is the Kummer confluent hypergeometric function .


  • 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.


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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  (33)

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  (2)

Real domain of Hypergeometric1F1:

Complex domain of Hypergeometric1F1:

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:


Function Identities and Simplifications  (3)

Argument simplification:

Sum of the Hypergeometric1F1 functions:

Recurrence identity:

Function Representations  (5)

Primary definition:

Relation to the LaguerreL polynomial:

Hypergeometric1F1 can be represented as a DifferentialRoot:

Hypergeometric1F1 can be represented in terms of MeijerG:

TraditionalForm formatting:

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:

Introduced in 1988