# HeunTPrime

HeunTPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunT function.

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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot HeunTPrime:

Series expansion of HeunTPrime:

## Scope(21)

### Numerical Evaluation(7)

Evaluate to high precision:

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

HeunTPrime can take one or more complex number parameters:

HeunTPrime can take complex number arguments:

Finally, HeunTPrime can take all complex number input:

Evaluate HeunTPrime efficiently at high precision:

Lists and matrices:

### Specific Values(1)

Value of HeunTPrime at origin:

### Visualization(5)

Plot the HeunTPrime function:

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

Plot HeunTPrime as a function of its second parameter :

Plot HeunTPrime as a function of and :

Plot the family of HeunTPrime functions for different values of the accessory parameter :

### Differentiation(1)

The derivatives of HeunTPrime are calculated using the HeunT function:

### Integration(3)

Integral of HeunTPrime gives back HeunT:

Definite numerical integral of HeunTPrime:

More integrals with HeunTPrime:

### Series Expansions(4)

Taylor expansion for HeunTPrime at origin:

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

Plots of the first three approximations for HeunTPrime around :

Series expansion for HeunTPrime at any ordinary complex point:

## Applications(1)

Use the HeunTPrime function to calculate the derivatives of HeunT:

## Properties & Relations(3)

HeunTPrime is analytic at the origin:

HeunTPrime can be calculated at any finite complex :

HeunTPrime is the derivative of HeunT:

## Possible Issues(1)

HeunTPrime calculations might take time for big arguments:

## Neat Examples(1)

The Schrödinger equation for the following infinite potential well can be solved in terms of HeunTPrime:

Plot the potential:

Construct the general solution of the Schrödinger equation:

Verify this solution by direct substitution: