WOLFRAM

gives the Neville theta function .

Details

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

Examples

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Basic Examples  (3)Summary of the most common use cases

Evaluate numerically:

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Plot over a subset of the reals::

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Plot over a subset of the complexes:

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Scope  (27)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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Evaluate to high precision:

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The precision of the output tracks the precision of the input:

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Complex number inputs:

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Evaluate efficiently at high precision:

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Compute average-case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix NevilleThetaS function using MatrixFunction:

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Specific Values  (3)

Values at corners of the fundamental cell:

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NevilleThetaS for special values of elliptic parameter:

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

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Visualization  (3)

Plot the NevilleThetaS functions for various values of the parameter:

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Plot NevilleThetaS as a function of its parameter :

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Plot the real part of TemplateBox[{z, {1, /, 2}}, NevilleThetaS]:

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Plot the imaginary part of TemplateBox[{z, {1, /, 2}}, NevilleThetaS]:

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Function Properties  (11)

The real domain of NevilleThetaS:

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The complex domain of NevilleThetaS:

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

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

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NevilleThetaS threads elementwise over lists:

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

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

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

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

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

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TemplateBox[{x, m}, NevilleThetaS] has no singularities or discontinuities for noninteger :

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

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TraditionalForm formatting:

Differentiation  (2)

The first-order derivatives:

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Higher-order derivatives:

Plot the higher-order derivatives:

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Series Expansions  (2)

Find the Taylor expansion using Series:

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Plots of the first three approximations around :

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The Taylor expansion for small elliptic parameter :

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The Taylor expansion around :

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Generalizations & Extensions  (1)Generalized and extended use cases

NevilleThetaS can be applied to power series:

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Applications  (7)Sample problems that can be solved with this function

Plot over the arguments' plane:

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Conformal map from a unit triangle to the unit disk:

Show points before and after the map:

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Uniformization of a Fermat cubic :

Plot the curve for real :

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Verify that points on the curve satisfy :

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

Plot the flow lines:

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Parametrize a lemniscate by arc length:

Show the classical and arc length parametrizations:

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Complex parametrization of a sphere:

The square of all points on the complex sphere is 1:

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Conformal map from an ellipse to the unit disk:

Visualize the map:

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Properties & Relations  (4)Properties of the function, and connections to other functions

Basic simplifications are automatically carried out:

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All Neville theta functions are a multiple of shifted NevilleThetaS:

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Use FullSimplify for expressions containing Neville theta functions:

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Numerically find a root of a transcendental equation:

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Possible Issues  (1)Common pitfalls and unexpected behavior

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

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

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2024_nevillethetas, author="Wolfram Research", title="{NevilleThetaS}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NevilleThetaS.html}", note=[Accessed: 10-January-2025 ]}

@misc{reference.wolfram_2024_nevillethetas, author="Wolfram Research", title="{NevilleThetaS}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/NevilleThetaS.html}", note=[Accessed: 10-January-2025 ]}

BibLaTeX

@online{reference.wolfram_2024_nevillethetas, organization={Wolfram Research}, title={NevilleThetaS}, year={1996}, url={https://reference.wolfram.com/language/ref/NevilleThetaS.html}, note=[Accessed: 10-January-2025 ]}

@online{reference.wolfram_2024_nevillethetas, organization={Wolfram Research}, title={NevilleThetaS}, year={1996}, url={https://reference.wolfram.com/language/ref/NevilleThetaS.html}, note=[Accessed: 10-January-2025 ]}