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The function defined by the contour integral J_(n,k)(z) =1/(2pii)int^((0+))t^(-n-1)(t+1/t)^kexp[1/2z(t-1/t)]dt, where int_((0+)) denotes the contour encircling the point z=0 ...

A statistical distribution for which the variables may take on a continuous range of values. Abramowitz and Stegun (1972, p. 930) give a table of the parameters of most ...

The operator partial^_ is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative d takes a function and yields a one-form. It decomposes ...

A Fréchet space is a complete and metrizable space, sometimes also with the restriction that the space be locally convex. The topology of a Fréchet space is defined by a ...

An equation of the form f(x,y,...)=0, where f contains a finite number of independent variables, known functions, and unknown functions which are to be solved for. Many ...

The Hilbert transform (and its inverse) are the integral transform g(y) = H[f(x)]=1/piPVint_(-infty)^infty(f(x)dx)/(x-y) (1) f(x) = ...

Let sumu_k be a series with positive terms and let f(x) be the function that results when k is replaced by x in the formula for u_k. If f is decreasing and continuous for ...

Given a Taylor series f(x)=f(x_0)+(x-x_0)f^'(x_0)+((x-x_0)^2)/(2!)f^('')(x_0)+... +((x-x_0)^n)/(n!)f^((n))(x_0)+R_n, (1) the error R_n after n terms is given by ...

If f(z) is continuous in a region D and satisfies ∮_gammafdz=0 for all closed contours gamma in D, then f(z) is analytic in D. Morera's theorem does not require simple ...

Numerical differentiation is the process of finding the numerical value of a derivative of a given function at a given point. In general, numerical differentiation is more ...