Computation of an integral of a rational function over the skeleton of unit polycylinder in Cn by Means of the Mellin transform Текст научной статьи по специальности «Математика»

УДК 517.55 + 519.1

Computation of an Integral of a Rational Function over the Skeleton of Unit Polycylinder in Cn by Means of the Mellin Transform

Georgy P. Egorychev* Viacheslav P. Krivokolesko^

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 30.11.2017, received in revised form 24.01.2018, accepted 06.03.2018 With the help of the Mellin transform we give a simple calculation of an integral of rational functions in several independent parameters aerlier appeared in [2]. The efficiency of this transform is due to the fact that calculation the degree of the polynomial acts as the degree of a monomial. In 2008, G. P. Egorychev and E.V. Zima [5] for the first time successfully used the Mellin transform in the theory of rational summation. The possibility of its application in the analysis and computation of integrals with different types of rational functions is discussed.

Keywords: integral representations, Mellin transform, combinatorial identities. DOI: 10.17516/1997-1397-2018-11-3-364-369.

Introduction

In [1] the author obtained an integral representation in bounded n-circular linearly convex domains with piecewise regular boundary. This integral representation is a sum of terms, each of which contains multiple integrals of the following form:

J = ^»-(¿F X I- I --. (1)

li.l- li.l-1 S ■ ■ ■ ■ ■ « n Kiz.i, + ... + alrnzn(n + C,)•,

j = 1

Here £ = ..., £n) G Cn, s = (siy ■ ■ ■, sn) G Nq , A = (a,q), a,q G C is a matrix of dimension m x n, t = (t1,... ,tn) G Nn.

We emphasize that for |£i| = l,. ■ ■, |£n| = 1 the complex parameters z = (z1,..zn) satisfy the conditions

m

X\(aj,izi £1 + ■ ■ ■ + a,,nZn£n + c, ) = 0- (2)

j-i

Condition (2) implies that for j = 1, ■ ■ ■ ,m the point z = (z1: ■ ■ ■, zn) G Cn is not in the union of a family of complex hyperplanes

{Vj} = {u = (ui, ■■■,Un) G Cn :a,,iui£i + ■ ■ ■+a,,nUn£n+c, =0, |£i| = 1,■■■, |£n| = 1}^ (3)

* gegorych@mail.ru tkrivokolesko@gmail.com © Siberian Federal University. All rights reserved

1. Necessary definitions and notation

Because with every point u = (u1;..., un ) G {Vj } all the points (u1e,.unel^n ), 0 < < 2n, l = 1,... ,n also belong to {Vj }, the family of complex hyperplanes {Vj } is a n-circular set. If a complex hyperplane a1u1 + ... + anun + c = 0 passes through the point u0 = (u0,..., vPn),

then the complex hyperplane a1 e

-i^-i

0ei^ = (u0ei^

ue

u1 e

,0 ei^ri

).

ui + ... + ane

-i^n

+ c = 0 passes through the point

Note that a complex hyperplane a1ei'1 u1 + ... + anei'nun + c = 0 has real dimension 2n — 2 and is the intersection of two mutually perpendicular real hyperplanes (of dimension 2n — 1). We find that for 0 ^ < 2n, l = 1,... ,n a family of complex hyperplanes a1ei'1 ui+ + ... + anev'nun+c = 0 is n-circular and is a ruled surface of real dimension 2n—1 which splits Cn = R2n into two disjoint sets.

If the point 2 = (z1?..., zn) G Cn do not belong to a family of complex hyperplanes a1 e1'1 u1 +

+.. .+anei^n un+c = 0, 0 < ^l < 2n, l = 1,.. .,n, then all points zeiv = (z1el

') GCn,

0 < < 2n, l = 1,.. .,n do not belong to this family of complex hyperplanes. That is, the set of points 2 = ... ,zn) £ Cn that do not belong to the family of complex hyperplanes is also an n-circular set. Now we can investigate the mutual position of two n-circular sets in Cn using the projection of Cn in R+ according to

n : (z1, ...,zn) ^ (|z1|,..., \zn\). Let a complex hyperplane V in Cn be given by the equation:

V = {u = (u1,. .., un) G Cn : a^1 + ... + anun + c = 0}.

