# 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 , where is the Pochhammer symbol.
• For certain special arguments, Hypergeometric0F1 automatically evaluates to exact values.
• Hypergeometric0F1 can be evaluated to arbitrary numerical precision.
• Hypergeometric0F1 automatically threads over lists.
• Hypergeometric0F1 can be used with Interval and CenteredInterval objects. »

# 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(38)

### Numerical Evaluation(5)

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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix Hypergeometric0F1 function using MatrixFunction:

### 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(2)

Solve the 1+1-dimensional Dirac equation:

Plot the solution:

Hypergeometric0F1 has the following infinite series:

## 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:

Wolfram Research (1988), Hypergeometric0F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric0F1.html (updated 2022).

#### Text

Wolfram Research (1988), Hypergeometric0F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric0F1.html (updated 2022).

#### CMS

Wolfram Language. 1988. "Hypergeometric0F1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Hypergeometric0F1.html.

#### APA

Wolfram Language. (1988). Hypergeometric0F1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric0F1.html

#### BibTeX

@misc{reference.wolfram_2024_hypergeometric0f1, author="Wolfram Research", title="{Hypergeometric0F1}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric0F1.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hypergeometric0f1, organization={Wolfram Research}, title={Hypergeometric0F1}, year={2022}, url={https://reference.wolfram.com/language/ref/Hypergeometric0F1.html}, note=[Accessed: 12-September-2024 ]}