The Asymptotic Expansion of Kummer Functions for Large Values of the $a$-Parameter, and Remarks on a Paper by Olver

0.46 MB·

by  Hans Volkmer

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 046, 22 pages
The Asymptotic Expansion of Kummer Functions
for Large Values of the a-Parameter,
and Remarks on a Paper by Olver(cid:63)
Hans VOLKMER
Department of Mathematical Sciences, University of Wisconsin-Milwaukee,
P.O. Box 413, Milwaukee, WI, 53201, USA
E-mail: [e-mail]
6
1 Received January 10, 2016, in final form May 01, 2016; Published online May 06, 2016
0
http://dx.doi.org/10.3842/SIGMA.2016.046
2

y
a Abstract. ItisshownthataknownasymptoticexpansionoftheKummerfunctionU(a,b,z)
M as a tends to infinity is valid for z on the full Riemann surface of the logarithm. A corre-
sponding result is also proved in a more general setting considered by Olver (1956).

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Key words: Kummer functions; asymptotic expansions

]
A
2010 Mathematics Subject Classification: 33B20; 33C15; 41A60
C
.
h
t
a 1 Introduction
m
[ Recently, the author collaborated on a project [1] investigating the maximal domain in which an

2 integral addition theorem for the Kummer function U(a,b,z) due to Magnus [2, 3] is valid. In
v this work it is important to know the asymptotic expansion of U(a,b,z) as a tends to infinity.
3
Such an expansion is well-known, and, for instance, can be found in Slater’s book [8]. Slater’s
6
2 expansion is in terms of modified Bessel functions K (z), and it is derived from a paper by
ν
2 Olver [5]. However, there are two problems when we try to use the known result. As Temme [9]
0
pointed out, there is an error in Slater’s expansion. Moreover, in all known results the range of
.
1
validity for the variable z is restricted to certain sectors in the z-plane.
0
6 Thepurposeofthispaperistwo-fold. Firstly,wecorrecttheerrorin[8],andweshowthatthe
1 corrected expansion based on [5] agrees with the result in [9] which was obtained in an entirely
:
v different way. Secondly, we show that the asymptotic expansion of U(a,b,z) as a tends to
i
X infinity is valid for z on the full Riemann surface of the logarithm. This is somewhat surprising
r becauseoftentherangeofvalidityofasymptoticexpansionsisrestrictedbyStokes’lines. Olver’s
a
results in [5] are valid for a more general class of functions (containing confluent hypergeometric
functions as a special case.) He introduces a restriction on argz, and on [5, p. 76] he writes “In
the case of the series with the basis function K we establish the asymptotic property in the
µ
range |argz| ≤ 3π. It is, in fact, unlikely that the valid range exceeds this ...”. However, we
2
show in this paper that the restriction |argz| ≤ 3π can be removed at least under an additional
2
assumption (2.4).
In Section 2 of this paper we review the results that we need from Olver [5]. We discuss
these results in Section 3. In Section 4 we prove that Olver’s asymptotic expansion holds on the
full Riemann surface of the logarithm. Sections 5, 6 and 7 deal with extensions to more general
values of parameters. In Section 8 we specialize to asymptotic expansions of Kummer functions.
In Section 9 we make the connection to Temme [9].
(cid:63)ThispaperisacontributiontotheSpecialIssueonOrthogonalPolynomials,SpecialFunctionsandApplica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 H. Volkmer
2 Olver’s work
Olver [5, (7.3)] considers the differential equation
1 (cid:18) µ2−1 (cid:19)
w(cid:48)(cid:48)(z) = w(cid:48)(z)+ u2+ +f(z) w(z). (2.1)
z z2
The function f(z) is even and analytic in a simply-connected domain D containing 0. It is
assumed that (cid:60)µ ≥ 0. The goal is to find the asymptotic behavior of solutions of (2.1) as
0 < u → ∞.
Olver [5, (7.4)] starts with a formal solution to (2.1) of the form
∞ ∞
(cid:88) As(z) z (cid:88) Bs(z)
w(z) = zZ (uz) + Z (uz) ,
µ u2s u µ+1 u2s
s=0 s=0
where either Z = I , Z = I or Z = K , Z = −K are modified Bessel functions.
