NevilleThetaS
✖
NevilleThetaS
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
-
- NevilleThetaS[z,m] is a meromorphic function of
and has a complicated branch cut structure in the complex
plane.
- For certain special arguments, NevilleThetaS automatically evaluates to exact values.
- NevilleThetaS can be evaluated to arbitrary numerical precision.
- NevilleThetaS automatically threads over lists.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Scope (27)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0e5cvs7fqvuevm-l274ju


https://wolfram.com/xid/0e5cvs7fqvuevm-wlv0g


https://wolfram.com/xid/0e5cvs7fqvuevm-b0wt9


https://wolfram.com/xid/0e5cvs7fqvuevm-hwtopb

The precision of the output tracks the precision of the input:

https://wolfram.com/xid/0e5cvs7fqvuevm-y7k4a


https://wolfram.com/xid/0e5cvs7fqvuevm-hfml09

Evaluate efficiently at high precision:

https://wolfram.com/xid/0e5cvs7fqvuevm-di5gcr


https://wolfram.com/xid/0e5cvs7fqvuevm-bq2c6r

Compute average-case statistical intervals using Around:

https://wolfram.com/xid/0e5cvs7fqvuevm-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0e5cvs7fqvuevm-thgd2

Or compute the matrix NevilleThetaS function using MatrixFunction:

https://wolfram.com/xid/0e5cvs7fqvuevm-o5jpo

Specific Values (3)
Values at corners of the fundamental cell:

https://wolfram.com/xid/0e5cvs7fqvuevm-srsfi

NevilleThetaS for special values of elliptic parameter:

https://wolfram.com/xid/0e5cvs7fqvuevm-gco8d


https://wolfram.com/xid/0e5cvs7fqvuevm-c4t14z

Find the first positive maximum of NevilleThetaS[x,1/2]:

https://wolfram.com/xid/0e5cvs7fqvuevm-f2hrld


https://wolfram.com/xid/0e5cvs7fqvuevm-fr3tb7

Visualization (3)
Plot the NevilleThetaS functions for various values of the parameter:

https://wolfram.com/xid/0e5cvs7fqvuevm-ecj8m7

Plot NevilleThetaS as a function of its parameter :

https://wolfram.com/xid/0e5cvs7fqvuevm-du62z6


https://wolfram.com/xid/0e5cvs7fqvuevm-ouu484


https://wolfram.com/xid/0e5cvs7fqvuevm-bzsmnc

Function Properties (11)
The real domain of NevilleThetaS:

https://wolfram.com/xid/0e5cvs7fqvuevm-cl7ele

The complex domain of NevilleThetaS:

https://wolfram.com/xid/0e5cvs7fqvuevm-de3irc


https://wolfram.com/xid/0e5cvs7fqvuevm-evf2yr


https://wolfram.com/xid/0e5cvs7fqvuevm-fphbrc

NevilleThetaS threads elementwise over lists:

https://wolfram.com/xid/0e5cvs7fqvuevm-71t8n

is an analytic function of
for
:

https://wolfram.com/xid/0e5cvs7fqvuevm-gva6yl

is neither non-decreasing nor non-increasing:

https://wolfram.com/xid/0e5cvs7fqvuevm-2ra8g


https://wolfram.com/xid/0e5cvs7fqvuevm-c9npzh


https://wolfram.com/xid/0e5cvs7fqvuevm-b5buvp


https://wolfram.com/xid/0e5cvs7fqvuevm-patce


https://wolfram.com/xid/0e5cvs7fqvuevm-bcrbvs

is neither non-negative nor non-positive for noninteger
:

https://wolfram.com/xid/0e5cvs7fqvuevm-dvzykj

has no singularities or discontinuities for noninteger
:

https://wolfram.com/xid/0e5cvs7fqvuevm-fyfbxx


https://wolfram.com/xid/0e5cvs7fqvuevm-5vh4e

is affine only for
and otherwise it is neither convex nor concave:

https://wolfram.com/xid/0e5cvs7fqvuevm-l0srvu

TraditionalForm formatting:

https://wolfram.com/xid/0e5cvs7fqvuevm-mw90mm

Differentiation (2)
Series Expansions (2)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0e5cvs7fqvuevm-ewr1h8

