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Krall and Fink (1949) defined the Bessel polynomials as the function y_n(x) = sum_(k=0)^(n)((n+k)!)/((n-k)!k!)(x/2)^k (1) = sqrt(2/(pix))e^(1/x)K_(-n-1/2)(1/x), (2) where ...

The hyperbolic cosine integral, often called the "Chi function" for short, is defined by Chi(z)=gamma+lnz+int_0^z(cosht-1)/tdt, (1) where gamma is the Euler-Mascheroni ...

The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. The "simple" derivative of a function f with respect to ...

Poisson's equation is del ^2phi=4pirho, (1) where phi is often called a potential function and rho a density function, so the differential operator in this case is L^~=del ...

A reflection relation is a functional equation relating f(-x) to f(x), or more generally, f(a-x) to f(x). Perhaps the best known example of a reflection formula is the gamma ...

The sigmoid function, also called the sigmoidal curve (von Seggern 2007, p. 148) or logistic function, is the function y=1/(1+e^(-x)). (1) It has derivative (dy)/(dx) = ...

If f is continuous on a closed interval [a,b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that ...

The j-function is the modular function defined by j(tau)=1728J(tau), (1) where tau is the half-period ratio, I[tau]>0, ...

A necessary and sufficient condition that there should exist at least one nondecreasing function alpha(t) such that mu_n=int_(-infty)^inftyt^ndalpha(t) for n=0, 1, 2, ..., ...

A totative is a positive integer less than or equal to a number n which is also relatively prime to n, where 1 is counted as being relatively prime to all numbers. The number ...