Search Results for ""

The (complete) gamma function Gamma(n) is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by Gamma(n)=(n-1)!, ...

A tree is a mathematical structure that can be viewed as either a graph or as a data structure. The two views are equivalent, since a tree data structure contains not only a ...

The great success mathematicians had studying hypergeometric functions _pF_q(a_1,...,a_p;b_1,...,b_q;z) for the convergent cases (p<=q+1) prompted attempts to provide ...

The Cauchy remainder is a different form of the remainder term than the Lagrange remainder. The Cauchy remainder after n terms of the Taylor series for a function f(x) ...

Darboux's formula is a theorem on the expansion of functions in infinite series and essentially consists of integration by parts on a specific integrand product of functions. ...

Given a series of positive terms u_i and a sequence of positive constants {a_i}, use Kummer's test rho^'=lim_(n->infty)(a_n(u_n)/(u_(n+1))-a_(n+1)) (1) with a_n=n, giving ...

Let u_k be a series with positive terms and suppose rho=lim_(k->infty)(u_(k+1))/(u_k). Then 1. If rho<1, the series converges. 2. If rho>1 or rho=infty, the series diverges. ...

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

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep ...

The important binomial theorem states that sum_(k=0)^n(n; k)r^k=(1+r)^n. (1) Consider sums of powers of binomial coefficients a_n^((r)) = sum_(k=0)^(n)(n; k)^r (2) = ...