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<!DOCTYPE html>
<html lang="el">
<head>
<meta charset="UTF-8"/>
<meta name="viewport" content="width=device-width,initial-scale=1.0"/>
<title>ἐπιψαύσεις</title>
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<body>
<nav id="toc"><div id="toc-meta"><a href="index.html" id="back-btn">← Αρχική</a><div id="doc-title">Μετασχηματισμοί</div><div id="doc-author">Κώστας Κούδας</div></div><div id="toc-title">Περιεχόμενα</div>
<a href="#μετασχηματισμοί" class="toc-item toc-h1" style="padding-left:8px">Μετασχηματισμοί</a>
<a href="#μετασχηματισμοί-fourier" class="toc-item toc-h2" style="padding-left:22px">Μετασχηματισμοί Fourier</a>
<a href="#μετασχηματισμός-με-φυσική-ερμηνεία-hatfsint_-inftyinfty-ft-e-2pi-i-st-d-t" class="toc-item toc-h3" style="padding-left:36px">Μετασχηματισμός με φυσική ερμηνεία: $\hat{f}(s)=\int_{-\infty}^{\infty} f(t) e^{-2\pi i st} d t$</a>
<a href="#μετασχηματισμός-τραχανά-μδε-hatfsfrac1sqrt2piint_-inftyinfty-ft-e--i-st-d-t" class="toc-item toc-h3" style="padding-left:36px">Μετασχηματισμός Τραχανά (Μ.Δ.Ε.): $\hat{f}(s)={\frac{1}{\sqrt{2\pi}}}\int_{-\infty}^{\infty} f(t) e^{- i st} d t$</a>
<a href="#μετασχηματισμός-ισοδύναμος-που-αναφέρει-ο-τραχανάς-hatfsint_-inftyinfty-ft-e-i-st-d-t" class="toc-item toc-h3" style="padding-left:36px">Μετασχηματισμός ισοδύναμος που αναφέρει ο Τραχανάς: $\hat{f}(s)=\int_{-\infty}^{\infty} f(t) e^{ i st} d t$</a>
<a href="#μετασχηματισμοί-laplace" class="toc-item toc-h2" style="padding-left:22px">Μετασχηματισμοί Laplace</a>
</nav>
<div id="toc-overlay"></div>
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<div id="notebook">
<div class="cell cout cout-md"><h1 id="μετασχηματισμοί">Μετασχηματισμοί</h1></div>
<div class="cell cout cout-md"><h2 id="μετασχηματισμοί-fourier">Μετασχηματισμοί Fourier</h2></div>
<div class="cell cin code"><pre><code>Clear["Global`*"]
p[x_] := Which[-1/2 < x < 1/2, 1, Abs[x] > 1/2, 0]
Plot[p[x], {x, -1, 1}]</code></pre></div>
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</svg></div>
<div class="cell cout cout-md"><h3 id="μετασχηματισμός-με-φυσική-ερμηνεία-hatfsint_-inftyinfty-ft-e-2pi-i-st-d-t">Μετασχηματισμός με φυσική ερμηνεία: $\hat{f}(s)=\int_{-\infty}^{\infty} f(t) e^{-2\pi i st} d t$</h3></div>
<div class="cell cin code"><pre><code>FourierTransform[1, x, s, FourierParameters -> {0, -2 Pi}]
FourierTransform[p[x], x, s, FourierParameters -> {0, -2 Pi}]
InverseFourierTransform[Sin[π s]/(π s), s, x, FourierParameters -> {0, -2 Pi}] // Simplify</code></pre></div>
<div class="cell cout"><div class="math-output"><span class="math-inline">\(DiracDelta[s]\)</span></div></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[\frac{\sin(\pi s)}{\pi s}\]</div></div></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[(\frac{1}{2}) (Sign[\frac{1}{2}-x]+Sign[\frac{1}{2}+x])\]</div></div></div>
<div class="cell cout cout-md"><h3 id="μετασχηματισμός-τραχανά-μδε-hatfsfrac1sqrt2piint_-inftyinfty-ft-e--i-st-d-t">Μετασχηματισμός Τραχανά (Μ.Δ.Ε.): $\hat{f}(s)={\frac{1}{\sqrt{2\pi}}}\int_{-\infty}^{\infty} f(t) e^{- i st} d t$</h3></div>
<div class="cell cin code"><pre><code>FourierTransform[p[x], x, s]
InverseFourierTransform[(Sqrt[2/π] Sin[s/2])/s, s, x] // Simplify
FourierTransform[p[x], x, s, FourierParameters -> {0, 1}]
InverseFourierTransform[(Sqrt[2/π] Sin[s/2])/s, s, x, FourierParameters -> {0, 1}] // Simplify</code></pre></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[\frac{(\sqrt{\frac{2}{\pi }}) \sin(\frac{s}{2})}{s}\]</div></div></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[(\frac{1}{2}) (Sign[\frac{1}{2}-x]+Sign[\frac{1}{2}+x])\]</div></div></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[\frac{(\sqrt{\frac{2}{\pi }}) \sin(\frac{s}{2})}{s}\]</div></div></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[(\frac{1}{2}) (Sign[\frac{1}{2}-x]+Sign[\frac{1}{2}+x])\]</div></div></div>
<div class="cell cout cout-md"><h3 id="μετασχηματισμός-ισοδύναμος-που-αναφέρει-ο-τραχανάς-hatfsint_-inftyinfty-ft-e-i-st-d-t">Μετασχηματισμός ισοδύναμος που αναφέρει ο Τραχανάς: $\hat{f}(s)=\int_{-\infty}^{\infty} f(t) e^{ i st} d t$</h3></div>
<div class="cell cin code"><pre><code>FourierTransform[p[x], x, s, FourierParameters -> {1, 1}]
InverseFourierTransform[(2 Sin[s/2])/s, s, x, FourierParameters -> {1, 1}] // Simplify</code></pre></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[\frac{2 \sin(\frac{s}{2})}{s}\]</div></div></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[(\frac{1}{2}) (Sign[\frac{1}{2}-x]+Sign[\frac{1}{2}+x])\]</div></div></div>
<div class="cell cout cout-md"><h2 id="μετασχηματισμοί-laplace">Μετασχηματισμοί Laplace</h2></div>
<div class="cell cin code"><pre><code>Clear["Global`*"]
f[x_] := x^2 Exp[x - a]
LaplaceTransform[f[x], x, s]
(* το f[s_] := LaplaceTransform[f[x],x,s] είναι πάρα πολύ ΑΡΓΟ *)
fL[t_] := Evaluate[%]</code></pre></div>
<div class="cell cout"><div class="math-output"><div class="math-display">\[\frac{2 (E^{-a})}{{(-1+s)}^{3}}\]</div></div></div>
<div class="cell cin code"><pre><code>InverseLaplaceTransform[(2 E^-a)/(-1 + s)^3, s, x]</code></pre></div>
<div class="cell cout"><div class="math-output"><span class="math-inline">\((E^{-a+x}) (x^{2})\)</span></div></div>
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