Problems based on Fourier Transform [Complex Fourier Transform]

II. (a) Problems based on Fourier Transform [Complex Fourier Transform]

Formula :

Example 4.2.a(2): Find the Fourier Transform of

 

Example 4.2.a(3) Find the Fourier transform of f (x) given by

 

Example 4.2.a(4): Show that the Fourier Transform of e 

(OR)

Show that   is self-reciprocal with respect to Fourier Transform.

Solution :

 

Example 4.2.a(5): Find the Fourier transform of f (x) defined by

 

Example 4.2.a(6): Show that the Fourier transform of

Example 4.2.a(7): Find the (complex) Fourier transform of

Example 4.2.a(8): Find the Fourier transform of 

 

Example 4.2.a(10) : Find the Fourier transform of e^-a2x2, a > 0,

Hence, show that e is self reciprocal under Fourier transform.

 

Example 4.2.a(11): Find the Fourier transform of Dirac delta function δ (t- a).

II. (b) Problems based on Fourier transform and its max inversion formula

Formula :

 

Example 4.2.b(1): Find the Fourier transform of the function

Example 4.2.b(3): Find Fourier Transform of e ^{a- |x|} and hence deduce that

 

Example 4.2.b(4): Find the Fourier transform of e^{-a- |x|} and hence find the fourier transform of e^{- |x|} cos 2x

 

II. (c) Problems based on inversion formula, Parseval's identity and Convolution theorem

Formula :

 

Example 4.2.c(1): Find the Fourier transform of   where a is a positive real number.

 

Example 4.2.c(2): Find the Fourier Transform of

 

Example 4.2.c(3): Show that the Fourier transform of

 

 

Example 4.2.c(5): Find the Fourier transform of

Example 4.2.c(6): Find the Fourier transform of f(x) given by

 

 

Example 4.2.c(9) : Verify convolution theorem for f(x) = g(x)=e-^x2

Definition: Convolution theorem for Fourier transforms. The Fourier transform of the convolution of f (x) and g (x) is the product of their Fourier transforms.

F [f(x) *g(x)] = F {f(x)} F {g (x)}

Given: f(x) = g(x) = e ^x2