6} \\, \\pi^{2}</script></html>"
      ],
      "text/plain": [
       "1/6*pi^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sum(1/k^2,k,1,oo).show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Αν αντί για $2$ θέσουμε στο εκθέτη της σειράς έναν περιττό θετικό ακέραιο, π.χ. 7, το Sage μας δίνει "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "zeta(7)\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\zeta(7)</script></html>"
      ],
      "text/plain": [
       "zeta(7)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z7 = sum(1/k^7,k,1,oo) ; print z7 ; z7.show() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Από την βοήθεια που μας δίνει το Sage για την συνάρτηση $\\zeta$, συμπεραίνουμε ότι πρόκειται για την συνάρτηση $\\zeta(s)$ του **Riemann**, η οποία ορίζεται αρχικά από την σχέση <br><br>\n",
    "$$ \\zeta(s) = \\sum_{k=1}^\\infty \\frac{1}{k^s} \\,,$$ <br>\n",
    "όπου $s$, είναι μιγαδικός αριθμός με πραγματικό μέρος ${\\mathrm{Re}} (s)>1$, και κατόπιν επεκτείνεται αναλυτικά στο $\\mathbb{C}\\setminus\\{1\\}$. Οπότε με την βοήθεια των σειρών Fourier για την συνάρτηση $f(x)=|x|$, βρήκαμε ότι $\\zeta(2)=\\frac{\\pi^2}{6}$. Γενικότερα, αν $s=2\\,n$ άρτιος θετικός ακέραιος, αποδεικνύεται ότι η τιμή της συνάρτησης $\\zeta(s)$ είναι ένας ρητός πολλαπλασισμένος με $\\pi^{2\\,n}$. Η συνάρτηση $\\zeta(s)$ είναι η πιο διάσημη συνάρτηση στην θεωρία αριθμών, και η πηγή του πιο εξέχοντος (και άλυτου μέχρι στιγμής) προβλήματος στα μαθηματικά, της **υπόθεσης του Riemann**. Η εύρεση όλων των μιγαδικών ριζών της συνάρτησης $\\zeta(s)$ του Riemann, έχουν επικηρυχθεί με $\\mbox{\\$}10^6$. <br><br>\n",
    "Στα επόμενα δυο σχήματα απεικονίζουμε την $2\\,\\pi$-περιοδική επέκταση της $f(x)=|x|$, καθώς και τους πρώτους οχτώ όρους στη σειρά Fourier. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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oq62hsjZGXW0NlbW+2rp6881u1+M7Kmtj1NXkqKzZYGDtEHW1dVTWofHMAK2jstZXWVdv\nuMGcKzRGZR0adbV1VNb0MbB2gLraHirr0B59tFxXL7lEuvBCt+sJBZV1MPautofKOjTqauuorOlj\nYO0AdbV9VNZq7F1tH5V1MOpq+6is1air7aOypouBtU3U1c5QWas9+qi0Y4c5pq62jspaRl3tDJW1\nGnW1fVTWdDGwtom62jkqq8He1c5RWcuoq52jshrU1c5RWdPDwNoG6mo6qKzGI4+U6+qHPyxNm+Z2\nPaGislJX00JlNairnaOypoeBtQ3U1fQUvbIePiz93d+VX2bvavuorNTVNBW9slJX00NlTQcDa4uo\nq+kqemWlrqaryJWVupquoldW6mp6qKzpYGBtEXU1fUWtrNTV9BW5slJX01fUykpdTR+VtXMMrC2g\nrmajqJWVupqNIlZW6mo2ilpZqavpo7J2joG1BdTV7BStsh46xDMDZKWIlZW6mp2iVVbqanaorJ1h\nYE2IupqtolXWRx6RXnvNHH/kI9LUqW7XE5siVVbqaraKVlmpq9mhsnaGgTUh6mr2ilJZDx1i72rW\nilRZqavZK0plpa5mj8raPgbWBKir+ShKZaWu5qMIlZW6mo+iVFbqavaorO1jYE2Aupqf2CtrbV1l\n72p2ilBZqav5ib2yUlfzQ2VtDwNrE9TVfMVeWSvr6kc/Kl1wgdv1xC7mykpdzVfslZW6mh8qa3sY\nWJugruYv1spa+8wA7F3NXsyVlbqav1grK3U1f1TW1jGwNkBddSPWyvrd70o7d5pj6mp+Yqys1FU3\nYq2s1NX8UVlbx8DaAHXVndgqK88M4E6MlZW66k5slZW66g6VtTWZDay7dpn/AaGirroVW2WlrroV\nU2WlrrpVW1nt/4tQUVfdia2ybtiQ7ffW1AfWI0ek666TRo+W/uzPpIGBtD9CPqir7sVSWXlmAPdi\nqqzUVfdiqazUVfdiqawDA2bmGz3azIBHjqT/MVIfWI85xkzYhw6ZP5cvT/sjZI+66odYKuvKleW6\nOnOm9MEPul1PUcVQWamrfqisrG+8EW5lpa66F0tlXb68evbLYmbKZEtA5f68O+8Mr7JSV/0RemVl\n76o/Yqis1FV/VH4th1hZqav+CL2yDgyYWc/K6jqXycB63nnSpZea4127pIceyuKjZIO66pfQK+vK\nldL//I85pq66t2hRuJWVuuqXP/1T6eMfN8chVlbqqj9Cr6y2rkpm9jvvvGw+TmY/dBVqZaWu+ifU\nykpd9c873yndeKM5Dq2yUlf9E+peVuqqf0KtrHnVVSnDgfW886S5c83x7t1hVFbqqp9CrazUVT+F\nuJeVuuqnUCsrddU/oVbWvOqqlPHzsIZWWamr/gqtsh48yDMD+CrEykpd9VdolZW66q/QKmuedVXK\neGCdNKm6sj74YJYfrTPUVb+FVlkr6+qsWdIHPuB0OagRUmWlrvottMpKXfVXaJU1z7oq5fCbrkKp\nrNRV/4VSWQ8elP7+78svs3fVPyFVVuqq/0KprNRV/4VSWfOuq1IOA+ukSdK8eeZ4zx4/Kyt1NQyh\nVFbqahhCqKzU1TCEUlmpq/4LpbLmXVelHAZWaXBl9e3RJ3U1HL5X1tq6yt5Vf4VQWamr4fC9slJX\nw+F7ZXVRV6WcBtZzz/W3slJXw+J7ZX344XJdnT1bmjLF6XLQhM+Vddcu6mpIfK+s1NVw+F5ZXdRV\nKaeBVfL3t4JQV8Pja2Vl72p4fK6s1NXw+FpZqavh8bWyuqqrUo4D67nnSvPnm2NfKit1NUy+VtaH\nH5Zef90cU1fD4WNl3bXLrEWirobE18pKXQ2Pr5XVVV2VchxYJf/2slJXw+VbZaWuhsvHykpdDZdv\nlZW6Gi7fKqvLuirlPLBOnFiurHv3lguCC9TVsPlWWamrYfOpslbW1RNPpK6GxrfKSl0Nl2+V1WVd\nlXIeWCV/9rJSV8PnS2Xlt1qFz6fKSl0Nny+VlboaPl8qq+u6KjkYWCdOlC67zBzv3evm0Sd1NQ6+\nVNYVK6T//V9z/LGPSeef72Yd6IwPlbW2rt50U/5rQOd8qazU1fD5Ulld11XJwcAquX/0SV2Nh+vK\nyt7VePhQWamr8XB9naOuxsN1ZfWhrkqOBtbKypr3o0/qalxcV1bqalxcVlbqalxcV1bqajxcV1Yf\n6qrkaGCV3O1lpa7Gx1Vlpa7Gx2Vlpa7Gx1Vlpa7Gx1Vl9aWuSg4H1tpHn9/+dvYfk7oaJ1eVtbKu\nzplDXY2Fi8pKXY2Tq8pKXY2Pq8rqS12VHA6sUvXJX7pU6u/P9uNRV+OVd2WlrsbLRWWlrsYr78pK\nXY1X3pXVp7oqOR5Y83z0SV2NW96Vdfny6ro6eXK2Hw/5yrOyUlfjlndlpa7GK+/K6lNdlaSuUsnt\nL/vassV8QUvSqadKO3ZIxx+f/sdZu1a65BJzfNZZ0tatDKyxOXxYGjtW+s1vzMvr1kkXXZT+xzl4\nUHrve8tfyM8/z8Aao7/+63JduPZa6d57k/293t5ejRw5Uj09PRoxYkTT97/++vJt33ZbddFAHF58\n0fywsWSuc6+9Jr3jHel/nHXrpIsvNsdjxkjbtjGwxubIEWncOOmVV8zLa9eW/5+naWDA/Kulvc79\n4hfuB1anhVWSzjlHuvxyc5zVXlbqajHkVVkrH3VSV+OVR2WlrhZDXpWVuhq/vCqrb3VV8qCwStWV\n9ZRTzKPPNCsrdbU4sq6stXXVh0edyE47lbWVwkpdLY6sKyt1tTiyrqw+1lXJg8IqVVfWffvSrazU\n1WLJurI+9JB/jzqRnSwrK3W1WLKurNTV4si6svpYVyVPCqskvfSSGVyldCsrdbV4sqqsvj7qRLZa\nraxJCyt1tXiyqqzU1eLJqrL6fJ3zorBK0oQJ0hVXmON9+6T77+/8NqmrxZRVZfX1USeylUVlpa4W\nU1aVlbpaPFlVVp+vc94UVin9ykpdLa60K6vPjzqRvVYqa5LCSl0trrQrK3W1uNKurL5f57wprFK6\nlZW6WmxpV1afH3Uie2lWVupqsaVdWamrxZV2ZfX9OudVYZWkX/3KDK6S9Md/bB59Dh/e+u1QV5FW\nZR0YMM8MsHu3efmFF6RJk1JbJgKRtLI2K6zUVaRVWamrSKuy+l5XJc8KqySNHy9deaU53r+/vcpK\nXYWUXmV96KHysDp3LsNqUaVRWamrkNKrrNRVpFVZfa+rkoeFVeq8slJXYXVaWamrqJSksjYqrNRV\nWJ1WVuoqrE4rawh1VfKwsEqdVVbqKip1Wlmpq6jUSWWlrqJSp5WVugqr08oaQl2VPC2skqmi48eb\n41YqK3UVtdqtrNRVDKVZZa1XWKmrqNVuZaWuola7lTWUuip5Wlgl6eyzpU98whzv3y/dd1/zv0Nd\nxVDarawPPlgeVufNY1iF0U5lpa5iKO1WVuoqarVbWUOpq5LHhVWqrqzvfre0c2fjykpdRT2tVtaB\nAWn0aGnPHvPyL38pnXtu5stEIBpV1qEKK3UV9bRaWamrqKfVyhpSXZU8LqxSdWX97W+lf/3X+u9L\nXUUjrVbWBx8sD6vz5jGsolorlZW6ikZarazUVdTTamUNqa5KnhdWyVTSCRPMSX/3u82jzxNOGPx+\n1FU0k7SyUleRRL3KWltYqatoJmllpa6imaSVNbS6KnleWKXBlXWovazUVSSRtLJSV5FEkspKXUUS\nSSsrdRXNJK2sodVVKYDCKjWvrNRVJNWssv7hD+aZAezA+p//WS4fQK2hKmtlYb3tthHUVSTSrLJS\nV5FUs8oaYl2VAiisUuO9rNRVtKJZZa2sq/PnM6yisUaVdfdu6iqSa1ZZqatIqlllDbGuSoEMrJI5\n+V1d5viuu6QDB6Tu7m4995y0caN5/VlnSZ/8pLs1+qq7u9v1EryyYIF5dClJzz0nrV9vjh9+uFtL\nlpTfb/Hi/NfmO+5L1d75TunGG83xoUOquv/84z9KBw+a4xtukN71rvzX5zvuT9Uqh4wlS8y/+HR3\nd2vdOlNYJVNXP/1pF6vzG/elap/6lPS+95njDRvK95+VK7ur/qUnpOtcMAPruHHlYfT//s9U1u7u\nbupqAnwhV6tXWb/xjW7qahPclwarraz2uXsfftj8SV2tj/tTtaEqa3d3N3U1Ae5L1epV1m98ozvI\nuioFNLBK5gu1srLu309dRXtqK+uzz0rbt5ffHtKjTrhVW1n/4R/KxxJ1Fa2pray//S11Fe2prazP\nPFP++Q0pvOtc4oE1zUcv7d5WbWV94YXy2zqtq2l9fj6cp6z4+Lm1ezu1lXXmTPPPb5J02WWd1VXX\nn1vWt5WWmM5TZWW1ZVXqvK768LlldTtpiuk81VbWyutcp3XV9eeW9W2lwcfPrd3bqa2ss2eXr3Od\n1lUX5ymogVWq3st69Kj5M4266vqOlfVtpcHHz62T26msrJUqv8Db4cPnluVtpSWm81RZWQ8fLr++\n07rqw+eW1e2kKbbzVPk96MgR82caddWHzy3L20qDj59bJ7dTWVkrdVpXXZynRI/VSqWSDh8+rN7e\n3o4WZXVyW3/yJ9Lll0tPPCFJhyX16uabzQ9huVpTFreT5m35djtp3lant3PzzdKf//nbt6Y5c3o1\nerTUydJ8+dyyuC3WVN/ChdI//ZN04ID5+8OH92rhQj/uS2neFmvK/nbOOMMUsCeflCqvc2+95W5N\nWdxOmrfl2+2keVtpXOcWLnz71jRzZq/GjPHre1OpVFKXrZF1JHoeVvu8ggAAAECa7G8GbCTRwFoq\nldTX15fawtKwapW0Zo30N38jvf/9rleDkG3ZYn6Ccs4c6QtfcL0ahOzAAemv/qpXjz/+Hr3yyi6d\nckrjb8BAI9/5jvTUU+b70znnuF4NQvbii+Z+NGuW9Bd/4Xo1g5100knpFFYAQDKVv+mqWTEAACQT\n1NNaAQAAoHgYWAEAAOA1BlYAAAB4jYEVAAAAXmNgBQAAgNcYWAEAAOC1Dn4rsR9uvfVWzZ07V1On\nTnW9FK+sX79e+/fv14EDB/T000/r1ltv1cSJE10vyzvPP/+8fvrTn6qvr08///nP9ZWvfEXTp093\nvSwABcL3oWS4rrUmtvko6MK6ceNGrVixQkfsL1vG26644goNDAzoc5/7nKZMmaJ58+a5XpJ3+vv7\ntXr1at10001avHixFi5cqNmzZ+uNN95wvTQABcH3oeS4riUX43wU7MDa19enl156SWPHjnW9FC+t\nX79eV1555dsvx3SnTcv27du1bNkyvfbaa5KkmTNnqr+/X5s2bXK8MgBFwfeh5LiuJRPrfBTswPrA\nAw/oi1/8our9oq7Fixdr/PjxOv300zV27Ni3/7zmmmvyXagj48aN0/HHHy9JWrNmje66665B71P0\nczR+/Hht2rRJZ5xxhiTp9ddfV1dXl84888yq9yv6eRoK5wTt4r5TLcn3Ic6Z0ey6xnkyGs1HQZ+j\nUoCefPLJ0rZt20qlUqk0Y8aM0vr166ve/sgjj5Q2btxYOnr0aOlb3/pWqVQqlZYuXZr7Ol17/vnn\nS7fffnvplltuKQ0MDFS9jXM02Gc/+9nSokWLql7HeRqMc9JYT09PSVKpp6fH9VK8w32nudrvQ5yz\navWua5wno9F8FPo58uqHru69917t3LlTXV1dVa8vlUrq6urS9OnTNWnSJP3+97/XnDlz6t7OZz7z\nGUnSs88+qwkTJmjr1q068cQTM117XpKco0svvVSSNHnyZE2ePFn333+/pk2bpnXr1mn48OGS4j5H\nUmvnSZKWL1+uU089VcuWLat6/9jPU60k561o5wTp4b7T2FDfhzhn1epd1zhP0htvvNFwPgr9HHk1\nsF533XVN32fFihXav3+/7rrrLpVKJb366qt67LHH1N/fr5kzZ1a97/Lly7Vy5Ur96Ec/ymrJuUty\njjZv3qz58+dr8+bNGjVqlGbMmKFrr71WTz/9tC6//PKq943xHEnJzpP14x//WEePHtWyZcs0MDCg\nffv2adSoUVXvE+t5qtXKeSvKOUH6uO8M1uz7UNHPWdLrWpHP0zPPPJNoPgr1HHk1sCZRu8/ivvvu\n01VXXTXoKUBefvll9ff3a9iwYRoYGNCWLVtyXKVbxx57rM4++2yddtppkqQdO3bouOOO0znnnFP1\nfkU+R9b+UXPVAAABOElEQVSGDRu0d+9eXXrppdq3b582b96sU045pepCwXkajHOCdnHfGazZ9yHO\nWbLrWtHPU5L5KORzFNzAau3Zs0d333239u/fr29+85t66623NGvWrLffvmbNGl199dWSpAkTJmjV\nqlWulpq7SZMm6fOf/7zuuecedXV1adOmTXrqqac0evToqvcr8jmSpJ07d2ru3Ll68803JZX/2bun\np6fq/Yp+nobCOUG7uO9US/J9iHOW7LrGeTIazUchn6OuUqnOj9kDAFrW29urkSNHqqenRyNGjHC9\nHACIAgMrAKSoVCqpr69PJ5100qAfXgMAtIeBFQAAAF4L9hcHAAAAoBgYWAEAAOA1BlYAAAB4jYEV\nAAAAXmNgBQAAgNcYWAEAAOA1BlYAAAB4jYEVAAAAXmNgBQAAgNf+H7geg6UvpciMAAAAAElFTkSu\nQmCC\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def abs_ext(x):\n",
    "    if   x % pi.n() == 0.0 :\n",
    "         y = 0\n",
    "    else:\n",
    "        y = f( ( x ) % (2*pi.n()) )\n",
    "    return y\n",
    "abs_ext = plot(abs_ext,-4*pi-pi/3 , 4*pi+pi/3, ymin=-0.3 , ymax=3.15, \\\n",
    "              ticks=[pi,[pi]], tick_formatter=[pi,pi], \\\n",
    "              thickness=2 , aspect_ratio=2 )\n",
    "abs_ext.show(figsize=7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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e5RMzkbghQ1itnzxZdSSOIhVWvVy5wjv0du3YwC8erUMHYNEirnrPlk11NNZU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\nRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pfabs8 = plot(fourier_abs,(x,-4*pi-pi/3,4*pi+pi/3) , ymin=-0.3 , ymax=3.15, \\\n",
    "               color='red' , aspect_ratio=2, thickness=1); \n",
    "(pfabs8).show(figsize=7, ticks=[pi,[pi]], tick_formatter=[pi,pi])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.6 Παραγώγιση και ολοκλήρωση σειρών Fourier"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.6.1 Ολοκλήρωση"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Η ολοκλήρωση είναι μια διαδικασία που το αποτέλεσμά της είναι μια συνάρτηση πιο ομαλή από αυτήν που ολοκληρώνουμε. Σε αντιδιαστολή, η παραγώγιση οξύνει τα προβλήματα που ενδεχομένως έχει η συνάρτηση που παραγωγίζουμε. Για παράδειγμα, το αόριστο ολοκλήρωμα της συνάρτησης ${\\mathrm{u}}(x)$ του παραδείγματος 8., η οποία παρουσιάζει ασυνέχεια στο $x=0$, είναι η $x\\,{\\mathrm{u}}(x) + c$, η οποία είναι συνεχής στο $x=0$. Από την άλλη, η παράγωγος της συνεχούς συνάρτησης $|x|$ είναι η $2\\,{\\mathrm{u}}(x)-1$ η οποία παρουσιάζει ασυνέχεια στο $x=0$. <br>\n",
