# HeunB

HeunB[q,α,γ,δ,ϵ,z]

gives the bi-confluent Heun function.

# Details • HeunB belongs to the Heun class of functions and occurs in quantum mechanics, mathematical physics and applications.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• HeunB[q,α,γ,δ,ϵ,z] satisfies the bi-confluent Heun differential equation .
• The HeunB function is the regular solution of the bi-confluent Heun equation that satisfies the condition HeunB[q,α,γ,δ,ϵ,0]=1.
• For certain special arguments, HeunB automatically evaluates to exact values.
• HeunB can be evaluated for arbitrary complex parameters.
• HeunB can be evaluated to arbitrary numerical precision.
• HeunB automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the HeunB function:

Series expansion of HeunB:

## Scope(23)

### Numerical Evaluation(7)

Evaluate to high precision:

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

HeunB can take one or more complex number parameters:

HeunB can take complex number arguments:

Finally, HeunB can take all complex number input:

Evaluate HeunB efficiently at high precision:

Lists and matrices:

### Specific Values(1)

Value of HeunB at origin:

### Visualization(5)

Plot the HeunB function:

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

Plot HeunB as a function of its second parameter :

Plot HeunB as a function of and :

Plot the family of HeunB functions for different accessory parameter :

### Function Properties(1)

HeunB can be simplified to Hypergeometric1F1 function in the following case:

### Differentiation(2)

The -derivative of HeunB is HeunBPrime:

Higher derivatives of HeunB are calculated using HeunBPrime:

### Integration(3)

Indefinite integrals of HeunB are not expressed in elementary or other special functions:

Definite numerical integral of HeunB:

More integrals with HeunB:

### Series Expansions(4)

Taylor expansion for HeunB at regular singular origin:

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

Plot the first three approximations for HeunB around :

Series expansion for HeunB at any ordinary complex point:

## Applications(3)

Solve the bi-confluent Heun differential equation using DSolve:

Plot the solution:

Directly solve the bi-confluent Heun differential equation:

Solve the class of confinement potentials for the radial Schrödinger equation in terms of HeunB functions:

Plot the potential for arbitrary parameters:

This general potential is solved in terms of HeunB functions:

## Properties & Relations(3)

HeunB is analytic at the origin:

HeunB can be calculated at any finite complex :

The derivative of HeunB is HeunBPrime:

## Possible Issues(1)

HeunB diverges for big arguments:

## Neat Examples(2)

Create a table of some special cases for HeunB :

The quantum-mechanical doubly anharmonic oscillator potential is:

Plot the potential:

The general solution of the Schrödinger equation is written in terms of HeunB functions:

Verify this solution by direct substitution: