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

Plot the imaginary part of TemplateBox[{{x, +, {i,  , y}}, {1, /, 2}}, NevilleThetaD]:

Function Properties  (12)

The real domain of NevilleThetaD:

The complex domain of NevilleThetaD:

Function range of TemplateBox[{x, 0}, NevilleThetaD]:

Function range of TemplateBox[{x, 1}, NevilleThetaD]:

NevilleThetaD is an even function:

NevilleThetaD threads elementwise over lists:

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

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

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

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

TemplateBox[{x, {1, /, m}}, NevilleThetaD] is non-negative:

TemplateBox[{x, {1, /, m}}, NevilleThetaD] does not have either singularity or discontinuity for noninteger m:

TemplateBox[{x, m}, NevilleThetaD] 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  (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:

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

Text

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

BibTeX

@misc{reference.wolfram_2021_nevillethetad, author="Wolfram Research", title="{NevilleThetaD}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NevilleThetaD.html}", note=[Accessed: 23-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_nevillethetad, organization={Wolfram Research}, title={NevilleThetaD}, year={1996}, url={https://reference.wolfram.com/language/ref/NevilleThetaD.html}, note=[Accessed: 23-October-2021 ]}

CMS

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

APA

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