    "\n",
    "Οπότε για μια συγκλίνουσα σειρά Fourier δεν θα έπρεπε να είχαμε κανένα πρόβλημα να την ολοκληρώσουμε όρο κατά όρο. Όμως υπάρχει ένα λεπτό σημείο που χρειάζεται αν προσέξουμε. Το ολοκλήρωμα μιας περοδικής συνάρτησης δεν είναι αναγκαστικά περιοδική συνάρτηση. Το ζήτημα  είναι ο σταθερός όρος $1$ στη σειρά Fourier. Έστω κι αν η σταθερή συνάρτηση είναι περιοδική συνάρτηση, το ολοκλήρωμά της, δηλαδή η $x$, δεν είναι με κανένα τρόπο περιοδική. Το πρόβλημα αυτό προφανώς δεν υπάρχει για τους όρους με τα ημίτονα και τα συνημίτονα των οποίων τα ολοκληρώματα είναι περιοδικές συναρτήσεις. Οπότε μόνο ο όρος <br><br>\n",
    "$$ \\frac{a_0}{2} = \\frac{1}{2\\,\\pi} \\, \\int_{\\pi}^\\pi f(x) \\, {\\mathrm d}\\,x\\,,$$ <br>\n",
    "δηλαδή η μέση τιμή της $f(x)$ στο διάστημα $[-\\pi,\\pi]$ μπορεί να δημιουργήσει κάποια δυσκολία στην ολοκλήρωση. Αν η μέση τιμή της $f(x)$ στο διάστημα $[-\\pi,\\pi]$ είναι μηδέν, τότε ισχύει το ακόλουθο:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Λήμμα 12.** Αν η $f(x)$ είναι $2\\,\\pi$ περιοδική συνάρτηση τότε το ολοκλήμωμά της \n",
    "$g(x) = \\int_{0}^x f(t) \\, {\\mathrm d}\\,t$, είναι περιοδική συνάρτηση αν και μόνο αν \n",
    "$\\int_{-\\pi}^\\pi f(x) \\, {\\mathrm d}\\,x=0$, δηλαδή η μέση τιμή της $f(x)$ στο $[-\\pi,\\pi]$ είναι μηδέν."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "**Θεώρημα 13.** Αν η $f(x)$ τμηματικά συνεχής και έχει μηδενική μέση τιμή στο $[-\\pi,\\pi]$, τότε η σειρά Fourier της $f(x)$, μπορεί να ολοκληρωθεί όρο προς όρο και είναι η σειρά Fourier <br><br>\n",
    "$$g(x) = \\int_{0}^x f(t) \\, {\\mathrm d}\\,t \\sim\n",
    "M + \\sum_{k=1}^\\infty \\left( -\\frac{b_k}{k} \\, \\cos k\\,x + \\frac{a_k}{k}\\,\\sin k\\,x \\,\\right)\\,,  $$<br>\n",
    "όπου $M$ είναι η μέση τιμή της $g(x)$ στο $[-\\pi,\\pi]$, $ M = \\int_{-\\pi}^\\pi g(x) \\, {\\mathrm d}\\,x\\,$. <br><br>\n",
    "**Γενικότερα**, αν η μέση τιμή της $f(x)$ δεν είναι μηδέν στο $[-\\pi,\\pi]$, τότε η σειρά που προκύπτει από την όρο προς όρο ολοκλήρωση της σειράς Fourier της $f(x)$, θα περιλαμβάνει και την ταυτοτική συνάρτηση $x$, δηλαδή <br><br>\n",
    "$$ g(x) = \\int_{0}^x f(t) \\, {\\mathrm d}\\,t \\sim  \n",
    "\\frac{a_0}{2}\\,x + M + \\sum_{k=1}^\\infty \\left( -\\frac{b_k}{k} \\, \\cos k\\,x + \\frac{a_k}{k}\\,\\sin k\\,x \\,\\right)\\,.  $$<br>\n",
    "Υπάρχουν δυο τρόποι για να ερμηνεύσουμε το αποτέλεσμα αυτό.\n",
    "- Μπορούμε να γράψουμε την σχέση\n",
    "$$ g(x) - \\frac{a_0}{2}\\,x  \\sim M + \\sum_{k=1}^\\infty \\left( -\\frac{b_k}{k} \\, \\cos k\\,x + \\frac{a_k}{k}\\,\\sin k\\,x \\,\\right)\\,,  $$<br>\n",
    "στην οποία το αριστερό μέλος είναι μια $2\\,\\pi$- περιοδική συνάρτηση.\n",
    "- Εναλλακτικά, μπορούμε να αντικαταστήσουμε στην παραπάνω σχέση, την σειρά Fourier της $x$ που βρήκαμε στο παράδειγμα 3. Το αποτέλεσμα θα είναι η $2\\,\\pi$-περιοδική επέκταση του ολοκληρώματος με το οποίο ορίζεται η $g(x)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.6.2 Παραγώγιση"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Η σειρά που προκύπτει παραγωγίζοντας την σειρά Fourier μιας συνάρτησης $f(x)$, όρο προς όρο, θα πρέπει να παραμένει τμηματικά συνεχής. Οπότε η $f(x)$ θα πρέπει  να είναι δυο φορές τμηματικά διαφορίσιμη για να μπορέσουμε να εφαρμόσουμε το Θεώρημα 7. της σύγκλισης. Πιο συγκεκριμένα"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Θεώρημα 14.** Αν η $f(x)$ έχει μια τμηματικά, δυο φορές διαφορίσιμη, τότε η σειρά Fourier της $f(x)$, μπορεί να παραγωγιστεί όρο προς όρο και είναι <br><br>\n",
