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 function. 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 research. 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 Transforms. Know More About Real World Problems with Rational Numbers
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. Learn More Rational Expressions Word Problems
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