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The amazing identity for all theta, where Gamma(z) is the gamma function. Equating coefficients of theta^0, theta^4, and theta^8 gives some amazing identities for the ...

A generalized hypergeometric function _pF_q(a_1,...,a_p;b_1,...,b_q;x) is a function which can be defined in the form of a hypergeometric series, i.e., a series for which the ...

Given a random variable X with continuous and strictly monotonic probability density function f(X), a quantile function Q_f assigns to each probability p attained by f the ...

A function f(x) is said to be antiperiodic with antiperiod p if -f(x)=f(x+np) for n=1, 3, .... For example, the sine function sinx is antiperiodic with period pi (as well as ...

The interesting function defined by the definite integral G(x)=int_0^xsin(tsint)dt, illustrated above (Glasser 1990). The integral cannot be done in closed form, but has a ...

The integral transform (Kf)(x)=int_0^infty((x-t)_+^(c-1))/(Gamma(c))_2F_1(a,b;c;1-t/x)f(t)dt, where Gamma(x) is the gamma function, _2F_1(a,b;c;z) is a hypergeometric ...

Any real function u(x,y) with continuous second partial derivatives which satisfies Laplace's equation, del ^2u(x,y)=0, (1) is called a harmonic function. Harmonic functions ...

A completely multiplicative function, sometimes known as linear or totally multiplicative function, is an arithmetic function f(n) such that f(mn)=f(m)f(n) holds for each ...

A symmetric function on n variables x_1, ..., x_n is a function that is unchanged by any permutation of its variables. In most contexts, the term "symmetric function" refers ...

The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathological continuous function which is nowhere differentiable. Its name ...