# Hypergeometric1F1

Hypergeometric1F1[a,b,z]

is the Kummer confluent hypergeometric function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The function has the series expansion .
• For certain special arguments, Hypergeometric1F1 automatically evaluates to exact values.
• Hypergeometric1F1 can be evaluated to arbitrary numerical precision.
• Hypergeometric1F1 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(33)

### 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 Hypergeometric1F1 efficiently at high precision:

Hypergeometric1F1 threads elementwise over list arguments and parameters:

### 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(2)

Real domain of Hypergeometric1F1:

Complex domain of Hypergeometric1F1:

### 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(5)

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(2)

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:

## Properties & Relations(2)

Integrate may give results involving Hypergeometric1F1:

Use FunctionExpand to convert confluent hypergeometric functions:

Introduced in 1988
(1.0)