gives the Neville theta function .

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

# 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(26)

### 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:

NevilleThetaD for special values of elliptic parameter:

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

### Visualization(3)

Plot the NevilleThetaD functions for various values of the parameter:

Plot NevilleThetaD as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(12)

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 non-negative: does not have either singularity or discontinuity for noninteger m: 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)

NevilleThetaD 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 :

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:

Cartesian coordinates of a pendulum:

Plot the time dependence of the coordinates:

Plot the trajectory:

## Properties & Relations(4)

Basic simplifications are automatically carried out:

All Neville theta functions are a multiple of shifted NevilleThetaD:

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: