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 $TemplateBox[{a, b, z}, Hypergeometric1F1]=sum_(k=0)^(infty)TemplateBox[{a, k}, Pochhammer]/TemplateBox[{b, k}, Pochhammer]z^k/k!$ , where 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

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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 (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:

Hypergeometric1F1 threads elementwise over list arguments and parameters:

Hypergeometric1F1 can be used with Interval and CenteredInterval objects:

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 :

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 :

Plot the imaginary part of :

Function Properties (9)

Real domain of Hypergeometric1F1:

Complex domain:

is an analytic function for real values of and :

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

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

is not injective:

is not surjective:

Hypergeometric1F1 is non-negative for specific values:

is neither non-negative nor non-positive:

has both singularity and discontinuity when is a negative integer:

is convex:

is neither convex nor concave:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Formula for the 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 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 (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:

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