INTEGRAL TRANSFORMS - SOUL OF MATHEMATICS

In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the inverse transform.

An integral transform is any transform T of the following form:

{\displaystyle (Tf)(u)=\int _{t_{1}}^{t_{2}}f(t)\,K(t,u)\,dt}

Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in their original representations. An integral transform “maps” an equation from its original “domain” into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform.

Table of Transforms

Transform	Symbol	K	t₁	t₂
Abel transform		$\frac{2t}{\sqrt{t^2-u^2}}$	u	$\infty$
Fourier transform	${\mathcal {F}}$	$e^{-2\pi iut}$	$-\infty$	$\infty$
Fourier sine transform	$\mathcal{F}_s$	$\sqrt{\frac{2}{\pi}} \sin(ut)$	$0$	$\infty$
Fourier cosine transform	$\mathcal{F}_c$	$\sqrt{\frac{2}{\pi}} \cos(ut)$	0	$\infty$
Hankel transform		$t\,J_\nu(ut)$	0	$\infty$
Hartley transform	${\mathcal {H}}$	$\frac{\cos(ut)+\sin(ut)}{\sqrt{2 \pi}}$	$-\infty$	$\infty$
Hermite transform		$e^{-x^{2}}H_{n}(x)$	$-\infty$	$\infty$
Hilbert transform	$\mathcal{H}il$	$\frac{1}{\pi}\frac{1}{u-t}$	$-\infty$	$\infty$
Jacobi transform		$(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)$
Laguerre transform		$e^{-x}\ x^{\alpha }\ L_{n}^{\alpha }(x)$	$0$	$\infty$
Laplace transform	${\mathcal {L}}$	e^−ut	0	$\infty$
Legendre transform	${\mathcal {J}}$	$P_{n}(x)\,$
Mellin transform	${\mathcal {M}}$	t^u−1	0	$\infty$
Two-sided Laplace transform	${\mathcal {B}}$	e^−ut	$-\infty$	$\infty$
Poisson kernel		$\frac{1-r^2}{1-2r\cos\theta +r^2}$	0	2π
Radon Transform	Rƒ		$-\infty$	$\infty$
Weierstrass transform	${\mathcal {W}}$	$\frac{e^{-\frac{(u-t)^2}{4}}}{\sqrt{4\pi}}\,$	$-\infty$	$\infty$

We will get into details of few of the most popular transforms like Laplace Transform, Fourier Transform an many more.

Laplace Transform