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  (20)

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)

NevilleThetaD for symbolic m:

Values at zero:

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

Different NevilleThetaD types give different symbolic forms:

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  (5)

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:

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)

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  (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)