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If f(x) is an even function, then b_n=0 and the Fourier series collapses to f(x)=1/2a_0+sum_(n=1)^inftya_ncos(nx), (1) where a_0 = 1/piint_(-pi)^pif(x)dx (2) = ...

The Fourier cosine transform of a real function is the real part of the full complex Fourier transform, F_x^((c))[f(x)](k) = R[F_x[f(x)](k)] (1) = ...

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 ...

If f(x) is an odd function, then a_n=0 and the Fourier series collapses to f(x)=sum_(n=1)^inftyb_nsin(nx), (1) where b_n = 1/piint_(-pi)^pif(x)sin(nx)dx (2) = ...

A Fourier series in which there are large gaps between nonzero terms a_n or b_n.

Given a semicircular hump f(x) = sqrt(L^2-(x-L)^2) (1) = sqrt((2L-x)x), (2) the Fourier coefficients are a_0 = 1/2piL (3) a_n = ((-1)^nLJ_1(npi))/n (4) b_n = 0, (5) where ...

A generalized Fourier series is a series expansion of a function based on the special properties of a complete orthogonal system of functions. The prototypical example of ...

For a power function f(x)=x^k with k>=0 on the interval [0,2L] and periodic with period 2L, the coefficients of the Fourier series are given by a_0 = (2^(k+1)L^k)/(k+1) (1) ...

The cosine function cosx is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let theta be an ...

Consider a square wave f(x) of length 2L. Over the range [0,2L], this can be written as f(x)=2[H(x/L)-H(x/L-1)]-1, (1) where H(x) is the Heaviside step function. Since ...