NevilleThetaC

NevilleThetaC[z,m]

gives the Neville theta function .

Details

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

Examples

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Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals::

Plot over a subset of the complexes:

Series expansion at the origin:

Scope(29)

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 NevilleThetaC function using MatrixFunction:

Specific Values(4)

Values at corners of the fundamental cell:

NevilleThetaC for special values of elliptic parameter:

Find the first positive maximum of NevilleThetaC[x,1/4]:

Different NevilleThetaC types give different symbolic forms:

Visualization(3)

Plot the NevilleThetaC functions for various values of the parameter:

Plot NevilleThetaC as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties(12)

The real domain of NevilleThetaC:

The complex domain of NevilleThetaC:

Approximate function range of :

Approximate function range of :

NevilleThetaC is an even function:

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, except for :

has no singularities or discontinuities except for :

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 m:

The Taylor expansion around :

Generalizations & Extensions(1)

NevilleThetaC can be applied to a power series:

Applications(4)

Plot over the plane:

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:

Uniformization of a Fermat cubic :

Plot the curve for real :

Verify that points on the curve satisfy :

Properties & Relations(3)

Basic simplifications are automatically carried out:

All Neville theta functions are a multiple of shifted NevilleThetaC:

Numerically find a root of a transcendental equation:

Possible Issues(1)

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

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_nevillethetac, author="Wolfram Research", title="{NevilleThetaC}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NevilleThetaC.html}", note=[Accessed: 14-September-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_nevillethetac, organization={Wolfram Research}, title={NevilleThetaC}, year={1996}, url={https://reference.wolfram.com/language/ref/NevilleThetaC.html}, note=[Accessed: 14-September-2024 ]}