# Hypergeometric0F1

Hypergeometric0F1[a,z]

is the confluent hypergeometric function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The function has the series expansion .
• For certain special arguments, Hypergeometric0F1 automatically evaluates to exact values.
• Hypergeometric0F1 can be evaluated to arbitrary numerical precision.
• Hypergeometric0F1 automatically threads over lists.

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(37)

### Numerical Evaluation(4)

Evaluate to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate Hypergeometric0F1 efficiently at high precision:

### Specific Values(4)

Evaluate symbolically for half-integer parameters:

Limiting value at infinity:

Find a zero of :

Heun functions can be reduced to hypergeometric functions:

### Visualization(3)

Plot the Hypergeometric0F1 function for various values of parameter :

Plot Hypergeometric0F1 as a function of its first parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

Real domain of :

Complex domain: is an analytic function when :

For negative values of , it may or may not be analytic: is neither non-decreasing nor non-increasing: is not injective: is not surjective: is surjective:

Note that the latter function grows very slowly as :

Hypergeometric0F1 is neither non-negative nor non-positive: has no singularities or discontinuities: is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Formula for the  derivative:

### Integration(3)

Indefinite integral of Hypergeometric0F1:

Definite integral:

Integral involving a power function:

### Series Expansions(3)

Taylor expansion for Hypergeometric0F1:

Plot the first three approximations for around :

General term in the series expansion of Hypergeometric0F1:

Series expansion for at infinity:

### Function Identities and Simplifications(3)

Product of the Hypergeometric0F1 functions:

Recurrence relation:

Use FunctionExpand to express Hypergeometric0F1 through other functions:

### Function Representations(5)

Series representation:

Relation to Hypergeometric1F1 function:

Hypergeometric0F1 can be represented as a DifferentialRoot:

Hypergeometric0F1 can be represented in terms of MeijerG:

## Applications(1)

Solve the 1+1-dimensional Dirac equation:

Plot the solution:

## Properties & Relations(2)

Use FunctionExpand to expand in terms of Bessel functions:

Hypergeometric0F1 can be represented as a DifferenceRoot:

## Neat Examples(1)

Continued fraction with arithmetic progression terms: