# Hypergeometric1F1

Hypergeometric1F1[a,b,z]

is the Kummer confluent hypergeometric function .

# 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(40)

### 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 Hypergeometric1F1 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 Hypergeometric1F1 function using MatrixFunction:

### Specific Values(4)

Hypergeometric1F1 automatically evaluates to simpler functions for certain parameters:

Limiting values at infinity for some case of Hypergeometric1F1:

Find a value of satisfying the equation :

Heun functions can be reduced to hypergeometric functions:

### Visualization(3)

Plot the Hypergeometric1F1 function:

Plot Hypergeometric1F1 as a function of its second parameter:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

Real domain of Hypergeometric1F1:

Complex domain:

is an analytic function for real values of and :

For positive values of , it may or may not be analytic:

Hypergeometric1F1 is neither non-decreasing nor non-increasing except for special values:

is not injective:

is not surjective:

Hypergeometric1F1 is non-negative for specific values:

is neither non-negative nor non-positive:

has both singularity and discontinuity when is a negative integer:

is convex:

is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Formula for the derivative:

### Integration(3)

Apply Integrate to Hypergeometric1F1:

Definite integral of Hypergeometric1F1:

More integrals:

### Series Expansions(4)

Taylor expansion for Hypergeometric1F1:

Plot the first three approximations for around :

General term in the series expansion of Hypergeometric1F1:

Expand Hypergeometric1F1 in a series around infinity:

Apply Hypergeometric1F1 to a power series:

### Integral Transforms(2)

Compute the Laplace transform using LaplaceTransform:

### Function Identities and Simplifications(3)

Argument simplification:

Sum of the Hypergeometric1F1 functions:

Recurrence identity:

### Function Representations(4)

Primary definition:

Relation to the LaguerreL polynomial:

Hypergeometric1F1 can be represented as a DifferentialRoot:

Hypergeometric1F1 can be represented in terms of MeijerG:

## Generalizations & Extensions(1)

Apply Hypergeometric1F1 to a power series:

## Applications(3)

Hydrogen atom radial wave function for continuous spectrum:

Compute the energy eigenvalue from the differential equation:

Closed form for Padé approximation of Exp to any order:

Compare with explicit approximants:

Solve a differential equation:

## Properties & Relations(2)

Integrate may give results involving Hypergeometric1F1:

Use FunctionExpand to convert confluent hypergeometric functions:

Wolfram Research (1988), Hypergeometric1F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric1F1.html (updated 2022).

#### Text

Wolfram Research (1988), Hypergeometric1F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric1F1.html (updated 2022).

#### CMS

Wolfram Language. 1988. "Hypergeometric1F1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Hypergeometric1F1.html.

#### APA

Wolfram Language. (1988). Hypergeometric1F1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric1F1.html

#### BibTeX

@misc{reference.wolfram_2024_hypergeometric1f1, author="Wolfram Research", title="{Hypergeometric1F1}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric1F1.html}", note=[Accessed: 19-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hypergeometric1f1, organization={Wolfram Research}, title={Hypergeometric1F1}, year={2022}, url={https://reference.wolfram.com/language/ref/Hypergeometric1F1.html}, note=[Accessed: 19-September-2024 ]}