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The two-argument Ramanujan function is defined by phi(a,n) = 1+2sum_(k=1)^(n)1/((ak)^3-ak) (1) = 1-1/a(H_(-1/a)+H_(1/a)+2H_n-H_(n-1/a)-H_(n+1/a)). (2) The one-argument ...

The convex hull of two or more functions is the largest function that is concave from above and does not exceed the given functions.

Given a function f(x)=f_0(x), write f_1=f^'(x) and define the Sturm functions by f_n(x)=-{f_(n-2)(x)-f_(n-1)(x)[(f_(n-2)(x))/(f_(n-1)(x))]}, (1) where [P(x)/Q(x)] is a ...

A "curve" (i.e., a continuous map of a one-dimensional interval) into a two-dimensional area (a plane-filling function) or a three-dimensional volume.

An anti-analytic function is a function f satisfying the condition (df)/(dz)=0. (1) Using the result ...

An asymmetrical apodization function defined by M(x,b,d)={0 for x<-b; (x-b)/(2b) for -b<x<b; 1 for b<x<b+2d; 0 for x<b+2d, (1) where the two-sided portion is 2b long (total) ...

The regularized gamma functions are defined by P(a,z) = (gamma(a,z))/(Gamma(a)) (1) Q(a,z) = (Gamma(a,z))/(Gamma(a)), (2) where gamma(a,z) and Gamma(a,z) are incomplete gamma ...

The sum of the aliquot divisors of n, given by s(n)=sigma(n)-n, where sigma(n) is the divisor function. The first few values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... ...

A function f(z) is said to be doubly periodic if it has two periods omega_1 and omega_2 whose ratio omega_2/omega_1 is not real. A doubly periodic function that is analytic ...

A Smarandache-like function which is defined where S_k(n) is defined as the smallest integer for which n|S_k(n)^k. The Smarandache S_k(n) function can therefore be obtained ...