# Integral transform

In mathematics, an integral transform is any transform T of the following form:

[itex] (Tf)(u) = \int_{t_1}^{t_2} f(t)\, K(t, u)\, dt.[itex]

The input of this transform is a function f, and the output is another function Tf.

There are several useful integral transforms. Each transform corresponds to a different choice of the function K, which is called the kernel of the transform.

Table of Integral Transforms
TransformSymbolKernelt1t2
Fourier transform

[itex]\mathcal{F}[itex]

[itex]\frac{e^{iut}}{\sqrt{2 \pi}}[itex]

[itex]-\infty\,[itex][itex]\infty\,[itex]
Mellin transform

[itex]\mathcal{M}[itex]

[itex]t^{u-1}\,[itex]

[itex]0\,[itex][itex]\infty\,[itex]
Two-sided Laplace transform

[itex]\mathcal{B}[itex]

[itex]e^{-ut}\,[itex]

[itex]-\infty\,[itex][itex]\infty\,[itex]
Laplace transform

[itex]\mathcal{L}[itex]

[itex]e^{-ut}\,[itex]

[itex]0\,[itex][itex]\infty\,[itex]
Hankel transform

[itex]t\,J_\nu(ut)[itex]

[itex]0\,[itex][itex]\infty\,[itex]
Abel transform

[itex]\frac{t}{\sqrt{t^2-u^2}}[itex]

[itex]u\,[itex][itex]\infty\,[itex]
Hilbert transform

[itex]\mathcal{H}[itex]

[itex]\frac{1}{\pi}\frac{1}{u-t}[itex]

[itex]-\infty\,[itex][itex]\infty\,[itex]
Identity transform

[itex]\delta (u-t)\,[itex]

[itex]t_1[itex]t_2>u\,[itex]

Although the properties of integral transforms vary widely, they have some properties in common. For example, every integral transform is a linear operator, since the integral is a linear operator, and in fact if the kernel is allowed to be a generalized function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem).

## Bibliography

• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.pt:Transformada integral

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