    "$$f'(x) \\sim\n",
    "\\sum_{k=1}^\\infty \\left( k\\,b_k\\, \\cos k\\,x - k\\,a_k\\,\\sin k\\,x \\,\\right)\\,.  $$<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Παράδειγμα 15.** Η παράγωγος της συνάρτησης $f(x)=|x|$ είναι η συνάρτηση προσήμου:<br><br>\n",
    "$$ \\frac{ \\mathrm{d} }{ \\mathrm{d} x} \\, f(x) = {\\mathrm{sign}} \\, (x)\\,  = \\begin{cases}\n",
    "+1 \\,,  & x > 0 \\,, \\\\ \n",
    "-1  \\,, & x < 0 \\,. \\end{cases} $$ <br>\n",
    "Παραγωγίζουμε όρο προς όρο την σειρά Fourier της $|x|$ που βρήκαμε στο Παράδειγμα 11. και έχουμε "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, \\sin\\left(7 \\, x\\right)}{7 \\, \\pi} + \\frac{4 \\, \\sin\\left(5 \\, x\\right)}{5 \\, \\pi} + \\frac{4 \\, \\sin\\left(3 \\, x\\right)}{3 \\, \\pi} + \\frac{4 \\, \\sin\\left(x\\right)}{\\pi}</script></html>"
      ],
      "text/plain": [
       "4/7*sin(7*x)/pi + 4/5*sin(5*x)/pi + 4/3*sin(3*x)/pi + 4*sin(x)/pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fourier_abs.diff(x).show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "δηλαδή \n",
    "$$ {\\mathrm{sign}} \\,x \\sim \\frac{4}{\\pi}\\, \\left( \\sin x + \\frac{\\sin 3\\,x}{3}+\\frac{\\sin 5\\,x}{5} + \\frac{\\sin 7\\,x}{7} + \\cdots   \\right)\\,. $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.7 Αλλαγή κλίμακας <br><br>\n",
    "\n",
    "Ορίσαμε την σειρά Fourier μιας συνάρτησης $f(x)$ στο διάστημα $[-\\pi,\\pi]$, με μήκος $2\\,\\pi$, που είναι ίσο με την περίοδο της περιοδικής επέκτασης της $f(x)$. Όμως, δεν υπάρχει κάποιος λόγος να θεωρούμε ότι υπάρχει κάτι το ιδιαίτερο με το μήκος του διαστήματος αυτού. Το αντίθετο μάλιστα, στα προβλήματα του πραγματικού κόσμου τα μήκη ράβδων, νημάτων κτλ. δεν έχουν μήκος $2\\,\\pi$.  Γι' αυτό τον λόγο κρίνεται σκόπιμο να ορίσουμε την σειρά Fourier μιας συνάρτησης $f(x)$ σε οποιοδήποτε συμμετρικό διάστημα $[-L,L]$, με $L$ θετικό πραγματικό αριθμό. <br>  \n",
    "Ο τρόπος που θα υλοποιήσουμε την μετατροπή αυτή είναι να τεντώσουμε κατάλληλα την μεταβλητή \n",
    "$y$ που διατρέχει το διάστημα $[-\\pi,\\pi]$, έτσι ώστε τα άκρα του διαστήματος $[-\\pi,\\pi]$ για τo $y$ να συμπέσουν με τα άκρα του διαστήματος $[-L,L]$ που διατρέχει το $x$. Αυτό επιτυγχάνεται με την αλλαγή μεταβλητής <br><br>\n",
    "$$x = \\frac{L}{\\pi} \\, y \\qquad -\\pi\\leq y \\leq \\pi\\,, \\qquad -\\pi \\leq x \\leq \\pi\\,. $$<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Με την αλλαγή αυτή μια συνάρτηση $F(y)$ που παριστάνεται με την σειρά Fourier <br><br>\n",
    "$$ F(y) \\sim \\frac{a_0}{2} +  \\sum_{k=1}^\\infty \\left( {a_k} \\, \\cos k\\,y + {b_k}\\,\\sin k\\,y \\,\\right)\\,,  $$<br>\n",
    "στο $[-\\pi,\\pi]$, με \n",
    "$$ \n",
    "a_k \\,= \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(y)\\,\\cos k\\,y \\, {\\mathrm{d}} y\\,,\\qquad \n",
    "b_k \\,= \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(y)\\,\\sin k\\,y \\, {\\mathrm{d}} y\\,, \n",
    "$$<br>\n",
    "μετατρέπεται κάτω από την αλλαγή κλίμακας $x=\\frac{L}{\\pi}\\,y$, σε μια συνάρτηση $f(x)$ που έχει την αναπαράσταση σε σειρά Fourier  <br><br>\n",
    "$$ f(x) \\sim \\frac{a_0}{2} +  \\sum_{k=1}^\\infty \\left( {a_k} \\, \\cos \\frac{k\\,\\pi\\,x}{L} + {b_k}\\,\\sin \\frac{k\\,\\pi\\,x}{L} \\,\\right)\\,,  $$<br>\n",
    "στο $[-L,L]$, με \n",
    "$$ \n",
    "a_k \\,= \\frac{1}{L} \\int_{-L}^{L} f(x)\\,\\cos \\frac{k\\,\\pi\\,x}{L} \\, {\\mathrm{d}} x\\,,\\qquad \n",