µ µ µ+1 µ+1 µ µ µ+1 µ+1
ThefunctionsA (z) = A (µ,z),B (z) = B (µ,z)aredefinedbyA (z) = 1,andthenrecursively,
s s s s 0
for s ≥ 0,
(cid:90) z(cid:18) 2µ+1 (cid:19)
2B (z) = −A(cid:48)(z)+ f(t)A (t)− A(cid:48)(t) dt, (2.2)
s s s t s
0
(cid:90)
2µ+1
2A (z) = B (z)−B(cid:48)(z)+ f(z)B (z)dz. (2.3)
s+1 z s s s
Theintegralin(2.3)denotesanarbitraryantiderivativeoff(z)B (z). ThefunctionsA (z),B (z)
s s s
are analytic in D, and they are even and odd, respectively.
If the domain D is unbounded, Olver [5, p. 77] requires that f(z) = O(|z|−1−α) as |z| → ∞,
where α > 0. In our application to the confluent hypergeometric equation in Section 8 the
function f(z) = z2 does not satisfy this condition. Therefore, throughout this paper, we will
take
D = {z: |z| < R }, (2.4)
0
where R is a positive constant. Olver [5, p. 77] introduces various subdomains D(cid:48), D , D of D.
0 1 2
We may choose D(cid:48) = {z: |z| ≤ R}, where 0 < R < R . The domain D comprises those points z
0 1
in D(cid:48) which can be joined to the origin by a contour which lies in D(cid:48) and does not cross either
the imaginary axis, or the line through z parallel to the imaginary axis. For our special D(cid:48) the
contour can be taken as the line segment connecting z and 0, so D = D(cid:48). The domain D
1 1
appears in Olver [5, Theorem D(i)]. According to this theorem, (2.1) has a solution W (u,z) of
1
the form
(cid:32)N−1 (cid:33) (cid:32)N−1 (cid:33)
(cid:88) As(z) z (cid:88) Bs(z)
W (u,z) = zI (uz) +g (u,z) + I (uz) +zh (u,z) , (2.5)
1 µ u2s 1 u µ+1 u2s 1
s=0 s=0
where
K
1
|g (u,z)|+|h (u,z)| ≤ for 0 < |z| ≤ R, u ≥ u . (2.6)
1 1 u2N 1
Remarks 2.1.
1. Theparameterµisconsideredfixed. WemaywriteW (u,µ,z)toindicatethedependence
1
of W on µ.
1
The Asymptotic Expansion of Kummer Functions 3
2. Every solution w(z) of (2.1) is defined on the Riemann surface of the logarithm over D.
Note that there is no restriction on argz in (2.6), see [5, p. 76].
3. The precise statement is this: for every positive integer N there are functions g , h and
1 1
positive constants K , u (independent of u,z) such that (2.5), (2.6) hold.
1 1
4. The functions A (z), B (z) are not uniquely determined because of the free choice of
s s
integration constants in (2.3). Even if we make a definite choice of these integration
constants, the solution W (u,z) is not uniquely determined by (2.5), (2.6). For example,
1
one can replace W (u,z) by (1+e−u)W (u,z).
1 1
5. Olver’s construction of W (u,z) is independent of N but may depend on R. In our appli-
1
cation to the confluent hypergeometric differential equation we have f(z) = z2. Then R
can be any positive number but W (u,z) may depend on the choice of R.
1
6. Olver has the term z in place of z in front of h in (2.5) but since we assume |z| ≤ R
1+|z| 1
this makes no difference.
For the definition of D we suppose that a is an arbitrary point of the sector |arga| < 1π
2 2
and (cid:15) > 0. Then D comprises those points z ∈ D(cid:48) for which |argz| ≤ 3π, (cid:60)z ≤ (cid:60)a, and
2 2
a contour can be found joining z and a which satisfies the following conditions:
(i) it lies in D(cid:48),
(ii) it lies wholly to the right of the line through z parallel to the imaginary axis,
(iii) it does not cross the negative imaginary axis if 1π ≤ argz ≤ 3π, and does not cross the
2 2
positive imaginary axis if −3π ≤ argz ≤ −π,
2 2
(iv) it lies outside the circle |t| = (cid:15)|z|.
