Confluent Hypergeometric Limit Function

Author

Eric W. Weisstein
January 16, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Sum

(∑_(n=0)^∞)((z^n)/(n! (Pochhammer[a, n])))==(Hypergeometric0F1[a, z])

V7

Sum[((z^n)/(Pochhammer[a,n]n!)),{n,0,∞}]

Limit

(lim_(q->∞)) (Hypergeometric1F1[q, a, (z/q)])==(Hypergeometric0F1[a, z])

Limit[Hypergeometric1F1[q,a,(z/q)],q->∞]

With[{a=.2,z=1.2},{Hypergeometric1F1[q,a,(z/q)]/.q->10^5,Hypergeometric0F1[a,z]}]

With[{a=.2+2I,z=1.2},{Hypergeometric1F1[q,a,(z/q)]/.q->10^5,Hypergeometric0F1[a,z]}]

ODE

a y'(z)+z y''(z)-y(z)==0

DSolve[z y''[z]+a y'[z]-y[z]==0,y[z],z]

FullSimplify[%]