Hypergeometric1F1

Hypergeometric1F1[a,b,z]

is the Kummer confluent hypergeometric function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The function has the series expansion .
  • For certain special arguments, Hypergeometric1F1 automatically evaluates to exact values.
  • Hypergeometric1F1 can be evaluated to arbitrary numerical precision.
  • Hypergeometric1F1 automatically threads over lists.

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  (39)

Numerical Evaluation  (4)

Evaluate to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate Hypergeometric1F1 efficiently at high precision:

Hypergeometric1F1 threads elementwise over list arguments and parameters:

Specific Values  (4)

Hypergeometric1F1 automatically evaluates to simpler functions for certain parameters:

Limiting values at infinity for some case of Hypergeometric1F1:

Find a value of satisfying the equation TemplateBox[{{1, /, 2}, {sqrt(, 2, )}, x}, Hypergeometric1F1]=2:

Heun functions can be reduced to hypergeometric functions:

Visualization  (3)

Plot the Hypergeometric1F1 function:

Plot Hypergeometric1F1 as a function of its second parameter:

Plot the real part of TemplateBox[{1, {sqrt(, 2, )}, {x, +, {ⅈ,  , y}}}, Hypergeometric1F1]:

Plot the imaginary part of TemplateBox[{1, {sqrt(, 2, )}, {x, +, {ⅈ,  , y}}}, Hypergeometric1F1]:

Function Properties  (9)

Real domain of Hypergeometric1F1:

Complex domain:

TemplateBox[{a, b, z}, Hypergeometric1F1] is an analytic function for real values of and b in TemplateBox[{}, Reals]:

For positive values of , it may or may not be analytic:

Hypergeometric1F1 is neither non-decreasing nor non-increasing except for special values:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, Hypergeometric1F1] is not injective:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, Hypergeometric1F1] is not surjective:

Hypergeometric1F1 is non-negative for specific values:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, Hypergeometric1F1] is neither non-negative nor non-positive:

TemplateBox[{a, b, z}, Hypergeometric1F1] has both singularity and discontinuity when is a negative integer:

TemplateBox[{{-, 2}, 1, z}, Hypergeometric1F1] is convex:

TemplateBox[{2, 1, z}, Hypergeometric1F1] is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Formula for the ^(th) derivative:

Integration  (3)

Apply Integrate to Hypergeometric1F1:

Definite integral of Hypergeometric1F1:

More integrals:

Series Expansions  (4)

Taylor expansion for Hypergeometric1F1:

Plot the first three approximations for TemplateBox[{{1, /, 2}, {sqrt(, 2, )}, x}, Hypergeometric1F1] around :

General term in the series expansion of Hypergeometric1F1:

Expand Hypergeometric1F1 in a series around infinity:

Apply Hypergeometric1F1 to a power series:

Integral Transforms  (2)

Compute the Laplace transform using LaplaceTransform:

HankelTransform:

Function Identities and Simplifications  (3)

Argument simplification:

Sum of the Hypergeometric1F1 functions:

Recurrence identity:

Function Representations  (4)

Primary definition:

Relation to the LaguerreL polynomial:

Hypergeometric1F1 can be represented as a DifferentialRoot:

Hypergeometric1F1 can be represented in terms of MeijerG:

Generalizations & Extensions  (1)

Apply Hypergeometric1F1 to a power series:

Applications  (2)

Hydrogen atom radial wave function for continuous spectrum:

Compute the energy eigenvalue from the differential equation:

Closed form for Padé approximation of Exp to any order:

Compare with explicit approximants:

Properties & Relations  (2)

Integrate may give results involving Hypergeometric1F1:

Use FunctionExpand to convert confluent hypergeometric functions:

Wolfram Research (1988), Hypergeometric1F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric1F1.html.

Text

Wolfram Research (1988), Hypergeometric1F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric1F1.html.

BibTeX

@misc{reference.wolfram_2021_hypergeometric1f1, author="Wolfram Research", title="{Hypergeometric1F1}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric1F1.html}", note=[Accessed: 15-May-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_hypergeometric1f1, organization={Wolfram Research}, title={Hypergeometric1F1}, year={1988}, url={https://reference.wolfram.com/language/ref/Hypergeometric1F1.html}, note=[Accessed: 15-May-2021 ]}

CMS

Wolfram Language. 1988. "Hypergeometric1F1." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Hypergeometric1F1.html.

APA

Wolfram Language. (1988). Hypergeometric1F1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric1F1.html