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Given a square complex or real matrix A, a matrix norm ||A|| is a nonnegative number associated with A having the properties 1. ||A||>0 when A!=0 and ||A||=0 iff A=0, 2. ...

The complex numbers are the field C of numbers of the form x+iy, where x and y are real numbers and i is the imaginary unit equal to the square root of -1, sqrt(-1). When a ...

The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by (x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1, (1) where ...

Given a Jacobi theta function, the nome is defined as q(k) = e^(piitau) (1) = e^(-piK^'(k)/K(k)) (2) = e^(-piK(sqrt(1-k^2))/K(k)) (3) (Borwein and Borwein 1987, pp. 41, 109 ...

Let the elliptic modulus k satisfy 0<k^2<1. (This may also be written in terms of the parameter m=k^2 or modular angle alpha=sin^(-1)k.) The incomplete elliptic integral of ...

The elliptic lambda function lambda(tau) is a lambda-modular function defined on the upper half-plane by lambda(tau)=(theta_2^4(0,q))/(theta_3^4(0,q)), (1) where tau is the ...

Jacobi's imaginary transformations relate elliptic functions to other elliptic functions of the same type but having different arguments. In the case of the Jacobi elliptic ...

The polar coordinates r (the radial coordinate) and theta (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by x = ...

Let G be an open subset of the complex plane C, and let L_a^2(G) denote the collection of all analytic functions f:G->C whose complex modulus is square integrable with ...

A set of numbers a_0, a_1, ..., a_(m-1) (mod m) form a complete set of residues, also called a covering system, if they satisfy a_i=i (mod m) for i=0, 1, ..., m-1. For ...