A <span class="mathjax-formula">$q$</span>-logarithmic analogue of Euler’s sine integral

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A q-logarithmic analogue of Euler's sine integral (*)

NOBUSHIGEKUROKAWA(**) - MASATOWAKAYAMA(***)

ABSTRACT- We study aq-logarithmic analogue of Euler's sine integral proved in 1769. It is evaluated by the quantum dilogarithm function.

1. Introduction.

In [E] 1769, Euler calculated the famous integral Z^p²

0

log (sinx)dx p 2log 2:

1:1

This appears frequently in calculus texts as an example of somewhat tricky calculation. We remark that a refined formulation of (1.1) is given by the formula

Z^p²

0

log (1 e^2ix)dx p²i 8 : 1:2

(*) Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012, and by Grant-in-Aid for Exploratory Research No. 15654003

Mathematics Subject Classification 2000: 11M36

(**) Indirizzo dell'A.: Department of Mathematics, Tokyo Institute of Technol- ogy, Oh-okayama Meguro, Tokyo, 152-0033 JAPAN. e-mail: [email protected] tech.ac.jp

(***) Indirizzo dell'A.: Faculty of Mathematics, Kyushu University, Hakozaki Fukuoka, 812-8581, JAPAN. e-mail: [email protected]

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In fact, taking the real part of (1.2), we have (1.1) because

Re Z^p²

0

log (1 e^2ix)dx Z^p²

0

logj1 e^2ixjdx

Z^p²

0

log (2 sinx)dx

Z^p²

0

log (sinx)dxp 2log 2 by noticing that Re ^p₈²ⁱ 0.

We formulate a slightly generalized version. We use Euler's dilogarithm function

Li₂(x)X¹

n1

xⁿ n² and the double sine function

F(x)e^xY¹

n1

1 _n^x 1_n^x _n

e^2x

due to HoÈlder [H]. We define

F1(x)F(x) ^2pi(1 e^2pix)^2pixe^p²^(x^{2 1}⁶⁾ for Im (x)0. Then we have the

THEOREM1.1. For0<a1 Z^p²

0

log (1 ae^2ix)dx 1

2ifLi₂( a) Li₂(a)g

1

2ilog F1 loga 2pi ¹₂

F1 loga 2pi

0

@

1 A:

In the case of a1 in Theorem 1.1 we obtain (1.2) from the facts F₁(0)e ^p⁶²andF₁¹₂ e^p¹²². We can easily show these values are obtained from the definition of F1(x). Actually, the value ofF1(0) is obvious from

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F(0)1. Also, using the resultF¹₂

p2

of HoÈlder in [H], we have F1 1

2 F 1 2

2pi2^pie^p¹²²e^p¹²²:

Now we present aq-analogue of Theorem 1.1 or (1.2). Letq>1. We put

`q(x)(q 1)X¹

m1

x 1 x 1q^m

forx2C, which defines a meromorphic function inx. The function`_q(x) is considered as aq-analogue of the usual logarithm

logxX¹

m1

( 1)ⁿ ¹(x 1)ⁿ n

expanded injx 1j<1, as the following calculation shows:

`_q(x)(q 1)X¹

m1

(x 1)q ^m 1(x 1)q ^m

(q 1)X¹

m1

X¹

n1

( 1)ⁿ ¹(x 1)ⁿq ^nm

(q 1)X

n1

( 1)ⁿ ¹(x 1)ⁿ

qⁿ 1 X

n1

( 1)ⁿ ¹(x 1)ⁿ [n]_q ; where [n]_q^q_qⁿ₁¹and lim_q#1[n]_qn.

We show that aq-analogue of (1.2) is expressed as Z^p²

0

`q(1 e^2ix)dxq 1

2i log Y¹

n1

1q ⁿ 1 q ⁿ

! : 1:3

To formulate the result neatly, we recall the quantum dilogarithm (see Kirillov [K]);

Li2;q(x)X

n1

xⁿ n[n]_q: We also introduce

F_q(x)Y¹

n1

(1 q ⁿe^2pix)^q ¹: Then we have the following theorem.

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THEOREM1.2. For0<a1, it holds that Z^p²

0

`q(1 ae^2ix)dx 1

2i

Li2;q( a) Li2;q(a)

1

2ilog Fq loga 2pi ¹₂

F_q^log_2pi^a 0

@

1 A:

If we puta1 in Theorem 1.2, the equation (1.3) follows immediately from the second expression. Moreover, we have the following limit formulas which allow us to obtain (1.2) as the limitq#1 of (1.3).

PROPOSITION1.3.

limq#1(q 1) log Y¹

n1

(1q ⁿ)

!

p² 1:4 12

and

limq#1(q 1) log Y¹

n1

(1 q ⁿ)

!

p² 6 : 1:5

Furthermore, we prove the following

THEOREM1.4. SupposeImx0. Then we have limq#1F_q(x)F₁(x):

This result indeed shows that ``limq#1 (Theorem 1.2) = Theorem 1.1''.

