# NevilleThetaS

NevilleThetaS[z,m]

gives the Neville theta function .

# 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

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## Basic Examples(3)

Evaluate numerically:

Plot over a subset of the reals::

Plot over a subset of the complexes:

## Scope(27)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix NevilleThetaS function using MatrixFunction:

### Specific Values(3)

Values at corners of the fundamental cell:

NevilleThetaS for special values of elliptic parameter:

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

### Visualization(3)

Plot the NevilleThetaS functions for various values of the parameter:

Plot NevilleThetaS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(11)

The real domain of NevilleThetaS:

The complex domain of NevilleThetaS:

Function range of :

Function range of :

is an analytic function of for :

is neither non-decreasing nor non-increasing:

is not injective:

is not surjective:

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

has no singularities or discontinuities for noninteger :

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

### Differentiation(2)

The first-order derivatives:

Higher-order derivatives:

Plot the higher-order derivatives:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The Taylor expansion for small elliptic parameter :

The Taylor expansion around :

## Generalizations & Extensions(1)

NevilleThetaS can be applied to power series:

## Applications(7)

Plot over the arguments' plane:

Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

Uniformization of a Fermat cubic :

Plot the curve for real :

Verify that points on the curve satisfy :

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

Plot the flow lines:

Parametrize a lemniscate by arc length:

Show the classical and arc length parametrizations:

Complex parametrization of a sphere:

The square of all points on the complex sphere is 1:

Conformal map from an ellipse to the unit disk:

Visualize the map:

## Properties & Relations(4)

Basic simplifications are automatically carried out:

All Neville theta functions are a multiple of shifted NevilleThetaS:

Use FullSimplify for expressions containing Neville theta functions:

Numerically find a root of a transcendental equation:

## Possible Issues(1)

Machine-precision input is insufficient to give a correct answer:

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.

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2024_nevillethetas, author="Wolfram Research", title="{NevilleThetaS}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NevilleThetaS.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_nevillethetas, organization={Wolfram Research}, title={NevilleThetaS}, year={1996}, url={https://reference.wolfram.com/language/ref/NevilleThetaS.html}, note=[Accessed: 12-September-2024 ]}