In mathematics, the Neville theta functions, named after
Eric Harold Neville
Eric Harold Neville, known as E. H. Neville (1 January 1889 London, England – 22 August 1961 Reading, Berkshire, England) was an English mathematician. A heavily fictionalised portrayal of his life is rendered in the 2007 novel ''The Indian ...
, are defined as follows:
:
:
:
:
where: ''K(m)'' is the complete
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
of the first kind,
, and
is the elliptic nome.
Note that the functions ''θ
p(z,m)'' are sometimes defined in terms of the nome ''q(m)'' and written ''θ
p(z,q)'' (e.g. NIST
). The functions may also be written in terms of the ''τ'' parameter ''θ
p(z, τ)'' where
.
Relationship to other functions
The Neville theta functions may be expressed in terms of the Jacobi theta functions
:
:
:
:
where
.
The Neville theta functions are related to the
Jacobi elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tri ...
. If pq(u,m) is a Jacobi elliptic function (p and q are one of s,c,n,d), then
:
Examples
*
*
*
*
Symmetry
*
*
*
*
Complex 3D plots
Implementation
NetvilleThetaC
,m NevilleThetaD
,m NevilleThetaN
,m and NevilleThetaS
,mare built-in functions of
Mathematica.
Notes
References
*
*
*{{MathWorld , urlname=NevilleThetaFunctions , title=Neville Theta Functions
Special functions
Theta functions
Elliptic functions
Analytic functions