NevilleThetaN

NevilleThetaN[z,m]

gives the Neville theta function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • NevilleThetaN[z,m] is a meromorphic function of z and has a complicated branch cut structure in the complex m plane.
  • For certain special arguments, NevilleThetaN automatically evaluates to exact values.
  • NevilleThetaN can be evaluated to arbitrary numerical precision.
  • NevilleThetaN 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  (19)

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 zero:

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

Different NevilleThetaN types give different symbolic forms:

Visualization  (3)

Plot the NevilleThetaN functions for various values of the parameter:

Plot NevilleThetaN as a function of its parameter :

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

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

Function Properties  (5)

The real domain of NevilleThetaN:

The complex domain of NevilleThetaN:

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

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

NevilleThetaN is an even function:

NevilleThetaN threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (2)

The first derivative:

Higher derivatives:

Plot the higher derivatives:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The Taylor expansion at a generic point:

Generalizations & Extensions  (1)

NevilleThetaN can be applied to power series:

Applications  (5)

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 :

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 arc length and classical parametrizations:

Properties & Relations  (3)

Basic simplifications are automatically carried out:

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:

Introduced in 1996
 (3.0)