Hypergeometric0F1Regularized

Hypergeometric0F1Regularized[a,z]

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  (31)

Numerical Evaluation  (4)

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:

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  TemplateBox[{{sqrt(, 2, )}, {x, +, {ⅈ,  , y}}}, Hypergeometric0F1Regularized]:

Plot the imaginary part of TemplateBox[{{sqrt(, 2, )}, {x, +, {ⅈ,  , y}}}, Hypergeometric0F1Regularized]:

Function Properties  (4)

Real domain of Hypergeometric0F1Regularized:

Complex domain of Hypergeometric0F1Regularized:

Hypergeometric0F1Regularized threads elementwise over lists:

TraditionalForm formatting:

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 ^(th) 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:

FourierSeries:

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.

Text

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

BibTeX

@misc{reference.wolfram_2020_hypergeometric0f1regularized, author="Wolfram Research", title="{Hypergeometric0F1Regularized}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric0F1Regularized.html}", note=[Accessed: 03-December-2020 ]}

BibLaTeX

@online{reference.wolfram_2020_hypergeometric0f1regularized, organization={Wolfram Research}, title={Hypergeometric0F1Regularized}, year={1996}, url={https://reference.wolfram.com/language/ref/Hypergeometric0F1Regularized.html}, note=[Accessed: 03-December-2020 ]}

CMS

Wolfram Language. 1996. "Hypergeometric0F1Regularized." Wolfram Language & System Documentation Center. Wolfram Research. 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