EllipticThetaPrime

EllipticThetaPrime[a,u,q]

gives the derivative with respect to u of the theta function .

EllipticThetaPrime[a,q]

gives the theta constant .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For certain special arguments, EllipticThetaPrime automatically evaluates to exact values.
  • EllipticThetaPrime can be evaluated to arbitrary numerical precision.
  • EllipticThetaPrime automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin with respect to q:

Scope  (15)

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)

Value at zero:

EllipticThetaPrime evaluates symbolically for special arguments:

Find a value of for which EllipticThetaPrime[3,x,1/2]=2:

Visualization  (2)

Plot the EllipticThetaPrime function for various parameters:

Plot the real part of TemplateBox[{4, z, {1, /, 3}}, EllipticThetaPrime]:

Plot the imaginary part of TemplateBox[{4, z, {1, /, 3}}, EllipticThetaPrime]:

Function Properties  (4)

Real and complex domains of EllipticThetaPrime:

EllipticThetaPrime is a periodic function with respect to :

EllipticThetaPrime threads elementwise over lists:

TraditionalForm formatting:

Integration  (2)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Generalizations & Extensions  (1)

EllipticThetaPrime can be applied to power series:

Applications  (3)

Solution of the 1D heat equation with Dirichlet boundary condition and initial condition :

Calculate the temperature gradient:

Plot the temperature gradient:

Electrostatic force in a NaCllike crystal with pointlike ions:

Plot the magnitude of the force in a plane through the crystal:

The canonical rotational distribution function for linear molecules :

Plot a numerical approximation of the partition function:

Possible Issues  (3)

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

Use arbitrary-precision arithmetic to obtain the correct result:

The first argument must be an explicit integer between 1 and 4:

Different argument conventions exist:

Neat Examples  (1)

Visualize a function with a boundary of analyticity:

Introduced in 1996
 (3.0)
 |
Updated in 2017
 (11.1)
2017
 (11.2)