Hypergeometric0F1

Hypergeometric0F1[a,z]

is the confluent hypergeometric function TemplateBox[{a, z}, Hypergeometric0F1].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The function has the series expansion TemplateBox[{a, z}, Hypergeometric0F1]=sum_(k=0)^(infty)1/TemplateBox[{a, k}, Pochhammer] z^k/k!, where TemplateBox[{a, k}, Pochhammer] is the Pochhammer symbol.
  • For certain special arguments, Hypergeometric0F1 automatically evaluates to exact values.
  • Hypergeometric0F1 can be evaluated to arbitrary numerical precision.
  • Hypergeometric0F1 automatically threads over lists.
  • Hypergeometric0F1 can be used with Interval and CenteredInterval objects. »

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 to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate Hypergeometric0F1 efficiently at high precision:

Hypergeometric0F1 threads elementwise over lists:

Hypergeometric0F1 can be used with Interval and CenteredInterval objects:

Specific Values  (4)

Evaluate symbolically for half-integer parameters:

Limiting value at infinity:

Find a zero of TemplateBox[{{sqrt(, 2, )}, x}, Hypergeometric0F1]:

Heun functions can be reduced to hypergeometric functions:

Visualization  (3)

Plot the Hypergeometric0F1 function for various values of parameter :

Plot Hypergeometric0F1 as a function of its first parameter :

Plot the real part of TemplateBox[{{sqrt(, 2, )}, z}, Hypergeometric0F1]:

Plot the imaginary part of TemplateBox[{{sqrt(, 2, )}, z}, Hypergeometric0F1]:

Function Properties  (9)

Real domain of TemplateBox[{a, z}, Hypergeometric0F1]:

Complex domain:

TemplateBox[{a, z}, Hypergeometric0F1] is an analytic function when :

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

TemplateBox[{1, z}, Hypergeometric0F1] is neither non-decreasing nor non-increasing:

TemplateBox[{1, z}, Hypergeometric0F1] is not injective:

TemplateBox[{1, z}, Hypergeometric0F1] is not surjective:

TemplateBox[{{1, /, 3}, z}, Hypergeometric0F1] is surjective:

Note that the latter function grows very slowly as :

Hypergeometric0F1 is neither non-negative nor non-positive:

TemplateBox[{1, z}, Hypergeometric0F1] has no singularities or discontinuities:

TemplateBox[{{1, /, 2}, z}, Hypergeometric0F1] is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of Hypergeometric0F1:

Definite integral:

Integral involving a power function:

Series Expansions  (3)

Taylor expansion for Hypergeometric0F1:

Plot the first three approximations for TemplateBox[{1, x}, Hypergeometric0F1] around :

General term in the series expansion of Hypergeometric0F1:

Series expansion for TemplateBox[{1, x}, Hypergeometric0F1] at infinity:

Function Identities and Simplifications  (3)

Product of the Hypergeometric0F1 functions:

Recurrence relation:

Use FunctionExpand to express Hypergeometric0F1 through other functions:

Function Representations  (5)

Series representation:

Relation to Hypergeometric1F1 function:

Hypergeometric0F1 can be represented as a DifferentialRoot:

Hypergeometric0F1 can be represented in terms of MeijerG:

TraditionalForm formatting:

Applications  (2)

Solve the 1+1-dimensional Dirac equation:

Plot the solution:

Hypergeometric0F1 has the following infinite series:

Properties & Relations  (2)

Use FunctionExpand to expand in terms of Bessel functions:

Hypergeometric0F1 can be represented as a DifferenceRoot:

Neat Examples  (1)

Continued fraction with arithmetic progression terms:

Wolfram Research (1988), Hypergeometric0F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric0F1.html (updated 2022).

Text

Wolfram Research (1988), Hypergeometric0F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric0F1.html (updated 2022).

CMS

Wolfram Language. 1988. "Hypergeometric0F1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Hypergeometric0F1.html.

APA

Wolfram Language. (1988). Hypergeometric0F1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric0F1.html

BibTeX

@misc{reference.wolfram_2023_hypergeometric0f1, author="Wolfram Research", title="{Hypergeometric0F1}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric0F1.html}", note=[Accessed: 19-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_hypergeometric0f1, organization={Wolfram Research}, title={Hypergeometric0F1}, year={2022}, url={https://reference.wolfram.com/language/ref/Hypergeometric0F1.html}, note=[Accessed: 19-March-2024 ]}