    "b_k \\,= \\frac{1}{L} \\int_{-L}^{L} f(x)\\,\\sin \\frac{k\\,\\pi\\,x}{L} \\, {\\mathrm{d}} x\\,.\n",
    "$$<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Παράδειγμα 16.** Θα βρούμε την σειρά Fourier της $f(x)=x$ στο διάστημα $[-1,1]$. Επειδή η $x$ είναι περιττή συνάρτηση αναμένουμε στην σειρά Fourier να μην εμφανίζονται συνημίτονα. \n",
    "Τροποποιούμε ανάλογα στο Sage τα ολοκληρώματα τα άκρα της ολοκλήρωσης, τους συντελεστές κτλ. και έχουμε"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x |--> x\n"
     ]
    }
   ],
   "source": [
    "f(x)=x ; print f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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csPLLL/Dmm9C7t90ROYvm+YuTObSxYK+8PHjsMejUCRo2NOOXlfhFxJ/U9vGz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\nLyLiQEr+IiIOpOQvIuJASv4iIg70/wBe8FvsYMp89AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pf = plot(f,(x,-1,1)); pf.show(figsize=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def inner_scaled(f,g,L):\n",
    "    return 1/L * integrate(f*g,x,-L,L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a0 =  0\n",
      "ak =  0\n",
      "bk =  -2*(-1)^k/(pi*k)\n"
     ]
    }
   ],
   "source": [
    "var('k')\n",
    "l = 1\n",
    "assume(k,'integer')\n",
    "a0 = inner_scaled(f(x),1,1); print str('a0 = '), a0; \n",
    "a(k) = inner_scaled(f(x),cos(k*x*pi/l),l) ; print str('ak = '), a(k);\n",
    "b(k) = inner_scaled(f(x),sin(k*x*pi/l),l); print str('bk = '), b(k);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{\\sin\\left(6 \\, \\pi x\\right)}{3 \\, \\pi} + \\frac{2 \\, \\sin\\left(5 \\, \\pi x\\right)}{5 \\, \\pi} - \\frac{\\sin\\left(4 \\, \\pi x\\right)}{2 \\, \\pi} + \\frac{2 \\, \\sin\\left(3 \\, \\pi x\\right)}{3 \\, \\pi} - \\frac{\\sin\\left(2 \\, \\pi x\\right)}{\\pi} + \\frac{2 \\, \\sin\\left(\\pi x\\right)}{\\pi}</script></html>"
      ],
      "text/plain": [
       "-1/3*sin(6*pi*x)/pi + 2/5*sin(5*pi*x)/pi - 1/2*sin(4*pi*x)/pi + 2/3*sin(3*pi*x)/pi - sin(2*pi*x)/pi + 2*sin(pi*x)/pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fourier6_x_scaled = a0/2 + sum( a(k)*cos(k*x*pi/l) + b(k)*sin(k*x*pi/l) for k in [1..6]); \n",
    "fourier6_x_scaled.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Στο παρακάτω γραφικό απεικονίζουμε το μερικό άθροισμα των έξι πρώτων όρων της σειράς Fourier της $f(x)=x$ στο διάστημα $[-1,1]$, μαζί με την $2$-περιοδική επέκταση της $f(x)$ στην οποία συγκλίνει η σειρά Fourier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def fx_ext(x):\n",
    "    y = f( ( x ) % (2.0*l.n() ) )\n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8AEy9eyqFVRS+QcONLSw1+fJB4cLSlrKgW7dudOjQgQULFuTaMZKGB4DF1YNQ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\nEREREZHL/h9O/1q82YMH7QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 13 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "px_ext = plot(fx_ext,-5.,5, ymin=-1.2 , ymax=1.2, \\\n",
    "              exclude=[i for i in [-4..5] ],  ticks=[1,1], tick_formatter=[1,1], \\\n",
    "              thickness=2 , aspect_ratio=1, plot_points=400 )\n",
    "px_points = point([(i,0) for i in [-3,-1,1,3]],size=20)\n",
    "pf_x_6 = plot(fourier6_x_scaled,(x,-5,5) , ymin=-1.2 , ymax=1.2,color='red' , aspect_ratio=1); \n",
    "(px_ext + px_points + pf_x_6 ).show(figsize=8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## 7.8 Το φαινόμενο Gibbs <br><br>\n",
