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 z and has a complicated branch cut structure in the complex m 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  (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  (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 TemplateBox[{{x, +, iy}, {1, /, 2}}, NevilleThetaC]:

Plot the imaginary part of TemplateBox[{{x, +, iy}, {1, /, 2}}, NevilleThetaC]:

Function Properties  (12)

The real domain of NevilleThetaC:

The complex domain of NevilleThetaC:

Approximate function range of TemplateBox[{x, 0}, NevilleThetaC]:

Approximate function range of TemplateBox[{x, 1}, NevilleThetaC]:

NevilleThetaC is an even function:

NevilleThetaC threads elementwise over lists:

TemplateBox[{x, m}, NevilleThetaC] is an analytic function of for :

TemplateBox[{x, {1, /, 3}}, NevilleThetaC] is neither non-decreasing nor non-increasing:

TemplateBox[{x, {1, /, 3}}, NevilleThetaC] is not injective:

TemplateBox[{x, {1, /, 3}}, NevilleThetaC] is not surjective:

TemplateBox[{x, m}, NevilleThetaC] is neither non-negative nor non-positive, except for :

TemplateBox[{x, m}, NevilleThetaC] has no singularities or discontinuities except for :

TemplateBox[{x, m}, NevilleThetaC] is affine only for and otherwise it is neither convex nor concave:

Format NevilleThetaC in TraditionalForm:

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  (3)

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:

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_2022_nevillethetac, author="Wolfram Research", title="{NevilleThetaC}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NevilleThetaC.html}", note=[Accessed: 11-August-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_nevillethetac, organization={Wolfram Research}, title={NevilleThetaC}, year={1996}, url={https://reference.wolfram.com/language/ref/NevilleThetaC.html}, note=[Accessed: 11-August-2022 ]}