InourspecialcaseD(cid:48) = {z: |z| ≤ R}wechoosea = R. If0 ≤ argz ≤ 3π and0 < |z| ≤ R, we
2
choose the contour starting at z moving in positive direction parallel to the imaginary axis until
we hit the circle |t| = R. Then we move clockwise along the circle |t| = R towards a. Taking
into account condition (iv), we see that D is the set of points z with −3π +δ ≤ z ≤ 3π −δ,
2 2 2
0 < |z| ≤ R, where δ > 0. The domain D appears in Olver [5, Theorem D(ii)]. According to
2
this theorem, (2.1) has a solution W (u,z) of the form
2
(cid:32)N−1 (cid:33)
(cid:88) As(z)
W (u,z) = zK (uz) +g (u,z)
2 µ u2s 2
s=0
(cid:32)N−1 (cid:33)
z (cid:88) Bs(z)
− K (uz) +zh (u,z) , (2.7)
u µ+1 u2s 2
s=0
where
K 3
2
|g (u,z)|+|h (u,z)| ≤ for 0 < |z| ≤ R, |argz| ≤ π−δ, u ≥ u . (2.8)
2 2 u2N 2 2
Note that in (2.8) there is a restriction on argz.
In the rest of this paper we choose the functions A (z) such that
s
A (0) = 0 if s ≥ 1. (2.9)
s
Then the functions A (z), B (z) are uniquely determined.
s s
4 H. Volkmer
3 Properties of solutions W and W
1 2
The differential equation (2.1) has a regular singularity at z = 0 with exponents 1±µ. Substi-
tuting x = z2 we obtain an equation which has a regular singularity at x = 0 with exponents
1(1 ± µ). Therefore, for every µ which is not a negative integer, (2.1) has a unique solution
2
W (z) = W (u,µ,z) of the form
+ +
∞
(cid:88)
W (z) = z1+µ c z2n,
+ n
n=0
where the c are determined by c = 1, and
n 0
n−1
(cid:88)
4n(µ+n)c = u2c + f c for n ≥ 1
n n−1 j n−1−j
j=0
when
∞
(cid:88)
f(z) = f z2n.
n
n=0
If µ is not an integer, then W (u,µ,z) and W (u,−µ,z) form a fundamental system of
+ +
solutions of (2.1). If (cid:60)µ ≥ 0, there is a solution W (z) linearly independent of W (z) such that
− +
W (z) = z1−µp(cid:0)z2(cid:1)+dlnzW (z),
− +
where p is a power series and d is a suitable constant. If µ (cid:54)= 0 we choose p(0) = 1. If µ is not
an integer then d = 0.
Lemma 3.1. Suppose (cid:60)µ ≥ 0. There is a function α(u) such that
W (u,z) = α(u)W (u,z),
1 +
and, for every N = 1,2,3,...,
2−µuµ (cid:18) (cid:18) 1 (cid:19)(cid:19)
α(u) = 1+O as 0 < u → ∞. (3.1)
Γ(µ+1) u2N
Proof. There are functions α (u), α (u) such that
+ −
W (u,z) = α (u)W (u,z)+α (u)W (u,z). (3.2)
1 + + − −
Suppose (cid:60)µ > 0. Then (3.2) implies
lim zµ−1W (u,z) = α (u). (3.3)
1 −
z→0+
We use [7, (10.30.1)]
2−ν
limI (z)z−ν = .
ν
z→0 Γ(ν +1)
Then (2.5), (2.6) give
lim zµ−1W (u,z) = 0. (3.4)
1
z→0+
It follows from (3.3), (3.4) that α (u) = 0.
−
The Asymptotic Expansion of Kummer Functions 5
Now suppose that (cid:60)µ = 0, µ (cid:54)= 0. Then we argue as before but instead of z → 0+ we
approach 0 along a spiral z = re±ir, 0 < r → 0, when ±(cid:61)µ > 0. Then along this spiral z2µ → 0.
We obtain again that α (u) = 0. In a similar way, we also show that α (u) = 0 when µ = 0.
− −
Therefore, (3.2) gives
lim z−µ−1W (z,u) = α (u)
1 +
z→0+
and, from (2.5), (2.6), (2.9)
2−µuµ (cid:18) (cid:18) 1 (cid:19)(cid:19)
lim z−µ−1W (u,z) = 1+O
z→0+ 1 Γ(µ+1) u2N
which implies (3.1) with α(u) = α (u). (cid:4)
+
Let us define
2−µuµ
W (u,µ,z) = W (u,µ,z).