    "\n",
    "Όπως έχουμε ήδη παρατηρήσει στα σημεία που η $f(x)$ δεν είναι συνεχής, η σειρά Fourier **αστοχεί** να ταυτιστεί με την πραγματική τιμή της $f(x)$. Μάλιστα αυτή η αστοχία αυτή δεν οφείλεται στην αποκοπή των όρων στην σειρά Fourier. Όσους όρους και να συμπεριλάβουμε στη σειρά Fourier, η τελευταία εξακολουθεί να αστοχεί και μάλιστα η αστοχία δείχνει να μην αλλάζει. Το φαινόμενο αυτό ονομάζεται **φαινόμενο Gibbs** και η απόδειξή του είναι αρκετά πονηρή. Αλλά με την βοήθεια του Sage είναι πολύ πιο εύκολο να την κατανοήσει κανείς. <br> <br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Θεωρούμε την συνάρτηση προσήμου ${\\mathrm{sign}}(x)$ του Παραδείγματος 15. Όπως είδαμε οι οχτώ πρώτοι όροι στη σειρά Fourier της ${\\mathrm{sign}}(x)$ είναι οι "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, \\sin\\left(7 \\, x\\right)}{7 \\, \\pi} + \\frac{4 \\, \\sin\\left(5 \\, x\\right)}{5 \\, \\pi} + \\frac{4 \\, \\sin\\left(3 \\, x\\right)}{3 \\, \\pi} + \\frac{4 \\, \\sin\\left(x\\right)}{\\pi}</script></html>"
      ],
      "text/plain": [
       "4/7*sin(7*x)/pi + 4/5*sin(5*x)/pi + 4/3*sin(3*x)/pi + 4*sin(x)/pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_sign = fourier_abs.diff(x); f_sign.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def sign_ext(x):\n",
    "    if   x % pi.n() == 0.0 :\n",
    "         y = 0\n",
    "    else:\n",
    "         y = sign( ( x ) % (2*pi.n() ) )\n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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QaN9e+3WBgUCrVtyOBN9ISwPi4viZasFiAWbOBJ57DggJ0ff+1xXnngscPMg/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\n",
      "text/plain": [
       "Graphics object consisting of 12 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "psign_ext = plot(sign_ext,-5*pi,5*pi, ymin=-1.2 , ymax=1.2, \\\n",
    "              exclude=[i*pi for i in [-5..5] ], ticks=[pi,1], tick_formatter=[pi,1/2], \\\n",
    "              thickness=2 , aspect_ratio=3 )\n",
    "psign_points = point([(i*pi,0) for i in [-5..5]],color = 'blue' , size=20)\n",
    "pf_sign = plot(f_sign,(x,-5*pi,5*pi) , ymin=-1.2 , ymax=1.2,color='red' , aspect_ratio=1); \n",
    "(psign_ext + pf_sign + psign_points ).show(figsize=8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Από το σχήμα, διαβάζοντας τις διαβαθμίσεις του γραφικού του Sage, παρατηρούμε ότι η τιμή της σειράς Fourier υπερβαίνει την τιμή που έχει η συνάρτηση για $x>0$, περίπου κατά $+0.2$, το μέγιστο. Αντίστοιχες διαπιστώσεις ισχύουν και αριστερά του σημείου $x=0$. Θέλουμε να βρούμε πόση ακριβώς  είναι η μεγαλύτερη αστοχία της σειράς Fourier, όσους όρους (άπειρους) και να συμπεριλάβουμε στην σειρά Fourier."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Τα πιθανά ακρότατα της σειράς Fourier  (ψάχνουμε για τοπικό μέγιστο στο διάστημα $(0,\\pi)$ ) είναι εκεί που μηδενίζεται η πρώτη παράγωγός της για πρώτη φορά, οπότε"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, \\cos\\left(7 \\, x\\right)}{\\pi} + \\frac{4 \\, \\cos\\left(5 \\, x\\right)}{\\pi} + \\frac{4 \\, \\cos\\left(3 \\, x\\right)}{\\pi} + \\frac{4 \\, \\cos\\left(x\\right)}{\\pi}</script></html>"
      ],
      "text/plain": [
       "4*cos(7*x)/pi + 4*cos(5*x)/pi + 4*cos(3*x)/pi + 4*cos(x)/pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_sign.diff().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "H μερικό άθροισμα της σειράς σειράς Fourier που προκύπτει από την παραγώγιση της σειράς Fourier της ${\\mathrm{sign}}(x)$, όρο προς όρο, είναι <br><br>\n",
