# NevilleThetaN

NevilleThetaN[z,m]

gives the Neville theta function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• NevilleThetaN[z,m] is a meromorphic function of and has a complicated branch cut structure in the complex plane.
• For certain special arguments, NevilleThetaN automatically evaluates to exact values.
• NevilleThetaN can be evaluated to arbitrary numerical precision.
• NevilleThetaN 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(4)

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:

### Specific Values(3)

Values at corners of the fundamental cell:

NevilleThetaN for special values of elliptic parameter:

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

### Visualization(3)

Plot the NevilleThetaN functions for various values of the parameter:

Plot NevilleThetaN as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(13)

The real domain of NevilleThetaN:

The complex domain of NevilleThetaN:

Function range of :

Function range of :

NevilleThetaN is an even function:

NevilleThetaN vanishes at :

is an analytic function of for :

is neither non-decreasing nor non-increasing:

is not injective:

is not surjective:

is non-negative for noninteger m:

has no singularities or discontinuities for noninteger m:

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

### Differentiation(2)

The firstorder 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)

NevilleThetaN can be applied to power series:

## Applications(5)

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 arc length and classical parametrizations:

## Properties & Relations(5)

Basic simplifications are automatically carried out:

All Neville theta functions are a multiple of shifted NevilleThetaN:

Use FullSimplify for expressions containing Neville theta functions:

Numerically find a root of a transcendental equation:

NevilleThetaN can be represented with related elliptic functions:

## Possible Issues(1)

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

Wolfram Research (1996), NevilleThetaN, Wolfram Language function, https://reference.wolfram.com/language/ref/NevilleThetaN.html.

#### Text

Wolfram Research (1996), NevilleThetaN, Wolfram Language function, https://reference.wolfram.com/language/ref/NevilleThetaN.html.

#### CMS

Wolfram Language. 1996. "NevilleThetaN." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NevilleThetaN.html.

#### APA

Wolfram Language. (1996). NevilleThetaN. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NevilleThetaN.html

#### BibTeX

@misc{reference.wolfram_2023_nevillethetan, author="Wolfram Research", title="{NevilleThetaN}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NevilleThetaN.html}", note=[Accessed: 27-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_nevillethetan, organization={Wolfram Research}, title={NevilleThetaN}, year={1996}, url={https://reference.wolfram.com/language/ref/NevilleThetaN.html}, note=[Accessed: 27-September-2023 ]}