3 +
Γ(µ+1)
Then Lemma 3.1 gives
(cid:18) (cid:19)
1
W (u,z) = α˜(u)W (u,z), where α˜(u) = 1+O .
3 1 u2N
Therefore, W admits the asymptotic expansion (2.5), (2.6), so we can replace W by W . Note
3 1 3
that in contrast to W , W is a uniquely defined function which is identified as a (Floquet)
1 3
solution of (2.1) and not by its asymptotic behavior as u → ∞.
Unfortunately, it seems impossible to replace W by an easily identifiable solution of (2.1).
2
However, we will now prove several useful properties of W .
2
Lemma 3.2. Suppose that (cid:60)µ ≥ 0. There is a function β(u) such that
W (cid:0)u,zeπi(cid:1)−eπi(1−µ)W (u,z) = β(u)W (u,z), (3.5)
2 2 3
and, for every N = 1,2,3,...,
(cid:18) (cid:18) (cid:19)(cid:19)
1
β(u) = πi 1+O as 0 < u → ∞. (3.6)
u2N
Proof. We set λ = eπi(1±µ). Equation (2.1) has a fundamental system of solutions W , W
± + −
such that
W (cid:0)zeπi(cid:1) = λ W (z), W (cid:0)zeπi(cid:1) = λ W (z)+ρW (z).
+ + + − − − +
Let w(z) = c W (z)+c W (z) be any solution of (2.1). Then
+ + − −
w(cid:0)zeπi(cid:1)−λ w(z) = ((λ −λ )c +ρc )W (z).
− + − + − +
If we apply this result to w = W we see that there is a function β(u) such that (3.5) holds.
2
Let z > 0 and set z = zeπi. We use (2.7) for z in place of z, and [7, (10.34.2)]
1 1
K (cid:0)zeπim(cid:1) = e−πiνmK (z)−πisin(πνm)I (z) (3.7)
ν ν ν
sin(πν)
6 H. Volkmer
with m = 1. Then
(cid:32)N−1 (cid:33)
(cid:88) As(z)
W (u,z ) = z(λ K (uz)+πiI (uz)) +g (u,z )
2 1 − µ µ u2s 2 1
s=0
(cid:32)N−1 (cid:33)
z (cid:88) Bs(z)
+ (−λ K (uz)+πiI (uz)) +zh (u,z ) .
u − µ+1 µ+1 u2s 2 1
s=0
Using (2.7) a second time, we find that
(cid:32)N−1 (cid:33)
(cid:88) As(z)
W (u,z )−λ W (u,z) = πizI (uz) +g (u,z )
2 1 − 2 µ u2s 2 1
s=0
(cid:32)N−1 (cid:33)
z (cid:88) Bs(z)
+πi I (uz) +zh (u,z )
u µ+1 u2s 2 1
s=0
z2
+λ zK (uz)(g (u,z )−g (u,z))−λ K (uz)(h (u,z )−h (u,z)).
− µ 2 1 2 − µ+1 2 1 2
u
We now expand the right-hand side of (3.5) using (2.5), and compare the expansions. Setting
z = R and dividing by RI (uR), we obtain
µ
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
1 1
(β(u)−πi) 1+O = O as 0 < u → ∞,
u u2N
where we used [7, (10.40.1)]
ex (cid:18) (cid:18)1(cid:19)(cid:19)
I (x) = √ 1+O as 0 < x → ∞, (3.8)
ν
2πx x
and [7, (10.40.2)]
(cid:114) (cid:18) (cid:18) (cid:19)(cid:19)
π 1
K (x) = e−x 1+O as 0 < x → ∞. (3.9)
ν
2x x
This proves (3.6). (cid:4)
Lemma 3.3.
(a) If (cid:60)µ > 0 then, for every N = 1,2,3,..., we have
(cid:12)(cid:12) (cid:32) N(cid:88)−1 B(cid:48)(0)(cid:33)(cid:12)(cid:12) (cid:18) u−µ (cid:19)
limsup(cid:12)zµ−1W (u,z)−Γ(µ)2µ−1u−µ 1−2µ s (cid:12) = O (3.10)
(cid:12) 2 u2s+2 (cid:12) u2N+2
z→0+ (cid:12) s=0 (cid:12)
as 0 < u → ∞.
(b) If (cid:60)µ = 0, µ (cid:54)= 0, (3.10) holds when we replace zµ−1W (u,z) by
2
zµ−1W (u,z)−Γ(−µ)2−µ−1uµz2µ.
2
(c) If µ = 0 then
(cid:12) (cid:12) (cid:18) (cid:19)
(cid:12)W2(u,z) (cid:12) 1
limsup(cid:12) +1(cid:12) = O .
(cid:12) zlnz (cid:12) u2N
z→0+
The Asymptotic Expansion of Kummer Functions 7
Proof. Suppose that (cid:60)µ > 0. Then we use [7, (10.30.2)]
lim xνK (x) = Γ(ν)2ν−1 for (cid:60)ν > 0.
ν
x→0+
It follows that
lim zµK (uz) = Γ(µ)2µ−1u−µ, (3.11)
µ
z→0+
1
lim zµ+1K (uz) = Γ(µ+1)2µu−µ−2. (3.12)
µ+1
z→0+ u
Using (2.7), (2.9), (3.11), (3.12), we obtain
(cid:12)(cid:12) (cid:32) N(cid:88)−1 B(cid:48)(0)(cid:33)(cid:12)(cid:12)
limsup(cid:12)zµ−1W (u,z)−Γ(µ)2µ−1u−µ 1−2µ s (cid:12)
(cid:12) 2 u2s+2 (cid:12)
z→0+ (cid:12) s=0 (cid:12)
≤ limsup(cid:12)(cid:12)Γ(µ)2µ−1u−µg2(u,z)−Γ(µ+1)2µu−µ−2h2(u,z)(cid:12)(cid:12).
z→0+
Now (2.8) gives (3.10) with N −1 in place of N. If (cid:60)µ = 0, µ (cid:54)= 0, then we use [5, (9.7)]
K (x) = Γ(µ)2µ−1x−µ+Γ(−µ)2−µ−1xµ+o(1) as 0 < x → 0
µ
and argue similarly. If µ = 0 we use [7, (10.30.3)]
K (x)
lim 0 = −1. (cid:4)
x→0+ lnx
Theorem 3.4. Suppose that (cid:60)µ ≥ 0 and µ is not an integer. There are functions γ(u), δ(u)
such that
W (u,z) = γ(u)W (u,µ,z)+δ(u)W (u,−µ,z), (3.13)
2 3 3
and, for every N = 1,2,3,...,
(cid:18) (cid:18) (cid:19)(cid:19)
π 1
γ(u) = − 1+O , (3.14)
2sin(πµ) u2N
π (cid:32) N(cid:88)−1 B(cid:48)(0) (cid:18) 1 (cid:19)(cid:33)
δ(u) = 1−2µ s +O . (3.15)
2sin(πµ) u2s+2 u2N+2
s=0
Proof. Sinceµisnotaninteger,W (u,µ,z)andW (u,−µ,z)arelinearlyindependentso(3.13)
3 3
holds for some suitable functions γ, δ. From (3.13) we get
W (cid:0)u,zeπi(cid:1)−eπi(1−µ)W (u,z) = γ(u)(cid:0)eπi(1+µ)−eπi(1−µ)(cid:1)W (u,z).
2 2 3
Comparing with Lemma 3.2, we find −2iγ(u)sin(πµ) = β(u). Now (3.6) gives (3.14).
Suppose that (cid:60)µ > 0. Then (3.13) yields
2µu−µ
lim zµ−1W (u,z) = δ(u) .
2
z→0+ Γ(1−µ)
Using Lemma 3.3(a) we obtain
(cid:32) N(cid:88)−1 B(cid:48)(0) (cid:18) 1 (cid:19)(cid:33) 2µu−µ
Γ(µ)2µ−1u−µ 1−2µ s +O = δ(u) .
u2s+2 u2N+2 Γ(1−µ)
s=0
Applying the reflection formula for the Gamma function, we obtain (3.15). If (cid:60)µ = 0, µ (cid:54)= 0,
the proof of (3.15) is similar. (cid:4)
8 H. Volkmer
4 Removal of restriction on argz
Using β(u) from Lemma 3.2 we define
πi
W (u,z) = W (u,z).
