Fourier Transform Sine
To learn Fourier Transform Sine ,let’s have a look at Fourier transform of a
function. Transformation refers to the change of one function into another
This can be achieved by various transformation formulae defined by different
mathematicians. Following are few important transformation formulae: Laplace
transform Fourier transform Hankel transform Mellin transform.
These all transformations are widely used in the field of engineering and
As we have already discussed about Laplace transform of a function, now let’s
discuss about another important and interesting transformation i.e. Fourier sine
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To study Fourier sine transform lets first define Fourier transform of a function: If
k(s,x) = eisx, then the Fourier transform of a function f(x) can be defined by the
formula F(s) = -∞∫ -∞f(x)eisxdx.
This is an definite integral which has -∞ and +∞ as lower and upper limits.
Fourier transform is classified into two transforms i.e. Fourier sine transform and
Fourier cosine transform, Now let’s define,
Fourier sine transform: Fourier sine transform is also given by a Fourier sine
integral which can be defined as F(x)= 2/π∞∫ 0 [sinλx]dx ∞∫ 0 f(t) sinλtdtdλ, now
to obtain the Fourier transform sine, we have to perform some changes in the
above Fourier sine integral,
First of all replacing λ by s, we get f(x)= 2/π∞∫ 0 [sinsx]dx ∞∫ 0 f(t) sinstdtds
Now, in this integral denote the value of the inner integral i.e. ∞∫ 0 [sinsx]dx by
F(s) we have F(x)= 2/π∞∫ 0 F(s)sinsxdx And F(s)= ∞∫ 0 [sinsx]dx In the above
two expressions the function defined by f(s) is known as the Fourier transform
sine of the function f(x) in the interval 0<x>∞.
To solve the problems involving “Fourier transform sine” we should have good
knowledge of integral calculus.
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