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Q_n^((alpha,beta))(x)=2^(-n-1)(x-1)^(-alpha)(x+1)^(-beta) ×int_(-1)^1(1-t)^(n+alpha)(1+t)^(n+beta)(x-t)^(-n-1)dt. In the exceptional case n=0, alpha+beta+1=0, a nonconstant ...

A group of linear fractional transformations which transform the arguments of Kummer solutions to the hypergeometric differential equation into each other. Define A(z) = 1-z ...

Given a function of the form y=Ax^B, (1) least squares fitting gives the coefficients as b = ...

Let F be a differential field with constant field K. For f in F, suppose that the equation g^'=f (i.e., g=intf) has a solution g in G, where G is an elementary extension of F ...

A robust estimation based on maximum likelihood arguments.

The Mathieu functions are the solutions to the Mathieu differential equation (d^2V)/(dv^2)+[a-2qcos(2v)]V=0. (1) Even solutions are denoted C(a,q,v) and odd solutions by ...

A surface given by the parametric equations x(u,v) = u (1) y(u,v) = v (2) z(u,v) = au^4+u^2v-v^2. (3)

A line of constant longitude on a spheroid (or sphere). More generally, a meridian of a surface of revolution is the intersection of the surface with a plane containing the ...

A Monge patch is a patch x:U->R^3 of the form x(u,v)=(u,v,h(u,v)), (1) where U is an open set in R^2 and h:U->R is a differentiable function. The coefficients of the first ...

Two real algebraic manifolds are equivalent iff they are analytically homeomorphic (Nash 1952).