On the Voros Coefficient for the Whittaker Equation with a Large Parameter Some Progress around Sato s Conjecture in Exact WKB Analysis

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1 Publ RIMS Kyoto Univ 47 20, DOI 02977/PRIMS/39 On the Voros Coefficient for the Whittaker Equation with a Large Parameter Some Progress around Sato s Conjecture in Exact WKB Analysis by Tatsuya Koike and Yoshitsugu Takei Abstract Generalizing Sato s conjecture for the Weber equation in exact WKB analysis, we explicitly determine the Voros coefficient of the Whittaker equation with a large parameter By using our results we also compute alien derivatives of WKB solutions of the Whittaker equation at the so-called fixed singular points of their Borel transform 200 Mathematics Subject Classification: 34M30, 34M37, 34M55, 34M60 Keywords: exact WKB analysis, Whittaker equation, Voros coefficient, alien derivative, Stokes automorphism Introduction In this article we study a Schrödinger equation d2 dx 2 + η2 Qx, η ψ = 0 η > 0 a large parameter with the potential Qx, η of the form Qx, η = 4 α x + η 2 β x 2 α, β complex constants, that is, the Whittaker equation with a large parameter η, from the viewpoint of exact WKB analysis This is a contribution to the special issue The golden jubilee of algebraic analysis Communicated by M Kashiwara Received January 29, 200 Revised April 2, 200 T Koike: Department of Mathematics, Graduate School of Science, Kobe University, -, Rokkodai, Nada-ku, Kobe , Japan; koike@mathkobe-uacjp Y Takei: RIMS, Kyoto University, Kyoto , Japan; takei@kurimskyoto-uacjp c 20 Research Institute for Mathematical Sciences, Kyoto University All rights reserved

2 376 T Koike and Y Takei In [AKT2 Aoki, Kawai and the second author of the present article studied analytic properties of the Borel transforms of WKB solutions of an MTP equation, that is, a Schrödinger equation with a merging pair of simple turning points Making use of the reduction of an MTP equation to a canonical one, we determined the location of fixed singular points ie, singular points whose relative locations with respect to the reference singular point are unchanged; cf [DDP, [DP of Borel transformed WKB solutions and succeeded in explicitly computing the alien derivative of WKB solutions at fixed singular points for an MTP equation In the case of an MTP equation the canonical equation is given by the Weber equation and what we call Sato s conjecture, that is, the explicit form of the Voros coefficient of the Weber equation in terms of the Bernoulli numbers cf [KT; see also [SS and [T for its proof, played an important role in studying analytic properties of Borel transformed WKB solutions See, eg, 23 in Section 2 for the definition of the Voros coefficient On the other hand, as is emphasized in [Ko and [Ko2, the behavior of a WKB solution of a Schrödinger equation near a simple pole is similar to that near a simple turning point Inspired by these results, in a joint work [KKKoT with Kamimoto and Kawai we extended the analysis developed in [AKT2 to an MPPT equation, that is, a Schrödinger equation with a merging pair of a simple pole and a simple turning point, again by making use of the reduction of an MPPT equation to a canonical one In the case of an MPPT equation the canonical equation is given by the Whittaker equation and this motivates our interest in the Whittaker equation, and the computer-assisted study of its Voros coefficient indicated that it should be possible to write it down again in terms of the Bernoulli numbers The aim of this paper is to show that it is really the case and to analyze the analytic structure of Borel transformed WKB solutions of the Whittaker equation by using the explicit form of the Voros coefficient thus obtained The paper is organized as follows: In Section 2 we present our main result, ie, the precise formulation of the counterpart of Sato s conjecture for the Whittaker equation In Section 3 we prove the main result The proof follows an idea used in [T Finally in Section 4, by using the main result we study the analytic structure of the Borel transform of WKB solutions of the Whittaker equation 2 Main Theorem The equation we discuss in this article is the Whittaker equation with a large parameter η: 2 { d2 dx 2 + η2 4 α x + β } η 2 x 2 ψ = 0

3 Voros Coefficient for the Whittaker Equation 377 Here α 0 and β are complex constants Our main interest is in the analysis of a WKB solution, that is, a formal solution of 2 of the form x 22 ψ = exp Sx, η dx, where 23 Sx, η = ηs x + S 0 x + η S x + is a formal solution of the Riccati equation 24 S 2 + ds dx = η2 4 α x + β η 2 x 2 associated to 2 Here we briefly review the construction of WKB solutions cf [KT2 By substituting 23 into 24, we obtain the following recursion relations: 25 S 2 = 4 α x, j=0 2S S 0 + ds dx = 0, 2S S + S0 2 + ds 0 dx = β x 2, n 2S S n + S j S n j + ds n = 0 n 2 dx It follows from 25 that the leading term S x of 23 should be 29 S x = ± 4 α x Once we fix the sign in 29, ie, the branch of /4 α/x, the higher order terms {S n } n 0 are determined uniquely and recursively by In particular, the sign in 29 is inherited only by the odd degree terms Therefore there exist two formal solutions of 24 of the form 20 S ± x, η = ±S odd x, η + S even x, η, where Since S odd x, η = η 2j+ S 2j x, S even x, η = η 2j S 2j x j=0 2 S even x, η = d 2 dx log S oddx, η j=0

4 378 T Koike and Y Takei see Remark 22 in [KT2, we thus obtain two WKB solutions of 2, x 22 ψ ± x, η = ± Sodd x, η exp S odd x, η dx x 0 In exact WKB analysis cf [V, [DP, [KT2, we give an analytic meaning to these WKB solutions 22 or 22 through the Borel resummation method As was explained in Introduction, the following theorem plays a crucial role in studying the analytic structure of Borel transformed WKB solutions Theorem 2 Let V be the Voros coefficient of the Whittaker equation 2, that is, 23 V := S odd x, η ηs x dx = η 2n S 2n x dx 4α n= Then the following relation holds as a formal power series in η : 24 V = 2n B 2n γ 2n 2n n= Here γ is a constant satisfying γγ + = β and B n z denotes the Bernoulli polynomial of degree n defined by n n! 25 B n z = k! n k! B k z n k, k=0 where {B k } k 0 are the Bernoulli numbers defined by the generating function 26 t e t = t n B n n! n=0 Remark i The lower endpoint x = 4α of the integration interval on the righthand side of 23 is a simple turning point of 2 Since S n x behaves like Ox 4α 3n/2 near x = 4α, S 2n x n 0 is not integrable near 4α We consider the integral in 23 as a contour integral along a path γ shown in Figure, ie, V = S odd x, η ηs x dx 2 γ 4α Figure

5 Voros Coefficient for the Whittaker Equation 379 Note also that, since we can show 27 S n x = O x n/2 x by induction on n, S n x with n is integrable near infinity ii The first few Bernoulli numbers are 28 B 0 =, B = 2, B 2k+ = 0 k =, 2,, B 2 = 6, B 4 = 30, B 6 = 42, B 8 = 30, B 0 = 5 66 iii The generating function of the Bernoulli polynomials is given by te tz e t = n=0 B n z tn n! Formula 28 is obtained by using the definition 25 of the Bernoulli polynomials and the generating function 26 of the Bernoulli numbers iv It is also known that the Bernoulli polynomial satisfies B n z = n B n z In view of this relation we can readily find that the 2n-th Bernoulli polynomial B 2n z is a function of zz Therefore the right-hand side of 24 is a function of β, not depending on the choice of a solution γ of γγ + = β Theorem 2 is the counterpart of Sato s conjecture for the Whittaker equation It is proved in Section 3 and by using Theorem 2 we study some analytic properties of the Borel transforms of WKB solutions in Section 4 Note that our results play an essential role in analyzing the fixed singular points of an MPPT equation in [KKKoT see also [KKKoT2 for a review 3 Proof of Main Theorem We prove Theorem 2 using the idea employed in [T Firstly, we derive the difference equation that the Voros coefficient satisfies In this derivation the raising and lowering operators with respect to the parameter α play a crucial role Then, as the difference equation determines a solution uniquely, we compute its solution to verify that it coincides with the right-hand side of 24 In this paper we use the Borel transformation technique to compute a unique solution of the difference equation 3 Derivation of the difference equation In this section the parameter α plays an important role To show the dependence on α more manifestly, we use the notation S ± x, α, η for solutions 20 of the

6 380 T Koike and Y Takei Riccati equation, ψ ± x, α, η for WKB solutions, etc in what follows We also set 3 Iα, η := S odd x, α, η ηs x, α dx = 2V In this subsection we prove Proposition 3 Iα, η formally satisfies γ Iα + η, η Iα, η = 2 + log α2 + η α η 2 β α [ 2 = 2 + log γ + γ log log + In Proposition 3 the word formally means the equality of formal power series with respect to η In fact, the right-hand side of 32 has a formal power series expansion + 6β α 2 η β α 3 η 3 0β2 + 20β α 4 η 4 + 5β2 + 5β + 2 5α 5 η 5 + by the formal use of the Taylor expansion Figure 2 To prove Proposition 3, we first note that S 2n x, α for n =, 2, are single-valued around x = 4α and their residues are zero cf 2 Therefore 34 Iα, η = S + x, α, η ηs x, α S 0 x, α dx γ In studying the right-hand side of 34, we also consider the following cut-off integrals: I x α, η = S + x, α, η dx, γ x Inα, x η = S n x, α dx n =, 0, γ x

7 Voros Coefficient for the Whittaker Equation 38 where γ x is the contour shown in Figure 2 From 34 and the definition of I x and I x n, we have 35 Iα, η = lim x [Ix α, η ηi x α ηi x 0 α We then determine the asymptotic behavior of I x α + η, η I x α, η Lemma 3 As x tends to infinity, we have 36 I x α + η, η I x α, η = log α2 + η α η 2 β x 2 + O x To prove Lemma 3, we use the raising and lowering operators of the Whittaker equation with respect to α: We also set Aα = η d dx + 2 x α, A α = η d dx 2 x + α Lα = d2 dx 2 η2 4 α x + β η 2 x 2 Lemma 32 If ψ is a solution of the Whittaker equation 2 ie Lαψ = 0, then Lα η Aαψ = 0 and Lα + η A αψ = 0 Proof By direct computations, we have η 2 x 2 Lα = Aα + η A α + cα = A α η Aα + cα η with Therefore cα = α 2 + η α η 2 β η 2 x 2 Lα + η A αψ = [A αaα + η + cαa αψ = [A αaα + η A α + cαa αψ = [A α{η 2 x 2 Lα cα} + cαa αψ = 0 By the same reasoning we can also prove η 2 x 2 Lα η Aαψ = 0 Proof of Lemma 3 Lemma 32 implies that there exists a constant Cη for which 37 η x ddx 2 x + α ψ + x, α, η = Cηψ + x, α + η, η Since the left-hand side of 37 is η xs + 2 x + α ψ + x, α, η,

8 382 T Koike and Y Takei the logarithmic derivatives of both sides of 37 give S + x, α + η, η = S + x, α, η + d dx log [ η xs + x, α, η 2 x + α Therefore we obtain 38 S + x, α + η, η dx S + x, α, η dx γ x γ x = log [η xs + x, α, η 2 x + α log [η xs x, α, η 2 x + α To determine the asymptotic behavior of the right-hand side of 38, we use the following explicit formulas as x which are easily obtained from the recursion relations 25 28: S x, α = 4 α x = 2 α x α2 x 2 2α3 x 3 + O x 4, S 0 x, α = 4 x = α x 4α x 2 + O x 3, S x, α = 5 6 x/2 x 4α 5/2 + 8 x /2 x 4α 3/2 + β + 3 x 3/2 x 4α /2 6 = β x 2 + O x 3 These asymptotic formulas together with 27 show S + x, α, η = η 2 α x α2 x 2 + α + η β 39 x 2 + O x 5/2, 30 S x, α, η = η 2 + αx + α2 x 2 + α η β x 2 + O x 5/2 The formula 36 readily follows from 38, 39 and 30 This completes the proof of Lemma 3 Next we determine the asymptotic behavior of I x α + η I x α and I x 0 α + η I x 0 α Lemma 33 i As x tends to infinity, we have I α+η x I α x = η [ 2+2+ log + ii I x 0 α + η I x 0 α = 0 +log α2 x 2 +O x

9 Voros Coefficient for the Whittaker Equation 383 Proof Since S 0 x is single-valued near x = 4α, I0 x α is 2πi times the residue of S 0 x at x = 4α Hence I0 x α = iπ/2 and thus we obtain ii To prove i, we compute the integral explicitly: I α x = S dx γ x = 2x 4 α [ x α log x 2α + 2x 4 α x [ + α log x 2α 2x 4 α x As x tends to infinity, we have 4 α x = 2 α x + α2 x 2 + O x 3, and hence Therefore we obtain x 2α + 2x 4 α x x 2α 2x I x α = = 2x 4α 2α2 x + O x 2, 4 α x = 2α2 x + O x 2 γx S x, α dx = x 2α + α log α2 x 2 + O x This asymptotic behavior of I x α verifies i We now prove Proposition 3 Proof of Proposition 3 Lemmas 3 and 32 give [I x α + η, η ηi x α + η I x 0 α + η [I x α, η ηi x α I x 0 α = [I x α + η, η I x α, η η[i α x + η I α x = log + + log α2 + η α η 2 β α 2 + O x This relation and 35 give 32 Since β = γγ +, we can also verify α 2 + η α η 2 β α 2 = γ + γ + This proves 33

10 384 T Koike and Y Takei 32 Determination of the Voros coefficient Let V α, η be the Voros coefficient defined by 23 From Proposition 3 and 3, we have 3 V α + η, η V α, η = + [ 2 log γ + γ + + log + In this subsection we verify that this difference equation determines V uniquely To this end we convert 3 to a difference equation with respect to η by using the homogeneity of V α, η Lemma 34 Let Then V 2n α = 4α S 2n x, α dx n =, 2, V 2n α = α 2n V 2n for n =, 2, Proof Using the recursion relations 25 28, we can verify by induction on n Therefore 4α for n =, 2, and entails S n αx, α = α n+ S nx, n =, 0,, 2, S 2n x, α dx = α Thanks to Lemma 34, n= 4 S 2n αt, α dt = α 2n S 2n t, dt V α, η = V 2n αη 2n = V 2n 2n, n= α + η η = + = α η + α 32 V α + η, η = V α, η + α From 32 and 3 we obtain the following difference equation for V with respect to η 4

11 Voros Coefficient for the Whittaker Equation 385 Proposition V α, η + α V α, η = + 2 log [ γ + γ + + log + We solve the difference equation 33 by using the Borel transformation We first note that 33 has a unique formal solution of the form V = V n η 2n = V η + V 3 η 3 + V 5 η 5 + n= Hence 33 determines V uniquely We now explicitly compute its Borel transform V n V B = 2n 2! y2n 2 n= by considering the Borel transform of 33 The Borel transform of the left-hand side of 33 is 34 e y/α V B α, y To compute the Borel transform of the right-hand side, we use Lemma 35 Let C be a non-zero complex constant Then i B[log + Cη y = e Cy y ii B[η log + Cη + + Cye Cy Cy = y 2 Here B denotes the Borel transformation with respect to η Lemma 35 can be proved by straightforward computations By using Lemma 35 we can compute the Borel transform of the right-hand side of 33 in the following way: [ 35 B + log + [ = B α η log + = α [ y 2 + y e y/α e y/α α y = α e y/α y 2 y, [ B log + α

