is the confluent hypergeometric function .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The function has the series expansion .
  • For certain special arguments, Hypergeometric0F1 automatically evaluates to exact values.
  • Hypergeometric0F1 can be evaluated to arbitrary numerical precision.
  • Hypergeometric0F1 automatically threads over lists.


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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  (30)

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 Hypergeometric0F1 efficiently at high precision:

Hypergeometric0F1 threads elementwise over lists:

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, )}, {x, +, {ⅈ,  , y}}}, Hypergeometric0F1]:

Plot the imaginary part of TemplateBox[{{sqrt(, 2, )}, {x, +, {ⅈ,  , y}}}, Hypergeometric0F1]:

Function Properties  (2)

Real domain of Hypergeometric0F1:

Complex domain of Hypergeometric0F1:

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  (1)

Solve the 1+1-dimensional Dirac equation:

Plot the solution:

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:

Introduced in 1988