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# Jacobi elliptic functions

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance. [1]

## Abel elliptic functions

Abel elliptic functions are holomorphic functions of one complex variable and with two periods.

## Alfred Cardew Dixon

Sir Alfred Cardew Dixon, 1st Baronet Warford FRS (22 May 1865 – 4 May 1936) was an English mathematician.

## Alfred George Greenhill

Sir (Alfred) George Greenhill, F.R.S. (29 November 1847 in London – 10 February 1927 in London), was a British mathematician.

## Angular eccentricity

Angular eccentricity is one of many parameters which arise in the study of the ellipse or ellipsoid.

## Carlson symmetric form

In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced.

## Christoph Gudermann

Christoph Gudermann (March 25, 1798 &ndash; September 25, 1852) was a German mathematician noted for introducing the Gudermannian function and the concept of uniform convergence, and for being the teacher of Karl Weierstrass, who was greatly influenced by Gudermann's course on elliptic functions in 1839–1840, the first such course to be taught in any institute.

## Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

## Complex torus

In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles).

## Conic section

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

## Cross-ratio

In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Dihedral group of order 6

In mathematics, the smallest non-abelian group has 6 elements.

## Dixon's elliptic functions

In mathematics, Dixon's elliptic functions, are two doubly periodic meromorphic functions on the complex plane that have regular hexagons as repeating units: the plane can be tiled by regular hexagons in such a way that the restriction of the function to such a hexagon is simply a shift of its restriction to any of the other hexagons.

## Domain coloring

In mathematics, domain coloring or a color wheel graph is a technique for visualizing complex functions, which assigns a color to each point of the complex plane.

## Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.

## Elliptic filter

An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband.

## Elliptic function

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions.

## Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

## Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

## Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

## J-invariant

In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for defined on the upper half-plane of complex numbers.

James Whitbread Lee Glaisher FRS FRSE FRAS (5 November 1848, Lewisham – 7 December 1928, Cambridge), son of James Glaisher the meteorologist and Cecilia Glaisher the photographer, was a prolific English mathematician and astronomer.

## Lagrange inversion theorem

In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.

## Lambert series

In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resummed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n).

## Map projection

A map projection is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.

## Naum Akhiezer

Naum Ilyich Akhiezer (Нау́м Ильи́ч Ахие́зер; 6 March 1901 – 3 June 1980) was a Soviet mathematician of Jewish origin, known for his works in approximation theory and the theory of differential and integral operators.

## Neville theta functions

In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows: where: K(m) is the complete elliptic integral of the first kind, K'(m).

## Nome (mathematics)

In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by.

## Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.

## Paul Émile Appell

Paul Appell (27 September 1855 in Strasbourg – 24 October 1930 in Paris), also known as Paul Émile Appel, was a French mathematician and Rector of the University of Paris.

## Peirce quincuncial projection

The Peirce quincuncial projection is a conformal map projection developed by Charles Sanders Peirce in 1879.

## Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely.

## Pendulum (mathematics)

The mathematics of pendulums are in general quite complicated.

## Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.

In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).

## Quarter period

In mathematics, the quarter periods K(m) and iK &prime;(m) are special functions that appear in the theory of elliptic functions.

## Ramanujan theta function

In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties.

## Schwarz–Christoffel mapping

In complex analysis, a Schwarz–Christoffel mapping is a conformal transformation of the upper half-plane onto the interior of a simple polygon.

## TeX

TeX (see below), stylized within the system as TeX, is a typesetting system (or "formatting system") designed and mostly written by Donald Knuth and released in 1978.

## Theta function

In mathematics, theta functions are special functions of several complex variables.

## Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

## Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles.

## Weierstrass's elliptic functions

In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass.

## Zeros and poles

In mathematics, a zero of a function is a value such that.

## References

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