12 386 T Koike and Y Takei [ 36 B 2 log γ + γ + = [log 2 B γ + 2 [ B log + γ + = 2 eγy/α y + 2 e γ+y/α y = y 2 eγy/α + e γ+y/α y We conclude from 34, 35 and 36 that the difference equation 33 becomes e y/α V B α, y = α e y/α y 2 2 eγy/α + e γ+y/α y after the Borel transformation This equation can be easily solved and we obtain Theorem 3 The Borel transform V B of the Voros coefficient V is given by 37 V B α, y = 2y e γy/α + e γ+y/α α e y/α y 2 Thus, to prove Theorem 2, it suffices to show that the Borel transform of the right-hand side of 24 coincides with the right-hand side of 37 The Borel transform of the right-hand side of 24 is, by its definition, 38 n= n=0 α 2n B 2n γ y 2n 2 = α 2n! y 2 n= B 2n γ 2n! y α 2n We then use 28 to obtain B 2n z 39 2n! t2n = [ te tz 2 e t + te tz e t = t2 etz + e tz e t From 38 and 39 we can confirm that the Borel transform of the right-hand side of 24 is exactly the same as the right-hand side of 37 This proves Theorem 2 Remark Since e γy/α + e γ+y/α = 2 [ey/α + e γy/α + e γy/α + e y/α e γy/α e γy/α = e y/α + cosh γy α + ey/α sinh γy α,

13 Voros Coefficient for the Whittaker Equation 387 we have the following expression for V B : V B = [ 2 y e y/α + γy cosh e y/α α + γy sinh y α 2α y 2 This expression was used in [KKKoT and [KKKoT2 The following lemma will be used in the subsequent section Proposition 33 V B has poles at y = 2mπiα m Z \ {0} and has no other singular points The poles are all simple with residues Res V B = e2mπiγ + e 2mπiγ y=2mπiα 4mπi 4 Analytic structure of the Borel transform of WKB solutions for the Whittaker equation In this section we derive some important analytic properties of the Borel transforms of WKB solutions of the Whittaker equation All of them are obtained as consequences of our main theorems Theorems 2 and 3 For simplicity we restrict our considerations in this section to the case where arg α = π/2 or arg α is sufficiently close to π/2 The configuration of Stokes curves, ie, integral curves of the direction field I S x dx = 0 emanating from turning points, of the Whittaker equation 2 for arg α = π/2 is shown in Figure 3 Note that the origin also plays the same role as a turning point cf [Ko and [Ko2 We also show in Figure 4 and Figure 6 resp, Figure 5 and Figure 7 the configuration of Stokes curves for arg α = π/2 ε resp, arg α = π/2 + ε for sufficiently small positive ε Figure 3 arg α = π/2

14 388 T Koike and Y Takei Figure 4 arg α = π/2 02 Figure 5 arg α = π/ Figure 6 arg α = π/2 045 Figure 7 arg α = π/ As we can observe from these figures, degeneration occurs in the configuration of Stokes curves for arg α = π/2 Figure 3, ie, there exists a Stokes curve connecting a turning point 4α and the origin in Figure 3 Although this degeneration is resolved in Figures 4 7, the topological configuration of Stokes curves is quite different between, for example, Figure 6 and Figure 7; in particular, the configuration of a Stokes curve emanating from 4α abruptly changes at arg α = π/2 We will see in 42 that this abrupt change of the configuration of Stokes curves is related to Stokes phenomena for WKB solutions that occur when α varies 4 Fixed singularities of the Borel transforms of WKB solutions Let us consider WKB solutions normalized at a simple turning point 4α: 4 ψ ± = [ x exp ± S odd dx Sodd 4α

15 Voros Coefficient for the Whittaker Equation 389 These WKB solutions ψ + and ψ can be expanded as with y ± x = ± ψ ± = e ηy±x x 4α n=0 S x dx = ± and the Borel transform ψ ±,B of ψ ± is given by 42 ψ ±,B x, y = n=0 ψ ±,n xη n /2 x 4α 4 α x dx ψ ±,n x Γn + /2 y + y ±x n /2 Theorem 4 Assume that the path of integration in 4 does not cross any Stokes curve for any α with arg α π/2 < ε ε > 0 Then the Borel transform of the WKB solution ψ +,B resp, ψ,b has singular points at y = y + x + 2mπiα resp, y = y x+2mπiα with m Z\{0} Furthermore, their alien derivatives there are given by y= y±x+2mπiαψ ±,B x, y = e2mπiγ + e 2mπiγ ψ +,B x, y 2mπiα 2m Note that the singular points described in Theorem 4 are fixed singular points cf [DDP, [DP of ψ + resp, ψ since their relative locations with respect to the reference singular point y = y + x resp, y = y x are unchanged when x varies In this subsection we prove Theorem 4 Since the same reasoning holds also for ψ,b, we only consider ψ +,B We refer the reader to [Sa for the definition of the alien derivative and the relevant alien calculus employed in what follows See also [AKT2, 3 where a similar discussion is given for the Weber equation To prove Theorem 4 we use the relation [ 43 ψ + = exp S odd ηs x dx ψ + = e V ψ +, 4α where ψ + is a WKB solution normalized at infinity: 44 ψ + = [ x [ x exp η S x dx exp S odd x, η ηs x dx Sodd 4α The formal relation 43 becomes ψ +,B = B[e V ψ +,B

16 390 T Koike and Y Takei after the Borel transformation, where denotes the convolution product for the variable y Since V B is singular at y = 2mπiα m Z \ {0} as we have seen in Section 3, we then find that y = 2mπiα is also a singular point of ψ +,B To compute the alien derivative of ψ + there, we use Lemma 4 The WKB solution 44 normalized at infinity is Borel summable if the path of integration from infinity to x in 44 does not cross any Stokes curve See [DP, Theorem 22c for the corresponding result for the Weber equation Lemma 4 for the Whittaker equation can be verified in a similar manner Under our assumption we can choose a path of integration from infinity to x in such a way that it does not cross any Stokes curve see Remark below Hence ψ +,B is holomorphic at y = y +x + 2mπiα This enables us to compute the alien derivative of ψ +,B at y = y + x + 2mπiα as follows: y= y+x+2mπiαψ +,B = y= y+x+2mπiα[be V ψ +,B This completes the proof of Theorem 4 = [ y= y+x+2mπiαbe V ψ +,B = [ y= y+x+2mπiαv B Be V ψ +,B = 2πi Res y=2mπiα [V Bα, y ψ +,B = e2mπiγ + e 2mπiγ ψ +,B 2m Remark If x does not lie on a Stokes curve for arg α = π/2, we can find a path of integration from infinity to x so that it does not cross any Stokes curve for all α with arg α π/2 being sufficiently small Examples of such paths are shown in Figures 8 0 For example, for x j j =, 2 in Figure 8, γ j gives a path satisfying the above condition 42 Derivation of the Stokes automorphism for WKB solutions It follows from Theorem 4 that, when arg α = π/2, the singular points y = y + x 2mπiα m =, 2, of the Borel transform of the WKB solution ψ +,B are located on 45 {y; Ry R y + x, Iy = I y + x} As 45 is nothing other than the path of integration for the Borel sum of ψ +, this implies that ψ + is not Borel summable when arg α = π/2 If we let the parameter α vary from arg α = π/2 ε to arg α = π/2 + ε, these singular points cross the path of integration for the Borel sum Therefore, we conclude that the WKB

17 Voros Coefficient for the Whittaker Equation 39 Figure 8 arg α = π/2 Figure 9 arg α = π/2 02 Figure 0 arg α = π/ solution ψ + suffers a Stokes phenomenon with respect to α at arg α = π/2 Such Stokes phenomena are well described by the Stokes automorphism cf [Sa In this subsection we compute the action of the Stokes automorphism on the WKB solutions ψ ± Let S denote the Stokes automorphism In our case, it is defined by [ Sψ + = exp m= [ Sψ = exp m= y= y+x 2mπiα y= y x 2mπiα ψ +, ψ Theorem 42 When arg α = π/2, Sψ ± are explicitly given by Sψ ± = e 2πi+γ /2 e 2πi γ /2 ψ ±

