Confluent Hypergeometric Function of the Second Kind

Author

Eric W. Weisstein
September 30, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/ConfluentHypergeometricFunctionoftheSecondKind.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Definition

(1/(Gamma[a]))Integrate[Exp[-z t](t^(a-1))(((1+t))^(b-a-1)),{t,0,∞}]

i=(1/(Gamma[a]))Integrate[Exp[-z t](t^(a-1))(((1+t))^(b-a-1)),{t,0,∞},Assumptions->Re[a]>0&&Re[z]>0]

TraditionalForm[%]

Alternate Forms

π csc(b π) ((((Subscript[, 1])(Subscript[(OverTilde[F]), 1])(a;b;z))/(Γ(a-b+1)))-(((z^(1-b)) (Subscript[, 1])(Subscript[(OverTilde[F]), 1])(a-b+1;2-b;z))/(Γ(a))))

FullSimplify[i]

TraditionalForm[%]

FullSimplify[Subtract@@{HypergeometricU[a,b,z],i}]

(z^(-a)) (Subscript[, 2])(Subscript[F, 0])(a,a-b+1;;-(1/z))

Exists as a formal power series only

diff=FullSimplify[Subtract@@{HypergeometricU[a,b,z],(z^(-a))HypergeometricPFQ[{a,1+a-b},{},-(1/z)]}]

N[HypergeometricPFQ[{1,1},{},1],20]

Derivative

D[HypergeometricU[a,b,z],z]

Integral

Integrate[HypergeometricU[a,b,z],z]//FullSimplify

TraditionalForm[%]

Series

Series[HypergeometricU[a,b,z],{z,0,1}]//FullSimplify

Normal[%]//FullSimplify

Series[HypergeometricU[a,b,z],{z,∞,2}]//FullSimplify

Series[(z^(-a))HypergeometricPFQ[{a,1+a-b},{},-(1/z)],{z,∞,2}]//FullSimplify

Simplify[((((1/z)))^a)-(z^(-a))]