We add one remark. Since Z^p²

0

log (sinx)dx1 4

Z^2p

0

logjsinxjdx1 4Re1

4 Z^2p

0

log (sinx)dx;

it may be also reasonable to think the quantity 1

4Re1 4

Z^2p

0

`_q(sinx)dx

as an analogue of (1.1). From this viewpoint, we obtain the following

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THEOREM1.5. For q>2 1

4 Re1 4

Z^2p

0

`q(sinx)dx p

2 X¹

n1 2nn

2 ⁿ [n]_q

(q 1)p 2

X¹

n1

1 1 2q ^m

p 1

( )

:

We note that, in the formula above, it is impossible to take the limitq#1 directly.

2. Proof of Theorem 1.1.

The former part of the theorem is straightforward. In fact, we calculate Z^p²

0

log (1 ae^2ix)dx Z^p²

0

X¹

n1

aⁿe^2inx n

! dx

X¹

n1

aⁿ n

Z^p²

0

e^2inxdx X¹

n1

aⁿ n

e^2inx 2in ^p₂

0

1 2i

X¹

n1

( a)ⁿ aⁿ n²

1 2i

Li2( a) Li2(a) :

To show the latter part of the theorem, it is sufficient to prove the following lemma.

LEMMA2.1. ForImx0

F₁(x)exp ( Li₂(e^2pix)):

This is essentially proved in [KOW] or [KK], where the theory of multiple sine functions is developed in extending the results to the double sine functions due to HoÈlder [H] (for the period (1;1)) and Shintani [S] (for the period (v1;v2)). For reader's convenience, we sketch the proof. See also ``Miscellaneous Examples'' Chap. VII-20 of Whittaker-Watson [WW], which treats indeed the double sine function F(x) of HoÈlder [H] (but without mentioning to this reference).

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PROOF OFLEMMA2.1. We first show that F(x)exp

Z^x

0

ptcot (pt)dt 0

@

1 A: 2:1

In fact, from the expression logF(x)xX¹

n1

n log 1 x n

log 1x n

2x

n o

we obtain

F⁰(x)

F(x) 1X¹

n1

n 1

x n 1 nx

2

1X¹

n1

2x²

x² n²pxcot (px):

Hence, usingF(0)1, we obtain (2.1).

It follows from (2.1) that logF(x)

Z^x

0

ptcot (pt)dt:

Suppose Im (x)0. Then we have logF(x)

Z^x

0

pite^pite ^pit e^pit e ^pitdt

Z^x

0

pit 12 e^2pit 1 e^2pit

dt

pix²

2 2piX¹

n1

Z^x

0

te^2pintdt

pi

2 x² 2piX¹

n1

xe^2pinx 2pin

e^2pinx 1 (2pin)²

pi

2 x² xX¹

n1

e^2pinx n 1

2pi X¹

n1

e^2pinx 1 n²

pi

2 x²xlog (1 e^2pix) 1

2piLi2(e^2pix)pi 12:

SinceF1(x)F(x) ^2pi(1 e^2pix)^2pixe^p²^(x^{2 1}⁶⁾, this shows the claim in Lem-

ma 2.1. p

Thus Theorem 1.1 follows.

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3. Proof of Theorem 1.2

The former part is exactly similar to the case of Theorem 1.1. Actually, we have

Z^p²

0

`q(1 ae^2ix)dx Z^p²

0

X¹

n1

aⁿe^2inx [n]_q

! dx

X¹

n1

aⁿ [n]_q

Z^p²

0

e^2inxdx 1 2i

X¹

n1

( a)ⁿ aⁿ [n]_qn

1 2i

Li_2;q( a) Li_2;q(a) :

To show the latter part, it suffices to use the following result.

LEMMA3.1. ForIm (x)0

F_q(x)exp ( Li_2;q(e^2pix)):

PROOF. This is easily seen as follows (see [KW] also):

logFq(x)(q 1)X¹

n1

log (1 q ⁿe^2pix)

(q 1)X¹

n1

X¹

m1

q ^nme^2pimx

m (q 1)X¹

m1

e^2pimx m(q^m 1)

X¹

m1

e^2pimx

m[m]_q Li_2;q(e^2pix):

Hence the claim follows. p

Thus, we obtain Theorem 1.2.

This is obvious from Lemma 2.1 and Lemma 3.1, because the formula limq#1Li2;q(x)Li2(x):

holds forjxj<1.

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5. Calculations of special values.

First, we notice that one can directly see that HoÈlder's result is equivalent to Euler's integral, that is,

F 1 2

p2

() (1:1) as follows. From (1.2) we see that

F 1

2 exp Z¹²

0

ptcot (pt)dt

!

exp

tlog (sinpt)¹₂

0

Z¹²

0

log (sinpt)dt

!

exp Z¹²

0

log (sinpt)dt

! :

Hence we observe that F 1

2

p2 ()

Z¹²

0

log (sinpt)dt 1 2log 2

() Z^p²

0

log (sinx)dx p 2log 2:

We now look at the casea1 of Theorem 1.2 and take the limitq#1.

Then we have (1.2) again by

limq#1(q 1) log Y¹

n1

1q ⁿ 1 q ⁿ

!

p² 4 :

To see this limit formula, it is enough to show that (1.4) and (1.5) in Pro- position 1.3.