4 2
β(u)
Then we have
W (cid:0)u,zeπi(cid:1) = eπi(1−µ)W (u,z)+πiW (u,z). (4.1)
4 4 3
Moreover, (3.6)showsthatW sharestheasymptoticexpansion(2.7), (2.8)withW . From(4.1)
4 2
we obtain
W (cid:0)u,zeπim(cid:1) = eπi(1−µ)mW (u,z)+πisin(π(µ+1)m)W (u,z) (4.2)
4 4 3
sin(π(µ+1))
foreveryintegerm. Wewilluse(4.2)andtheasymptoticexpansions(2.5), (2.7)for|argz| ≤ 1π
2
to prove that in (2.8) we can remove the restriction on argz completely.
Theorem 4.1. Suppose that (cid:60)µ ≥ 0. For every N = 1,2,3,..., W (u,z) can be written as the
2
right-hand side of (2.7), and (2.8) holds without a restriction on argz:
K
2
|g (u,z)|+|h (u,z)| ≤ for 0 < |z| ≤ R, u ≥ u .
2 2 u2N 2
Proof. Without loss of generality we replace W by W . We assume that |argz| ≤ 1π, 0 <
2 4 2
|z| ≤ R, u > 0, m is an integer and z := zeπim. We insert (2.5), (2.7) on the right-hand side
1
of (4.2). Using (3.7) we obtain
N−1 N−1
(cid:88) As(z1) z1 (cid:88) Bs(z1)
W (u,z ) = z K (uz ) − K (uz ) +f(u,z), (4.3)
4 1 1 µ 1 u2s u µ+1 1 u2s
s=0 s=0
where
f = E g +E h +E g +E h ,
1 2 2 2 3 1 4 1
with
z2
E (u,z) = e−πi(µ+1)mzK (uz), E (u,z) = −e−πi(µ+1)m K (uz),
1 µ 2 µ+1
u
sin(π(µ+1)m) sin(π(µ+1)m)z2
E (u,z) = πi zI (uz), E (u,z) = πi I (uz).
3 µ 4 µ+1
sin(π(µ+1)) sin(π(µ+1)) u
We will construct functions G (u,z) and H (u,z) such that
j j
z2
E (u,z) = z K (uz )G (u,z)− 1K (uz )H (u,z)
j 1 µ 1 j µ+1 1 j
u
for j = 1,2,3,4. Then (4.3) becomes
(cid:32)N−1 (cid:33)
(cid:88) As(z1)
W (u,z ) = z K (uz ) +g (u,z)
4 1 1 µ 1 u2s 3
s=0
(cid:32)N−1 (cid:33)
z1 (cid:88) Bs(z1)
− K (uz ) +z h (u,z) , (4.4)
u µ+1 1 u2s 1 3
s=0
The Asymptotic Expansion of Kummer Functions 9
where
g = G g +G h +G g +G h , h = H g +H h +H g +H h .
3 1 2 2 2 3 1 4 1 3 1 2 2 2 3 1 4 1
We now use [7, (10.28.2)]
1
K (x)I (x)+K (x)I (x) = . (4.5)
µ µ+1 µ+1 µ
x
From (4.5) and the relation
I (cid:0)zeπim(cid:1) = eπiµmI (z) (4.6)
µ µ
we obtain
uz K (uz )eπi(µ+1)mI (uz)+uz K (uz )eπiµmI (uz) = 1.
1 µ 1 µ+1 1 µ+1 1 µ
Therefore, we can choose
G (u,z) = uzK (uz)I (uz), H (u,z) = −u2K (uz)I (uz).
1 µ µ+1 1 µ µ
We set
1+2|x|
l (x) = ln , l (x) = 1 if µ (cid:54)= 0,
0 µ
|x|
and note the estimates [5, (9.12)]
Cl (x) C|x|l (x)
µ µ
|I (x)K (x)| ≤ , |I (x)K (x)| ≤ , (4.7)
µ µ 1+|x| µ+1 µ 1+|x|2
C C
|I (x)K (x)| ≤ , |I (x)K (x)| ≤ (4.8)
µ+1 µ+1 µ µ+1
1+|x| |x|
valid when |argx| ≤ 1π with C independent of x. At this point we assume that µ (cid:54)= 0 (the case
2
µ = 0 is mentioned at the end of the proof). The estimates (4.7) give
|G (u,z)| ≤ C, |H (u,z)| ≤ Cu2. (4.9)
1 1
Similarly, we choose
G (u,z) = −z2K (uz)I (uz), H (u,z) = uzK (uz)I (uz).
