# 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(21)

### 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 :

Plot the imaginary part of :

### Function Properties(10)

Real and complex domains of EllipticThetaPrime:

EllipticThetaPrime is a periodic function with respect to : is an analytic function of x:

For example, has no singularities or discontinuities: is neither nondecreasing nor nonincreasing: is not injective: is not surjective: is neither non-negative nor non-positive: is neither convex nor concave:

### 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(4)

Verify Jacobi's triple product identity through a series expansion:

Green's function for the 1D heat equation with Dirichlet boundary conditions and initial condition :

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(4)

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:

EllipticThetaPrime has the attribute NHoldFirst:

Different argument conventions exist for theta functions:

## Neat Examples(1)

Visualize a function with a boundary of analyticity: