# HeunDPrime

HeunDPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunD function.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• HeunDPrime belongs to the Heun class of functions.
• For certain special arguments, HeunDPrime automatically evaluates to exact values.
• HeunDPrime can be evaluated for arbitrary complex parameters.
• HeunDPrime can be evaluated to arbitrary numerical precision.
• HeunDPrime automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot HeunDPrime:

Series expansion of HeunDPrime:

## Scope(23)

### Numerical Evaluation(8)

Evaluate to high precision:

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

HeunDPrime can take one or more complex number parameters:

HeunDPrime can take complex number arguments:

Finally, HeunDPrime can take all complex number input:

Evaluate HeunDPrime efficiently at high precision:

Lists and matrices:

Evaluate HeunDPrime for points on the real negative axis, bypassing irregular singular origin:

### Specific Values(2)

Value of HeunDPrime at :

Value of HeunDPrime at origin is undetermined:

### Visualization(5)

Plot the HeunDPrime function:

Plot the absolute value of the HeunDPrime function for complex parameters:

Plot HeunDPrime as a function of its second parameter :

Plot HeunDPrime as a function of and :

Plot the family of HeunDPrime functions for different accessory parameter :

### Differentiation(1)

The derivatives of HeunDPrime are calculated using the HeunD function:

### Integration(3)

Integral of HeunDPrime gives back HeunD:

Definite numerical integral of HeunDPrime:

More integrals with HeunDPrime:

### Series Expansions(4)

Taylor expansion for HeunDPrime at point :

Coefficient of the second term in the series expansion of HeunDPrime at :

Plots of the first three approximations for HeunDPrime around :

Series expansion for HeunDPrime at any ordinary complex point:

## Applications(1)

Use the HeunDPrime function to calculate the derivatives of HeunD:

## Properties & Relations(3)

HeunDPrime is analytic at the point :

Origin is a singular point of the HeunDPrime function:

Except for this singular point, HeunDPrime can be calculated at any finite complex :

HeunDPrime is the derivative of HeunD:

## Possible Issues(1)

HeunDPrime diverges for big arguments: