Complex (or infinite) Fourier Transform

FOURIER TRANSFORM PAIR:

Fourier Transform: [Complex Fourier Transform]

Definition: The complex (or infinite) Fourier Transform

The complex (or infinite) Fourier Transform of f(x) is given by

Then the function f(x) is the inverse Fourier Transform of F(s) and is given by

The above (1) & (2) are jointly called Fourier transform pair.

OTHER FORMATS OF FOURIER TRANSFORM PAIR

Note: Whatever definitions or format we use, there will be a difference in constant factor while finding F(s) = F f(x)]. But this will be adjusted while expressing f(x) as a Fourier integral.

For example,  is equal to π/2, whatever definitions or format we use.

Fourier Transform [Complex Fourier Transform]

Problems based on Fourier Transform [Complex Fourier Transform]

Formula:

 F(s) = F f(x)]

Example 1: Find the Fourier Transform of

Solution: The given function can be written as

Example 2: Find the Fourier Transform of

Solution: The given function can be written as

Example 3: Find the Fourier transform of f(x) given by

Solution: The given function can be written as

Definition: Self reciprocal:

If a transformation of a function f(x) is equal to f(s) then the function f(x) is called self reciprocal.

Example 4: Show that the Fourier Transform of is .

(OR)

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

Solution:

is self reciprocal with respect to Fourier transform.

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

Solution: We know that, F[s] = F[f(x)]

Example 6: Show that the Fourier transform of

f(x) = |x| for |x| < a

= 0 for |x| > a, a > 0

is √(2/π) [ (sa sin sa + cos sa ‒ 1) /s² ]

Solution: Given function can be written as

f(x) = |x| for ‒a<x<a

 = 0 for |x| > a, a > 0

Example 7: Find the (complex) Fourier transform of

Solution:

Example 8: Find the Fourier transform of

Solution: We know that,

Example 9: Find the Fourier transform of 1/√|x|

Solution: We know that,

Example 10: Find the Fourier transform of , a > 0, Hence, show  that e is self reciprocal under Fourier transform.

Solution:

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

Solution: The Dirac delta function is defined as