Physics, asked by sairam6805, 1 year ago

What is the relation between fourier and laplace transform?

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Answered by subhrajit74
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The Laplace transform evaluated at s=jωs=jω is equal to the Fourier transform if its region of convergence (ROC) contains the imaginary axis. This is also true for the bilateral (two-sided) Laplace transform, so the function need not be one-sided.

As for real and imaginary parts, since ss is a complex variable, both the Laplace and the Fourier transform generally have real and imaginary parts. Take as a simple example the function x(t)=e−atu(t)x(t)=e−atu(t), with a>0a>0, where u(t)u(t) is the unit step function. The Laplace transform is

XL(s)=1s+a(1)
(1)XL(s)=1s+a
Since a>0a>0, the ROC of XL(s)XL(s) contains the imaginary axis, and the Fourier transform of x(t)x(t) is simply obtained by evaluating XL(s)XL(s) on the imaginary axis s=jωs=jω:

XF(ω)=XL(jω)=1jω+a(2)
(2)XF(ω)=XL(jω)=1jω+a
Since s=σ+jωs=σ+jω is generally complex, not only the Fourier transform but also the Laplace transform (1)(1) has a real and an imaginary part:

XL(σ+jω)=1σ+jω+a=σ+a(σ+a)2+ω2−jω(σ+a)2+ω2(3)
(3)XL(σ+jω)=1σ+jω+a=σ+a(σ+a)2+ω2−jω(σ+a)2+ω2
Only when evaluated on the real axis s=σs=σ (ω=0ω=0) does the imaginary part vanish.

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