# 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(19)

### 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 zero:

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

Different NevilleThetaN types give different symbolic forms:

### 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(5)

The real domain of NevilleThetaN:

The complex domain of NevilleThetaN:

Function range of :

Function range of :

NevilleThetaN is an even function:

### Differentiation(2)

The first derivative:

Higher derivatives:

Plot the higher derivatives:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The Taylor expansion at a generic point:

## 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(3)

Basic simplifications are automatically carried out:

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:

Introduced in 1996
(3.0)