Fourier Cosine Transform (FCT)

FOURIER COSINE TRANSFORM:

The infinite Fourier cosine transform of f(x) is defined by

The inverse Fourier cosine transform Fc [f(x)] is defined by

Formula:

Problems based on Fourier Cosine Transform

Example 1: Find the Fourier cosine transform of

Solution:

We know that,

Example 2: Find the Fourier cosine transform of e^-ax/x and hence, find .

Solution: We know that,

Example 3: Find the Fourier cosine transform of e-ax, a>0.

Solution:

We know that,

Example 4: Find the Fourier cosine transform of the function 3e^‒5x + 5e^‒2x.

Solution:

Let f(x)= 3e^‒5x + 5e^‒2x

We know that,

Example 5: Find the Fourier cosine transform of

Solution: We know that,

Example 6: Find the Fourier cosine transform of f(x) = x.

Solution: We know that,

Example 7: Find the Fourier cosine transform of e^‒ax cos ax.

Solution: We know that,

Example 8: Show that e^-x2/2 is self‒reciprocal under Fourier cosine transform.

Solution:

We know that,

Hence, f(x) = e^-x2/2 is self reciprocal with respect to Fourier cosine transform.

Example 9: Find the Fourier cosine transform of e^‒ax sin ax.

Solution: We know that,

Example 10: Evaluate F_C [x^n‒1] if 0 < x < 1.

Deduce that 1/√x is self reciprocal under Fourier cosine transform.

Solution:

Hence, 1/√x is self reciprocal under Fourier cosine transform.

Note: Equating imaginary part, we get

Example 11: Find the Fourier cosine transform of

Solution: We know that,

Example 12: Find the Fourier cosine transform of 1/[a²+x²]

Solution:

We know that,

Example 13: Find the Fourier cosine transform of .

Solution: We know that,