Hypergeometric1F1Regularized

Hypergeometric1F1Regularized[a,b,z]

is the regularized confluent hypergeometric function .

Details

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  (32)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (7)

Hypergeometric1F1Regularized for symbolic a:

Limiting values at infinity:

Values at zero:

Find a value of x for which Hypergeometric1F1Regularized[1/2,1,x ]=0.4:

Evaluate symbolically for integer parameters:

Evaluate symbolically for half-integer parameters:

Hypergeometric1F1Regularized automatically evaluates to simpler functions for certain parameters:

Visualization  (3)

Plot the Hypergeometric1F1Regularized function:

Plot Hypergeometric1F1Regularized as a function of its second parameter :

Plot the real part of TemplateBox[{{1, /, 2}, {sqrt(, 2, )}, {x, +, iy}}, Hypergeometric1F1Regularized]:

Plot the imaginary part of TemplateBox[{{1, /, 2}, {sqrt(, 2, )}, {x, +, iy}}, Hypergeometric1F1Regularized]:

Function Properties  (4)

Real domain of Hypergeometric1F1Regularized:

Complex domain of Hypergeometric1F1Regularized:

Hypergeometric1F1Regularized threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to b when a=1:

First derivative with respect to z when a=1:

Higher derivatives with respect to b when a=1:

Higher derivatives with respect to z when a=1 and b=2:

Plot the higher derivatives with respect to z when a=1 and b=2:

Formula for the ^(th) derivative with respect to z when a=1:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (6)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

FourierSeries:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications  (2)

Recurrence relation:

Use FunctionExpand to express Hypergeometric1F1Regularized through other functions:

Generalizations & Extensions  (1)

Series expansion at infinity:

Properties & Relations  (3)

With a numeric second parameter, gives the ordinary hypergeometric function:

Hypergeometric1F1Regularized can be represented as a DifferentialRoot:

Hypergeometric1F1Regularized can be represented in terms of MeijerG:

Neat Examples  (1)

Visualize the confluence relation :

Introduced in 1996
 (3.0)