Hypergeometric1F1 [a,b,z]
is the Kummer confluent hypergeometric function TemplateBox[{a, b, z}, Hypergeometric1F1].
Hypergeometric1F1
Hypergeometric1F1 [a,b,z]
is the Kummer confluent hypergeometric function TemplateBox[{a, b, z}, Hypergeometric1F1].
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The function has the series expansion TemplateBox[{a, b, z}, Hypergeometric1F1]=sum_(k=0)^(infty)TemplateBox[{a, k}, Pochhammer]/TemplateBox[{b, k}, Pochhammer]z^k/k!, where TemplateBox[{a, k}, Pochhammer] is the Pochhammer symbol.
- For certain special arguments, Hypergeometric1F1 automatically evaluates to exact values.
- Hypergeometric1F1 can be evaluated to arbitrary numerical precision.
- Hypergeometric1F1 automatically threads over lists.
- Hypergeometric1F1 can be used with Interval and CenteredInterval objects. »
Examples
open all close allBasic 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 (40)
Numerical Evaluation (5)
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:
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
Or compute average-case statistical intervals using Around :
Compute the elementwise values of an array:
Or compute the matrix Hypergeometric1F1 function using MatrixFunction :
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, )}, z}, Hypergeometric1F1]:
Plot the imaginary part of TemplateBox[{1, {sqrt(, 2, )}, z}, 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)
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 :
Function Identities and Simplifications (3)
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 (3)
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:
Solve a differential equation:
Properties & Relations (2)
Integrate may give results involving Hypergeometric1F1 :
Use FunctionExpand to convert confluent hypergeometric functions:
Tech Notes
Related Guides
Related Links
History
Introduced in 1988 (1.0) | Updated in 2021 (13.0) ▪ 2022 (13.1)
Text
Wolfram Research (1988), Hypergeometric1F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric1F1.html (updated 2022).
CMS
Wolfram Language. 1988. "Hypergeometric1F1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. 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
BibTeX
@misc{reference.wolfram_2025_hypergeometric1f1, author="Wolfram Research", title="{Hypergeometric1F1}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric1F1.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_hypergeometric1f1, organization={Wolfram Research}, title={Hypergeometric1F1}, year={2022}, url={https://reference.wolfram.com/language/ref/Hypergeometric1F1.html}, note=[Accessed: 05-December-2025]}