18 392 T Koike and Y Takei Proof We compute the action of the Stokes automorphism in the formal model Let be the sum of the alien derivatives: = We then obtain ψ + = m= y= y+x 2mπiα B [ y= y+x 2mπiαψ +,B m= e 2mπiγ + e 2mπiγ = B ψ +,B x, y + 2mπiα 2m m= e 2mπiγ + e 2mπiγ = e 2mπi ψ + 2m m= = 2 log[ e2πi+γ e 2πi γ ψ + Here B denotes the Borel transformation Therefore we obtain Sψ + = e ψ + = e 2πi+γ /2 e 2πi γ /2 ψ + The same computation can also be done for ψ 43 Borel sum of the Voros coefficient In this final subsection we determine the Borel sum of the Voros coefficient Theorem 43 Let ɛ > 0 be sufficiently small Then the Borel sum of the Voros coefficient V B for arg α = π/2 ε is given by 46 Γ γ log + Γ + γ log + log + γ + log, 2 2π 2 2π 2 and for arg α = π/2 + ε it is 47 2 Γ γ log Γ + γ log 2π 2 2π log + γ + log + πi 2 Proof Making a change of variables w = y/α, we obtain αv B = [ e w 2w e w eγw + e w e γw w 2 = 2w e w + 2 e γw + w 2w + sinhγw 2w + coshγw w 2 e w + 2 w e γw

19 Voros Coefficient for the Whittaker Equation 393 Let S[V denote the Borel sum of the Voros coefficient V, ie, 0 S[V α, η := 0 e yη V B α, y dy = 0 e αwη αv B α, αw dw To compute S[V α, η, we make use of the following formula: e t + 2 tθ dt Γθ e = log θ log θ + θ Rθ > 0 t t 2π 2 cf [E, Vol I, Chapter I, 9, 5 We also use the following formulas for Laplace transforms of hyperbolic functions: 0 0 pt sinhat e dt = t 2 log p + a p a, pt coshat e t 2 dt = a 2 log p + a p a + p 2 log Then, when arg α = π/2 ε, we find S[V α, η = [ Γηα γ ηα γ 2 2π 2 = [ Γηα + γ 2π ηα + γ 2 a2 p 2 logηα γ + ηα γ logηα + γ + ηα + γ ηα + γ log ηα γ + γ ηα + γ log 2 ηα γ + ηα 2 log γ2 η 2 α 2 Γ γ log + Γ + γ log 2π 2 2π + log + γ + log 2 When arg α = π/2 ε, we can compute S[V α, η in a similar manner by using the relation S[V α, η = S[V α, η Using the concrete expressions 46 and 47 for the Borel sum of V given by Theorem 43, we can prove Theorem 42 without resorting to the alien calculus We conclude this final section with a different proof of Theorem 42 which is based on Theorem 43 Since both ψ + and V are Borel summable for arg α = π/2 ± ε, it follows from 43 that S[ψ + arg α=π/2 ε = exp [ S[V arg α=π/2 ε S[ψ + arg α=π/2 ε, S[ψ + arg α=π/2+ε = exp [ S[V arg α=π/2+ε S[ψ + arg α=π/2+ε,

20 394 T Koike and Y Takei where S denotes the Borel sum Note that ψ + is Borel summable for arg α [π/2 ε, π/2 + ε for a sufficiently small positive ε in view of Lemma 4 Thus we have S[ψ + arg α=π/2+ε = S[ψ + arg α=π/2 ε Therefore, we obtain 48 S[ψ + arg α=π/2+ε S[ψ + arg α=π/2 ε = exp [ S[V arg α=π/2+ε S[V arg α=π/2 ε The right-hand side of 48 can be explicitly computed thanks to Theorem 43: 49 S[V arg α=π/2+ε S[V arg α=π/2 ε Since and = 2 log Γ γγ + γγ γγ + γ 2π 2 2 logγ2 α 2 η 2 + πi Γ γγ + γ 2π Γ + γγ γ 2π the right-hand side of 49 becomes = = + γ 2 sin π γ γ 2 sin π + γ, 2 log 4α2 η 2 γ 2 sin π + γ sin π γ γ 2 α 2 η 2 + πi Hence we obtain = 2 log[eiπ+γ e iπ+γ e iπ γ e iπ γ + πi = 2 log[ e2iπ+γ e 2iπ γ S[ψ + arg α=π/2+ε = e 2iπ+γ /2 e 2iπ γ /2 S[ψ + arg α=π/2 ε This proves Theorem 42 Acknowledgments We would like to express our sincere gratitude to Professor Takahiro Kawai and Professor Takashi Aoki for their kind encouragement and stimulating discussions The research of the authors has been supported in part by JSPS grants-in-aid No , No and No S

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