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13 Confluent Hypergeometric FunctionsKummer Functions

§13.7 Asymptotic Expansions for Large Argument

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§13.7(i) Poincaré-Type Expansions

As x

13.7.1 M(a,b,x)exxa-bΓ(a)s=0(1-a)s(b-a)ss!x-s,

provided that a0,-1,.

As z

13.7.2 M(a,b,z)ezza-bΓ(a)s=0(1-a)s(b-a)ss!z-s+e±πiaz-aΓ(b-a)s=0(a)s(a-b+1)ss!(-z)-s,
-12π+δ±phz32π-δ,

unless a=0,-1, and b-a=0,-1,. Here δ denotes an arbitrary small positive constant. Also,

13.7.3 U(a,b,z)z-as=0(a)s(a-b+1)ss!(-z)-s,
|phz|32π-δ.

§13.7(ii) Error Bounds

See accompanying text
Figure 13.7.1: Regions R1, R2, R¯2, R3, and R¯3 are the closures of the indicated unshaded regions bounded by the straight lines and circular arcs centered at the origin, with r=|b-2a|. Magnify
13.7.4 U(a,b,z)=z-as=0n-1(a)s(a-b+1)ss!(-z)-s+εn(z),

where

13.7.5 |εn(z)|, β-1|εn(z)|2αCn|(a)n(a-b+1)nn!za+n|exp(2αρC1|z|),

and with the notation of Figure 13.7.1

13.7.6 Cn =1,
χ(n),
(χ(n)+σν2n)νn,

according as

13.7.7 z R1,
z R2R¯2,
z R3R¯3,

respectively, with

13.7.8 σ =|(b-2a)/z|,
ν =(12+121-4σ2)-1/2,
χ(n) =πΓ(12n+1)/Γ(12n+12).

Also, when zR1R2R¯2

13.7.9 α =11-σ,
β =1-σ2+σ|z|-12(1-σ),
ρ =12|2a2-2ab+b|+σ(1+14σ)(1-σ)2,

and when zR3R¯3 σ is replaced by νσ and |z|-1 is replaced by ν|z|-1 everywhere in (13.7.9).

For numerical values of χ(n) see Table 9.7.1.

Corresponding error bounds for (13.7.2) can be constructed by combining (13.2.41) with (13.7.4)–(13.7.9).

§13.7(iii) Exponentially-Improved Expansion

Let

13.7.10 U(a,b,z)=z-as=0n-1(a)s(a-b+1)ss!(-z)-s+Rn(a,b,z),

and

13.7.11 Rn(a,b,z)=(-1)n2πza-bΓ(a)Γ(a-b+1)(s=0m-1(1-a)s(b-a)ss!(-z)-sGn+2a-b-s(z)+(1-a)m(b-a)mRm,n(a,b,z),)

where m is an arbitrary nonnegative integer, and

13.7.12 Gp(z)=ez2πΓ(p)Γ(1-p,z).

(For the notation see §8.2(i).) Then as z with ||z|-n| bounded and a,b,m fixed

13.7.13 Rm,n(a,b,z)={O(e-|z|z-m),|phz|π,O(ezz-m),π|phz|52π-δ.

For proofs see Olver (1991b, 1993a). For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).