Hypergeometric1F1Regularized

Hypergeometric1F1Regularized[a,b,z]

is the regularized confluent hypergeometric function .

Details

Examples

open allclose all

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  (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  (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 TemplateBox[{{1, /, 2}, {sqrt(, 2, )}, {x, +, iy}}, Hypergeometric1F1Regularized]:

Plot the imaginary part of TemplateBox[{{1, /, 2}, {sqrt(, 2, )}, {x, +, iy}}, Hypergeometric1F1Regularized]:

Function Properties  (10)

Hypergeometric1F1Regularized is defined for all real and complex values:

Hypergeometric1F1Regularized threads elementwise over lists:

TemplateBox[{a, b, z}, Hypergeometric1F1Regularized] is an analytic function:

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

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, Hypergeometric1F1Regularized] is not injective:

TemplateBox[{1, 2, z}, Hypergeometric1F1Regularized] is not surjective:

Hypergeometric1F1Regularized is non-negative for specific values:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, Hypergeometric1F1Regularized] is neither non-negative nor non-positive:

TemplateBox[{a, b, z}, Hypergeometric1F1Regularized] has no singularities or discontinuities:

TemplateBox[{{-, 2}, 1, z}, Hypergeometric1F1Regularized] is convex:

TemplateBox[{2, 1, z}, Hypergeometric1F1Regularized] is neither convex nor concave:

TraditionalForm formatting:

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 ^(th) 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:

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 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.

Text

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

BibTeX

@misc{reference.wolfram_2021_hypergeometric1f1regularized, author="Wolfram Research", title="{Hypergeometric1F1Regularized}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html}", note=[Accessed: 24-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_hypergeometric1f1regularized, organization={Wolfram Research}, title={Hypergeometric1F1Regularized}, year={1996}, url={https://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html}, note=[Accessed: 24-October-2021 ]}

CMS

Wolfram Language. 1996. "Hypergeometric1F1Regularized." Wolfram Language & System Documentation Center. Wolfram Research. 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