# Hypergeometric1F1Regularized

Hypergeometric1F1Regularized[a,b,z]

is the regularized 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(6)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate 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 Hypergeometric1F1Regularized function using MatrixFunction:

### Specific Values(7)

Hypergeometric1F1Regularized for symbolic a:

Limiting values at infinity:

Values at zero:

Find a value of x for which Hypergeometric1F1Regularized[1/2,1,x ]=0.4:

Evaluate symbolically for integer parameters:

Evaluate symbolically for half-integer parameters:

Hypergeometric1F1Regularized automatically evaluates to simpler functions for certain parameters:

### Visualization(3)

Plot the Hypergeometric1F1Regularized function:

Plot Hypergeometric1F1Regularized as a function of its second parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(10)

Hypergeometric1F1Regularized is defined for all real and complex values:

Hypergeometric1F1Regularized threads elementwise over lists:

is an analytic function:

Hypergeometric1F1Regularized is neither non-decreasing nor non-increasing except for special values:

is not injective:

is not surjective:

Hypergeometric1F1Regularized is non-negative for specific values:

is neither non-negative nor non-positive:

has no singularities or discontinuities:

is convex:

is neither convex nor concave:

### Differentiation(3)

First derivative with respect to b when a=1:

First derivative with respect to z when a=1:

Higher derivatives with respect to b when a=1:

Higher derivatives with respect to z when a=1 and b=2:

Plot the higher derivatives with respect to z when a=1 and b=2:

Formula for the derivative with respect to z when a=1:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

### Series Expansions(6)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

### Function Identities and Simplifications(2)

Recurrence relation:

Use FunctionExpand to express Hypergeometric1F1Regularized through other functions:

## Generalizations & Extensions(1)

Series expansion at infinity:

## Properties & Relations(3)

With a numeric second parameter, gives the ordinary hypergeometric function:

Hypergeometric1F1Regularized can be represented as a DifferentialRoot:

Hypergeometric1F1Regularized can be represented in terms of MeijerG:

## Neat Examples(1)

Visualize the confluence relation :

Wolfram Research (1996), Hypergeometric1F1Regularized, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html (updated 2022).

#### Text

Wolfram Research (1996), Hypergeometric1F1Regularized, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html (updated 2022).

#### CMS

Wolfram Language. 1996. "Hypergeometric1F1Regularized." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html.

#### APA

Wolfram Language. (1996). Hypergeometric1F1Regularized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html

#### BibTeX

@misc{reference.wolfram_2024_hypergeometric1f1regularized, author="Wolfram Research", title="{Hypergeometric1F1Regularized}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html}", note=[Accessed: 17-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hypergeometric1f1regularized, organization={Wolfram Research}, title={Hypergeometric1F1Regularized}, year={2022}, url={https://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html}, note=[Accessed: 17-September-2024 ]}