    "$$ {\\mathrm{sign}}_Ν'(x) \\sim \\frac{4}{\\pi} \\, \\left( \\, \\cos x + \\cos 3\\,x+ \\cdots + \\cos (2\\,N-1)\\,x\\, \\,\\right) \\,.$$ <br>\n",
    "Είναι αξιοσημείωτο γεγονός ότι το άθροισμα αυτό έχει κλειστό τύπο τονοποίο γνωρίζει το Sage!! Πραγματικά"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, k, N)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('x,k,N')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, \\csc\\left(x\\right) \\sin\\left(2 \\, N x\\right)}{\\pi}</script></html>"
      ],
      "text/plain": [
       "2*csc(x)*sin(2*N*x)/pi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sum_1 = 4/pi * sum(cos((2*k-1)*x), k, 1, N);\n",
    "sum_2 = sum_1({arctan2(sin(2*x),cos(2*x)) : 2*x}).trig_simplify();\n",
    "sum_3 = sum_2.trig_reduce();  sum_3.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Δηλαδή  \n",
    "$$ {\\mathrm{sign}}_Ν'(x) = \\frac{2}{\\pi} \\, \\frac{\\sin 2\\,N\\,x}{\\sin x}\\,.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Η ψηλότερη κορυφή στην αστοχία στο διάστημα $(0,\\pi)$ είναι εκεί όπου μηδενίζεται για πρώτη φορά η παραπάνω παράγωγος, δηλαδή για $x=\\frac{\\pi}{2\\,N}$. Θέτοντας την τιμή αυτή για το $x$ στο μερικό άθροισμα της ${\\mathrm{sign}}_N(x)$ βρίσκουμε ότι <br><br> \n",
    " $$ {\\mathrm{sign}}_Ν\\left(\\frac{\\pi}{2\\,N}\\right) = \n",
    " \\frac{2}{\\pi} \\, \\frac{\\pi}{N}\\, \\left( \\, \\frac{\\sin (\\pi/2\\,N)}{\\pi/2\\,N} + \\frac{\\sin (3\\,\\pi/2\\,N)}{3\\,\\pi/2\\,N} + \\cdots + \n",
    " \\frac{\\sin \\big(2\\,N-1)\\,\\pi/2\\,N\\big)}{(2\\,N-1)\\pi/2\\,N} \\right)\\,. $$ <br>\n",
    "Το παραπάνω άθροισμα είναι ένα άθροισμα Riemann, όπου ως ενδιάμεσα σημεία παίρνουμε τα μέσα των διαστημάτων \n",
    "$ 0 = x_0 < x_1 < x_2 < \\cdots < x_{N-1} < x_{N} = \\pi $. Οπότε καθώς το $N$ απειρίζεται, το παραπάνω άθροισμα γίνεται το παρακάτω ορισμένο ολοκλήρωμα Riemann <br><br>\n",
    "$$ \\lim_{N\\rightarrow \\infty} {\\mathrm{sign}}_Ν\\left(\\frac{\\pi}{2\\,N}\\right)  = \\frac{2}{\\pi}\\,  \\int_0^\\pi \\frac{\\sin x}{x} \\, {\\mathrm{d}} \\,x \\,, \\qquad \\Delta\\,x = \\frac{\\pi}{N}\\,.$$<br>\n",
    "Με άλλα λόγια, καθώς το $N$ απειρίζεται, η τιμή της σειρά Fourier της ${\\mathrm{sign}}(x)$ στο σημείο με την μεγαλύτερη αστοχία τείνει στην τιμή του παραπάνω ολοκληρώματος. Το ολοκλήρωμα αυτό υπολογίζεται **μόνο** με αριθμητική προσέγγιση, οπότε με την βοήθεια του Sage (σε ακρίβεια e-13) βρίσκουμε"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.17897974447217\n"
     ]
    }
   ],
   "source": [
    "astoxia = 2/pi * integrate(sin(x)/x,(x,0,pi)) ; print astoxia.n()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Δηλαδή η τιμή της σειράς Fourier της ${\\mathrm{sign}}(x)$ στο σημείο με την μεγαλύτερη αστοχία \n",
    "είναι 1.17897974447217, που σημαίνει ότι η σειρά Fourier αστοχεί να βρει την σωστή τιμή της $f(x)$ που προσεγγίζει, περίπου κατά 0.18, το οποίο είναι **9% του άλματος** από το -1 στο 1. Το ίδιο συμβαίνει για όλες τις ασυνεχείς συναρτήσεις που παρουσιάζουν άλματα σε σημεία, δηλαδή \n",
    "\n",
    "> Στο σημείο ασυνέχειας $x_0$ μιας συνάρτησης $f(x)$, η σειρά Fourier της $f(x)$ αστοχεί να βρει την σωστή τιμή κατά **9% του άλματος** που παρουσιάζει η $f(x)$ στο σημείο $x_0$. Υπενθυμίζεται ότι το άλμα της $f(x)$  στο $x_0$ είναι ο αριθμός $\\,\\,f(x_0^+)-f(x_0^-)$. "
   ]
  }
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