2 µ+1 µ+1 2 µ+1 µ
The estimates (4.8) give
|G (u,z)| ≤ C|z|2, |H (u,z)| ≤ C. (4.10)
2 2
It follows from (3.7) that
z2
E (u,z) = −E (u,z)+z K (uz ), E (u,z) = −E (u,z)− 1K (uz ).
3 1 1 µ 1 4 2 µ+1 1
u
Therefore, we can choose
G (u,z) = 1−G (u,z), H (u,z) = −H (u,z),
3 1 3 1
G (u,z) = −G (u,z), H (u,z) = 1−H (u,z).
4 2 4 2
10 H. Volkmer
From (4.9), (4.10), we get
|G (u,z)| ≤ C +1, |H (u,z)| ≤ Cu2, (4.11)
3 3
|G (u,z)| ≤ C|z|2, |H (u,z)| ≤ C +1. (4.12)
4 4
The estimates (4.9), (4.10), (4.11), (4.12) give
|g (u,z)| ≤ C|g (u,z)|+C|z|2|h (u,z)|+(C +1)|g (u,z)|+C|z|2|h (u,z)|,
3 2 2 1 1
|h (u,z)| ≤ Cu2|g (u,z)|+C|h (u,z)|+Cu2|g (u,z)|+(C +1)|h (u,z)|.
3 2 2 1 1
Since we assumed that
K
|g (u,z)|+|h (u,z)|+|g (u,z)|+|h (u,z)| ≤
1 1 2 2 u2N
for |argz| ≤ 1π, 0 < |z| ≤ R, u ≥ u , the expansion (4.4) has the desired form with N replaced
2 0
by N −1.
Suppose µ = 0. We use [7, (10.31.2)]
(cid:18) (cid:18)1 (cid:19) (cid:19) 1x2 (cid:18) 1(cid:19) (cid:0)1x2(cid:1)2
K (x) = − ln x +γ I (x)+ 4 + 1+ 4 +··· . (4.13)
0 2 0 (1!)2 2 (2!)2
It follows from (4.13) that there exist positive constants r > 0, D > 0 such that
|K (x)| 1
0 ≤ D for 0 < |x| ≤ r, |argx| ≤ π, m ∈ Z.
|K (xeπim)| 2
0
Then we set
K (uz)
0
G (u,z) = , H (u,z) = 0 if 0 < |uz| ≤ r
1 1
K (uz )
0 1
with G and H the same as before when |uz| > r. The estimates (4.9) are valid with a suitable
1 1
constant C. The rest of the proof is unchanged. This completes the proof of the theorem. (cid:4)
5 Extension to complex u
So far we considered only 0 < u → ∞. Now we set u = teiθ, where t > 0 and θ ∈ R. In (2.1) we
substitute z = e−iθx, w˜(x) = w(z). Then we obtain the differential equation
d2 w˜(x) = 1 d w˜(x)+(cid:18)t2+ µ2−1 +e−2iθf(cid:0)e−iθx(cid:1)(cid:19)w˜(x). (5.1)
dx2 xdx x2
Assuming (cid:60)µ ≥ 0, we can apply Olver’s theory to this equation, and obtain functions W˜ (t,x)
1
and W˜ (t,x). Since we assumed that f(z) is analytic in the disk {z: |z| < R }, the new function
2 0
f˜(x) = e−2iθf(e−iθx) is analytic in the same disk. Therefore, the domains D , D are the same
1 2
as before. The functions A˜ (x), B˜ (x) that appear in place of A (z), B (z) satisfy
s s s s
A˜ (x) = e−2siθA (z), B˜ (x) = e−(2s+1)iθB (z),
s s s s
so
A˜ (x) A (z) B˜ (x) B (z)
s s s s
= , = .
t2s u2s t2s+1 u2s+1