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The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly ...

F_x[cos(2pik_0x)](k) = int_(-infty)^inftye^(-2piikx)((e^(2piik_0x)+e^(-2piik_0x))/2)dx (1) = 1/2int_(-infty)^infty[e^(-2pii(k-k_0)x)+e^(-2pii(k+k_0)x)]dx (2) = ...

F_x[sin(2pik_0x)](k) = int_(-infty)^inftye^(-2piikx)((e^(2piik_0x)-e^(-2piik_0x))/(2i))dx (1) = 1/2iint_(-infty)^infty[-e^(-2pii(k-k_0)x)+e^(-2pii(k+k_0)x)]dx (2) = ...

The Hartley Transform is an integral transform which shares some features with the Fourier transform, but which, in the most common convention, multiplies the integral kernel ...

The Hankel transform (of order zero) is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel and also called the ...

Simplemindedly, a number theoretic transform is a generalization of a fast Fourier transform obtained by replacing e^(-2piik/N) with an nth primitive root of unity. This ...

The Gelfand transform x|->x^^ is defined as follows. If phi:B->C is linear and multiplicative in the senses phi(ax+by)=aphi(x)+bphi(y) and phi(xy)=phi(x)phi(y), where B is a ...

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of ...

A general integral transform is defined by g(alpha)=int_a^bf(t)K(alpha,t)dt, where K(alpha,t) is called the integral kernel of the transform.

The so-called generalized Fourier integral is a pair of integrals--a "lower Fourier integral" and an "upper Fourier integral"--which allow certain complex-valued functions f ...