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Let n be an integer variable which tends to infinity and let x be a continuous variable tending to some limit. Also, let phi(n) or phi(x) be a positive function and f(n) or ...

The backward difference is a finite difference defined by del _p=del f_p=f_p-f_(p-1). (1) Higher order differences are obtained by repeated operations of the backward ...

Bailey's transformation is the very general hypergeometric transformation (1) where k=1+2a-b-c-d, and the parameters are subject to the restriction b+c+d+e+f+g-m=2+3a (2) ...

(dy)/(dx)+p(x)y=q(x)y^n. (1) Let v=y^(1-n) for n!=1. Then (dv)/(dx)=(1-n)y^(-n)(dy)/(dx). (2) Rewriting (1) gives y^(-n)(dy)/(dx) = q(x)-p(x)y^(1-n) (3) = q(x)-vp(x). (4) ...

In order to find a root of a polynomial equation a_0x^n+a_1x^(n-1)+...+a_n=0, (1) consider the difference equation a_0y(t+n)+a_1y(t+n-1)+...+a_ny(t)=0, (2) which is known to ...

A black bishop graph is a graph formed from possible moves of a bishop chess piece, which may make diagonal moves of any length on a chessboard (or any other board), when ...

If a continuous function defined on an interval is sometimes positive and sometimes negative, it must be 0 at some point. Bolzano (1817) proved the theorem (which effectively ...

The term Borel hierarchy is used to describe a collection of subsets of R defined inductively as follows: Level one consists of all open and closed subsets of R, and upon ...

For a general quadrilateral with sides of length a, b, c, and d, the area K is given by (1) where s=1/2(a+b+c+d) (2) is the semiperimeter, A is the angle between a and d, and ...

Using a Tschirnhausen transformation, the principal quintic form can be transformed to the one-parameter form w^5-10cw^3+45c^2w-c^2=0 (1) named after Francesco Brioschi ...