PROOF OFPROPOSITION 1.3. Let us show these limit formulas by em- ploying the modular formD(t) of Ramanujan defined by

D(t)e^2pitY¹

n1

(1 e^2pint)²⁴

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for Imt>0. This has the functional equation, i.e. the automorphy ofD(see, e.g., [We]);

D 1

t

t¹²D(t):

Forq>1, taket^{log (q}_2pi¹⁾. Then, sinceD(t)q ¹Q₁

n1(1 q ⁿ)²⁴, we have log Y¹

n1

(1 q ⁿ)

!

1

24logD(t) 1 24logq

1

24log t ¹²D 1 t

1 24logq

1

2logt 1 24

2pi

t log Y¹

n1

(1 e ^2pin^t )

!

1 24logq:

Hence it follows that (q 1) log Y¹

n1

(1 q ⁿ)

!

p² 6

q 1 logq

q 1

2 logt(q 1) log Y¹

n1

(1 e ^2pin^t )

!

q 1 24 logq:

Takingq#1, we have q 1

logq !1; (q 1) logt!0; e ^2pi^t !0 and (q 1) logq!0:

Therefore we obtain

limq#1(q 1) log Y¹

n1

(1 q ⁿ)

!

p² 6 :

(This process shows that one may have more detailed asymptotics.) Then, the relation

(q 1) log Y¹

n1

(1q ⁿ)

!

(q 1) log Y¹

n1

1 q ²ⁿ 1 q ⁿ

!

1

q1(q² 1) log Y¹

n1

(1 q ²ⁿ)

!

(q 1) log Y¹

n1

(1 q ⁿ)

!

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yields

limq#1(q 1) log Y¹

n1

(1q ⁿ)

!

1 2

p² 6

p² 12:

This completes the proof of Proposition 1.3. p

Forq>2 we calculate Z^2p

0

`_q( sinx)dxX¹

n1

( 1)ⁿ ¹ [n]_q

Z^2p

0

(sinx 1)ⁿdx

X¹

n1

1 [n]q

Z^2p

0

(1 sinx)ⁿdx:

A simple calculation shows the following lemma.

LEMMA6.1.

Z^2p

0

(1 sinx)ⁿdx2p 2n n 2 ⁿ: 6:1

PROOF. We calculate as Z^2p

0

(1 sinx)ⁿdx Z^2p

0

sinx 2 cosx

2

_2n

dx Z^2p

0

2 p sin x

2 p 4

_2n

dx

2ⁿ¹ Z^3p⁴

p4

sin²ⁿudu2ⁿ² Z^p²

0

sin²ⁿudu:

Therefore, by the well-known evaluation Z^p²

0

sin²ⁿudup 2

2n n 2 ²ⁿ;

we get (6.1). p

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From this lemma we get the former part of Theorem 1.4. We next show X¹

n1 2nn

2 ⁿ

[n]_q (q 1)X¹

m1

n 1 1 2q ^m p 1o

:

Recalling the binomial expansion (1 x) ¹²X¹

n0

2n

n 2 ²ⁿxⁿ (jxj<1);

we can calculate as X¹

n1 2nn

2 ⁿ

[n]_q (q 1)X¹

n1

2n

n 2 ⁿ(qⁿ 1) ¹

(q 1)X¹

n1

X¹

m1

2n

n 2 ⁿq ^nm(q 1)X¹

n1

X¹

m1

2n

n 2 ²ⁿ(2q ^m)ⁿ

(q 1)X¹

m1

n 1 1 2q ^m p 1o

: Thus the theorem follows.

REFERENCES

[E] L. EULER, De summis serierum numeros Bernoullianos involventium.

Novi comm. acad. scient. Petropolitanae, 14 (1769), pp. 129-167 (Opera Omnia I-15, pp.91-130).

[H] O. HOÈLDER,Ueber eine transcendente Function. GoÈttingen Nachrichten, 16(1886), pp. 514-522.

[K] A. N. KIRILLOV,Dilogarithm identities, Progr. Theoret. Phys. Suppl.,118 (1995), pp. 61-142.

[KK] N. KUROKAWA - S. KOYAMA, Multiple sine functions. Forum Math., 15 (2003), pp. 839-876.

[KOW] N. KUROKAWA- H. OCHIAI - M. WAKAYAMA,Multiple trigonometryand zeta funcitons.J. Ramanujan Math. Soc.,17(2002), pp. 101-113.

[KW] N. KUROKAWA- M. WAKAYAMA,Absolute tensor products.Internat. Math.

Res. Notices,5(2004), pp. 249-260.

[S] T. SHINTANI,On a Kronecker limit formula for real quadratic fields. J.

Fac. Sci. Univ. Tokyo,24(1977), pp. 167-199.

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[We] A. WEIL,Sur une formule classique. J. Math. Soc. Japan,20(1968), pp.

400-402.

[WW] E.T. WHITTAKER - G. N. WATSON, A Course of Modern Analysis,Cam- bridge Univ. Press (Third ed.) 1920.

Manoscritto pervenuto in redazione l'8 dicembre 2004, modificato il 20 aprile 2005.