Plots of the first three approximations around :

https://wolfram.com/xid/0e5cvs7fqvuevm-binhar

The Taylor expansion for small elliptic parameter :

https://wolfram.com/xid/0e5cvs7fqvuevm-jwxla7


https://wolfram.com/xid/0e5cvs7fqvuevm-ot937

Generalizations & Extensions (1)Generalized and extended use cases
NevilleThetaS can be applied to power series:

https://wolfram.com/xid/0e5cvs7fqvuevm-dyr330

Applications (7)Sample problems that can be solved with this function
Plot over the arguments' plane:

https://wolfram.com/xid/0e5cvs7fqvuevm-1ida1

Conformal map from a unit triangle to the unit disk:

https://wolfram.com/xid/0e5cvs7fqvuevm-b4mjvn
Show points before and after the map:

https://wolfram.com/xid/0e5cvs7fqvuevm-d5shw5

https://wolfram.com/xid/0e5cvs7fqvuevm-bh08vt

Uniformization of a Fermat cubic :

https://wolfram.com/xid/0e5cvs7fqvuevm-ku82s

https://wolfram.com/xid/0e5cvs7fqvuevm-iwlvkn

Verify that points on the curve satisfy :

https://wolfram.com/xid/0e5cvs7fqvuevm-f37x35

Current flow in a rectangular conducting sheet with voltage applied at a pair of opposite corners:

https://wolfram.com/xid/0e5cvs7fqvuevm-gmmm

https://wolfram.com/xid/0e5cvs7fqvuevm-op1mqd

Parametrize a lemniscate by arc length:

https://wolfram.com/xid/0e5cvs7fqvuevm-b9pj2c
Show the classical and arc length parametrizations:

https://wolfram.com/xid/0e5cvs7fqvuevm-f0sed9

https://wolfram.com/xid/0e5cvs7fqvuevm-bgqae0

Complex parametrization of a sphere:

https://wolfram.com/xid/0e5cvs7fqvuevm-vre9j
The square of all points on the complex sphere is 1:

https://wolfram.com/xid/0e5cvs7fqvuevm-dszg0y


https://wolfram.com/xid/0e5cvs7fqvuevm-gprxeo

Conformal map from an ellipse to the unit disk:

https://wolfram.com/xid/0e5cvs7fqvuevm-f3w5wn

https://wolfram.com/xid/0e5cvs7fqvuevm-exvlr9

Properties & Relations (4)Properties of the function, and connections to other functions
Basic simplifications are automatically carried out:

https://wolfram.com/xid/0e5cvs7fqvuevm-ojczlk


https://wolfram.com/xid/0e5cvs7fqvuevm-by5pcs

All Neville theta functions are a multiple of shifted NevilleThetaS:

https://wolfram.com/xid/0e5cvs7fqvuevm-b1re76


https://wolfram.com/xid/0e5cvs7fqvuevm-dnppg4


https://wolfram.com/xid/0e5cvs7fqvuevm-l5x9z

Use FullSimplify for expressions containing Neville theta functions:

https://wolfram.com/xid/0e5cvs7fqvuevm-65dsy

Numerically find a root of a transcendental equation:

https://wolfram.com/xid/0e5cvs7fqvuevm-gst1er

Wolfram Research (1996), NevilleThetaS, Wolfram Language function, https://reference.wolfram.com/language/ref/NevilleThetaS.html.
Text
Wolfram Research (1996), NevilleThetaS, Wolfram Language function, https://reference.wolfram.com/language/ref/NevilleThetaS.html.
Wolfram Research (1996), NevilleThetaS, Wolfram Language function, https://reference.wolfram.com/language/ref/NevilleThetaS.html.
CMS
Wolfram Language. 1996. "NevilleThetaS." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NevilleThetaS.html.
Wolfram Language. 1996. "NevilleThetaS." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NevilleThetaS.html.
APA
Wolfram Language. (1996). NevilleThetaS. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NevilleThetaS.html
Wolfram Language. (1996). NevilleThetaS. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NevilleThetaS.html
BibTeX
@misc{reference.wolfram_2025_nevillethetas, author="Wolfram Research", title="{NevilleThetaS}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NevilleThetaS.html}", note=[Accessed: 01-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_nevillethetas, organization={Wolfram Research}, title={NevilleThetaS}, year={1996}, url={https://reference.wolfram.com/language/ref/NevilleThetaS.html}, note=[Accessed: 01-May-2025
]}