# Hypergeometric0F1Regularized

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(38)

### Numerical Evaluation(5)

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:

Hypergeometric0F1Regularized can be used with Interval and CenteredInterval objects:

### Specific Values(6)

Hypergeometric0F1Regularized for symbolic a:

Limiting value at infinity:

Values at zero:

Find a value of x for which Hypergeometric0F1Regularized[10,x ]=0.000001:

Evaluate symbolically for integer parameters:

Evaluate symbolically for half-integer parameters:

### Visualization(3)

Plot the Hypergeometric0F1Regularized function for various values of parameter :

Plot Hypergeometric0F1Regularized as a function of its first parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(10)

is defined for all real and complex values:

is an analytic function:

is neither non-decreasing nor non-increasing:

is not injective:

is not surjective:

is surjective:

Note that the latter function grows very slowly as :

Hypergeometric0F1Regularized is neither non-negative nor non-positive:

has no singularities or discontinuities:

is neither convex nor concave:

### Differentiation(3)

First derivative with respect to z when a=5/2:

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

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

Formula for the derivative with respect to z when a=1/2:

### 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 relations:

Use FunctionExpand to express Hypergeometric0F1Regularized through other functions:

## Generalizations & Extensions(1)

Series expansion at infinity:

## Applications(1)

Probability that in a voting game with 2 candidates, and the number of votes being two independent Poisson random variables with means p and q, candidate 1 gets k more votes than candidate 2 out of n:

Plot distribution for almost even odds:

## Properties & Relations(4)

Hypergeometric0F1Regularized can be represented as a DifferentialRoot:

Hypergeometric0F1Regularized can be represented in terms of MeijerG:

Hypergeometric0F1Regularized can be represented as a DifferenceRoot:

General term in the series expansion of Hypergeometric0F1Regularized:

## Neat Examples(1)

Visualize confluence relation :

Wolfram Research (1996), Hypergeometric0F1Regularized, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric0F1Regularized.html (updated 2022).

#### Text

Wolfram Research (1996), Hypergeometric0F1Regularized, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric0F1Regularized.html (updated 2022).

#### CMS

Wolfram Language. 1996. "Hypergeometric0F1Regularized." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Hypergeometric0F1Regularized.html.

#### APA

Wolfram Language. (1996). Hypergeometric0F1Regularized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric0F1Regularized.html

#### BibTeX

@misc{reference.wolfram_2022_hypergeometric0f1regularized, author="Wolfram Research", title="{Hypergeometric0F1Regularized}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric0F1Regularized.html}", note=[Accessed: 01-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_hypergeometric0f1regularized, organization={Wolfram Research}, title={Hypergeometric0F1Regularized}, year={2022}, url={https://reference.wolfram.com/language/ref/Hypergeometric0F1Regularized.html}, note=[Accessed: 01-June-2023 ]}