Fourier Integral Theorem, Fourier Sine/Cosine Transforms & Inversion Theorems

Explanation

Fourier series requires periodicity. The Fourier transform extends harmonic analysis to non-periodic transient functions on (-∞, ∞) by taking the asymptotic limit of Fourier series as the fundamental period approaches infinity. In engineering, it transforms differential equations in the time domain into algebraic relationships in the continuous frequency domain.

Key points

• Fourier Cosine Integral: f(x) = (2/π) ∫_0^∞ cos(ωx) [ ∫_0^∞ f(t) cos(ωt) dt ] dω.
• Fourier Sine Integral: f(x) = (2/π) ∫_0^∞ sin(ωx) [ ∫_0^∞ f(t) sin(ωt) dt ] dω.
• Fourier Transform F{f(t)} = F(s) = (1/√(2π)) ∫_{-∞}^∞ f(t) e^{-ist} dt.
• Inverse Fourier Transform: f(t) = (1/√(2π)) ∫_{-∞}^∞ F(s) e^{ist} ds.
• Fourier Sine Transform: F_s{f(t)} = F_s(s) = √(2/π) ∫_0^∞ f(t) sin(st) dt; Inverse: f(t) = √(2/π) ∫_0^∞ F_s(s) sin(st) ds.
• Fourier Cosine Transform: F_c{f(t)} = F_c(s) = √(2/π) ∫_0^∞ f(t) cos(st) dt; Inverse: f(t) = √(2/π) ∫_0^∞ F_c(s) cos(st) ds.
• Properties: Linearity, Shifting F{f(t - a)} = e^{-isa} F(s), Modulation F{f(t) cos at} = [F(s - a) + F(s + a)] / 2, Derivative F{f'(t)} = is F(s).

Example

Code examples

// Fourier Sine Transform & Integral Evaluation:
// Find Fourier Sine Transform of f(t) = e^{-at} (a > 0)
// Step 1: Definition:
//    Fs(s) = √(2/π) ∫_0^∞ e^{-at} sin(st) dt
// Step 2: Use standard integration formula ∫ e^{-at} sin(st) dt = e^{-at}(-a sin st - s cos st) / (a^2 + s^2):
//    Fs(s) = √(2/π) [ e^{-at} (-a sin st - s cos st) / (a^2 + s^2) ]_0^∞
//          = √(2/π) [ 0 - (-s / (a^2 + s^2)) ] = √(2/π) * [ s / (a^2 + s^2) ]
//
// Step 3: By Fourier Sine Inversion Formula:
//    f(t) = √(2/π) ∫_0^∞ Fs(s) sin(st) ds
//    e^{-at} = √(2/π) ∫_0^∞ [ √(2/π) * (s / (a^2 + s^2)) ] sin(st) ds
//    e^{-at} = (2/π) ∫_0^∞ [ (s sin st) / (a^2 + s^2) ] ds
//
// Step 4: Rearranging gives the famous Definite Integral:
//    ∫_0^∞ [ (s sin st) / (s^2 + a^2) ] ds = (π / 2) e^{-at}  (for a > 0, t > 0)

Exam questions

• [8 Marks] Find the Fourier transform of f(x) = 1 - x² for |x| ≤ 1 and f(x) = 0 for |x| > 1. Hence evaluate ∫_0^∞ [(x cos x - sin x) / x³] cos(x/2) dx.
• [8 Marks] Using Fourier integral theorem, show that ∫_0^∞ [cos(ωx) / (1 + ω²)] dω = (π / 2) e^{-x} for x ≥ 0.
• [7 Marks] Find the Fourier cosine transform of e^{-x²}.