# Hypergeometric0F1Regularized

is the regularized confluent hypergeometric function .

# Details # 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:

Hypergeometric0F1Regularized threads elementwise over lists: 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 :