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Integral transforms are mathematical operations that convert functions from one domain to another. They are widely used in various branches of mathematics and physics to simplify problems and analyze functions in different representations. The most common integral transforms include the Fourier transform, Laplace transform, and the Z-transform.
Fourier Transform: The Fourier transform converts a function of time into a function of frequency. It is widely used in signal processing, image processing, and quantum mechanics. The Fourier transform of a function f(t) is given by:
F(w) = ∫[from -∞ to +∞] f(t) * e^(-iwt) dt
Here, F(w) is the Fourier transform of f(t), and e^(-iwt) is the complex exponential function.
Laplace Transform: The Laplace transform converts a function of time into a function of a complex variable s. It is particularly useful in solving linear ordinary differential equations. The Laplace transform of a function f(t) is given by:
F(s) = ∫[from 0 to +∞] f(t) * e^(-st) dt
Here, F(s) is the Laplace transform of f(t), s is a complex variable, and e^(-st) is the complex exponential function.
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Dr Ajay Kumar Sharma, Dr Vinod Kumar
Dr Ajay Kuamr Sharma Working as an Head,Department of Mathematics in M.G.College of Science Gadchandur(MH) .
