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13 Confluent Hypergeometric FunctionsKummer Functions

§13.7 Asymptotic Expansions for Large Argument

Contents
  1. §13.7(i) Poincaré-Type Expansions
  2. §13.7(ii) Error Bounds
  3. §13.7(iii) Exponentially-Improved Expansion

§13.7(i) Poincaré-Type Expansions

As x

13.7.1 𝐌(a,b,x)exxabΓ(a)s=0(1a)s(ba)ss!xs,

provided that a0,1,.

As z

13.7.2 𝐌(a,b,z)ezzabΓ(a)s=0(1a)s(ba)ss!zs+e±πiazaΓ(ba)s=0(a)s(ab+1)ss!(z)s,
12π+δ±phz32πδ,

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

13.7.3 U(a,b,z)zas=0(a)s(ab+1)ss!(z)s,
|phz|32πδ.

§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=|b2a|. Magnify
13.7.4 U(a,b,z)=zas=0n1(a)s(ab+1)ss!(z)s+εn(z),

where

13.7.5 |εn(z)|,β1|εn(z)|2αCn|(a)n(ab+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 z𝐑1,z𝐑2𝐑¯2,z𝐑3𝐑¯3,

respectively, with

13.7.8 σ =|(b2a)/z|,
ν =(12+1214σ2)1/2,
χ(n) =πΓ(12n+1)/Γ(12n+12).

Also, when z𝐑1𝐑2𝐑¯2

13.7.9 α =11σ,
β =1σ2+σ|z|12(1σ),
ρ =12|2a22ab+b|+σ(1+14σ)(1σ)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)=zas=0n1(a)s(ab+1)ss!(z)s+Rn(a,b,z),

and

13.7.11 Rn(a,b,z)=(1)n2πzabΓ(a)Γ(ab+1)(s=0m1(1a)s(ba)ss!(z)sGn+2abs(z)+(1a)m(ba)mRm,n(a,b,z)),

where m is an arbitrary nonnegative integer, and

13.7.12 Gp(z)=ez2πΓ(p)Γ(1p,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|zm),|phz|π,O(ezzm),π|phz|52πδ.

For proofs see Olver (1991b, 1993a). For the special case phz=±π see Paris (2013). For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).