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NevilleThetaD

NevilleThetaD[z,m]

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.
NevilleThetaD 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 (28)

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 NevilleThetaD function using MatrixFunction:

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)

The real domain of NevilleThetaD:

The complex domain of NevilleThetaD:

Function range of :

Function range of :

NevilleThetaD is an even function:

NevilleThetaD threads elementwise over lists:

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:

TraditionalForm formatting:

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 (8)

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:

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

Plot the flow lines:

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:

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