# Fourier Transforms

This page contains Theory, Formulae, Problems and Solutions in respective order. Subtopics for the kind of problems are listed when needed.

• Fourier Transformation
• Fourier Integral
• Parseval’s Identity

Fourier Transformation

Problem 1) Find the Fourier Transform of f(x)=1 for  |x|<1, f(x)=0 for |x|>1 and Hence, evaluate $\int^\infty_0 \frac{\sin{x}}{x} \,dx = \pi$

Problem 2 ) Find Fourier Transform of f(x)= 1-x^2 for |x|<=1, =0 for |x|>=1. Hence, evaluate $\int^\infty_0 \frac{ x\cos{x}-\sin{x}}{x^3}\cos{\frac{\pi}{2}} dx$

Problem 3 ) Find the Fourier Transform of $\exp{-a^{2}x^{2}}$ ,a>0. Hence, deduce that $\exp{\frac{-x^{2}}{2}}$ is self-reciprocal in respect of Fourier Transform. b) Find the Fourier transform of (i) $\exp{-2(x-3)^{2}}$  (ii) $\exp{x^{2}}\cos{3x}$

Solution:

Problem 4 ) Find the Cosine Fourier Transform of $\exp{-x^{2}}$

Problem 5 ) Find Fourier Sine Transform of $\exp{-|x|}$. Hence, show that $\int^\infty_0 \frac{x\sin{mx}}{1+x^{2}} \,dx = \frac{\pi\exp{-m}}{2}$ ,m>0.

Solution:

Problem 6 ) Find the Cosine Transform of f(x)=x, for 0<x<1 f(x)=2-x, for 1<x<2 f(x)=0, for x>2.

Solution:

Problem 7 ) Find the Fourier Sine Transform of $\exp{\frac{-ax}{x}}$.

Solution:

Problem 8 ) Find the Fourier Cosine Transform of f(x) = $\frac{1}{1+x^{2}}$. Hence, derive Fourier Sine Transform of $\phi(x)= \frac{x}{1+x^{2}}$

Problem 9 )  Find Fourier Cosine Transform and Fourier Sine Transform of $x^{n-1}$, n>0. $F_{s}(x^{n-1}) = \int^\infty_0 x^{n-1}\sin{sx} \,dx$ $F_{c}( x^{n-1}) = \int^\infty_0 x^{n-1}\cos{sx} \,dx$

Problem 10 )  Show that $F_{s}[xf(x)] = -\frac{d}{ds} [F_{c}(s)]$ ; $F_{c}[xf(x)] = \frac{d}{ds}[F_{s}(s)]$ b) Find Fourier Cosine Transform and Fourier Sine Transform of $x\exp{-ax}$

Solutions:

Problem 11 ) If Fourier Sine Transform of $f(x) = \frac{(1-\cos{n\pi})}{n^{2}\pi^{2}}$ (0<x<1). Find f(x).

FOURIER INTEGRAL

Problem 12 )  Express as Fourier Integral f(x) = 1 for |x|<1, f(x)  = 0 for |x|>1. Hence, evaluate $\int^\infty_0 \frac{\sin{\lambda}\cos{\lambda x}}{\lambda} \,d\lambda$

Solution:

Problem 13 ) Prove that $\int^\infty_0 \frac{\sin{\pi\lambda}\sin{\lambda}}{1-\lambda^{2}}\,d\lambda$ = $\frac{\pi}{2}\sin{x}$ for 0<x<$\pi$ and is equal to 0 otherwise.

Solution:

Problem 14 ) Find Fourier Transform of f(x) = 1 for |x|<a and f(x) = 0 for |x|>a. Hence, show $F[ f(x)[1+\frac{\cos({\pi}x)}{a}] = \frac{2\pi^{2}\sin{as}}{s(\pi^{2}-a^{2}s)}$

Problem 15 )  Find Fourier Transform of $f(x) = a^{2}-x^{2}$ for |x|<a and is equal to 0 otherwise.

Problem 16 ) Given $F(e^{-x^{2}}) = \sqrt{\pi}e^{\frac{s^{2}}{4}}$ Find Fourier Transform of i) $f(x) = e^{\frac{-x^{2}}{3}}$ ii) $e^{-4(x-3)^{2}}$

Solution:

Problem 17 ) Find Fourier Sine Transform of $\frac{1}{x(x^{2}+a^{2})}$

Solution:

Problem 18 ) Find Fourier Cosine Transform of $e^{-x^{2}}$ and hence, evaluate Fourier Sine Transform of $xe^{-x^{2}}$.

Problem 20 ) Obtain Fourier Sine Transform of i) $f(x) = \sin{x}$ for 0<x<a, and is equal to 0 otherwise. &   ii) f(x)= 4x, for 0<x<1 f(x)= 4-x, for 1<x<4 f(x)= 0, x>4.

PARSEVAL’S IDENTITY

Problem 21 ) Using Parseval’s Identity prove that, (i) $\int^\infty_0 \frac{1}{(a^{2}+t^{2})(b^{2}+t^{2})}\,dt = \frac{\pi}{2ab(a+b)}$ (ii) $\int^\infty_0 \frac{t^{2}}{(t^{2}+1)^{2}}\,dt = \frac{\pi}{4}$ (iii) $\int^\infty_0 \frac{\sin{t}}{t(a^{2}+t^{2})}\,dt = \frac{\pi(1-e^{-a^{2}})}{2a^{2}}$

Problem 22 ) If f(x) = 1, |x|<a and is equal to 0 otherwise, also $F(s) = \frac{2\sin{sa}}{s} (s\neq{0})$ Using this prove that $\int^\infty_0 \frac{(\sin{(ax)})^{2}}{x^{2}}\,dx = \frac{\pi{a}}{2}$ Hint: Use Parseval’s Identity.

Problem 23 ) Using Parseval’s identity show that (i) $\int^\infty_0 \frac{1}{(t^{2}+1)^{2}}\,dt = \frac{\pi}{4}$ (ii) $\int^\infty_0 \frac{t^{2}}{(4+t^{2})(9+t^{2})}\,dt=\frac{\pi}{10}$

Problem 24 )  Evaluate $\int^\infty_0 [\frac{1-\cos(x)}{x}]^{2} \,dx$ and $\int^\infty_0 [\frac{\sin(x)}{x}]^{2}\,dx$

Problem 25 ) Using information that $\int^\infty_0 [\frac{\sin{x}}{x}]^{2}e^{isx}\,dx = \pi(1-|s|) ,|s|<1$ and is equal to 0 otherwise. Find $\int^\infty_{-\infty} [\frac{sin(x)}{x}]^{4}\,dx = \frac{\pi}{3}$.