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 Solution

 

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

Solution :

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}}

Solution :

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}}

Solution :

  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

Solutions :

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).

Solution :                                            

                                          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)}

Solution :

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

Solution :

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}}.

Solution :

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.

Solution:  

                                                                                                     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}}

Solution :

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.

Solution:

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}

Solution :

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

Solution :

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}.

Solution :

 

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