(4)

(5)

A description of the projection (5) to R++ of a complex hyperplane of a1u1 + ... + anun + c = 0 in Cn is given in [2] (Proposition 4.3). Let \V\ = n(V). It is given by the system of inequalities:

|V| =

+ |a1 ||u11 - |a2||u2| - |a3||u31 - ... - \anl\uri\ - |c| < 0,

— |a1 ||u11 + |a2||u2| - |a3||u31 - ... - \an\\un\ - \c\ < 0,

— |a1 ||u11 - |a2||u2| - |a3||u31 - ... - \an\\un\ + \c\ < 0.

(6)

If 1 = 1,..., |£n| = 1, then the projection n maps each hyperplane of {V} = {a1^1u1 +... + +an£nun + c = 0} to the same projection in R+, i.e. n(V)=n({V}). The system of inequalities (6) 'splits' the points of R+ into n +1 disjoint parts:

|a1 ||u 1 \ - |a2||u2| - |a3||u3| |a1 ||u 1 \ + |a2||u2| - |a3||u3|

. . . an un

an un

■|a1 ||u 1 \ - |a2||u2| - |a3||u3| "|a1 ||u 1 \ - |a2||u2| - |a3||u3|

+ |an||un| -

|c| > 0, (n1)

|c| > 0, (n2)

\c\ > 0, (n„)

|c| > 0. (nn+1)

(7)

If ak =0 in (5), then nk = 0. If in (5) c = 0, then nn+1 = 0.

If the point z = (z1,... ,zn) G Cn does not belong to the family of complex hyperplanes (6), then n(z) belongs to one of n1;..., nn+1.

Consider a family of complex hyperplanes Vj = {z G Cn : aj,1u1 +... + aj,nun + cj = 0}, j = 1,... ,m. Let (nk )j be the set of points of R+ defined by the inequality |aj,1||u1| — |aji2||u2| — — ... — |aji(fc_1)||ufc_1| + ^j,j,k Huk | — |aj,(fc+1)||ufc+1| — ... — ^nHu^ — c > 0.

u

n

n

zn e

au

nn

Let for an n-circular set G the set |G| = n(G) in R+ belong to (nfcl)i n ... n {nkm)m. Then the collection of sets V1,... ,Vm; G consider the corresponding set of numbers (k1,... ,km), where 1 ^ kj < n +1. The result of calculation of the integral (1) depends on this set of numbers.

Note that the number of variants of mutual location of the point z = (z1,... ,zn) that does not belong to the family of the complex hyperplanes {Vij,..., {Vm} equals to (n + 1)m.

One approach to evaluation of the integral (1) is the method of binomial expansion of fractions 1/(a,ji1z1£1 + ... + a,jnnzn£n + cj)tj based on one of the inequalities (7).

In [3] the integral (1) is computed in the case when the corresponding set of numbers is (n + 1,... ,n +1), that is, when

-\aj,i\\zi\ - ■■■ - \aj,n\\zn\ + jcj > 0, j = l,...,m.

(8)

Here in Theorem 1 we compute (1) also for the case (n + 1,... ,n + 1) using the method of coefficients [4], and the Mellin transform for the function under the integral sign. Note that the calculation results in both cases coincide.

2. Proof of the theorem

Theorem 1. If the conditions (2) and (8) are satisfied, then the following formula is valid:

J

(-l)(Sl + '" + Sn) • zl1 • ... • ZSnn

(ti - 1)! • ... • (tm - 1)! • 41 • ... • cm

111 + ••• ml =Sl 111 lln+--- + lmn = Sn li __n 11 . . n m1 __n 1

x E

l11,...,lm1eNo

En 1 n . . n^mn

a1n . . . amn

. , , , , U ! • • 1 !

"11« • • • vmi- , , ,—*R\T L1n' • • • lmn•

l1nv?lmnfcNo

x (ti - 1 + (1n + ...+ 1in))! (tm - 1+- {1m1 + ...+ 1mn))! (9)

(l11+...+l1n) C1

If = 1,..., \£n\ = 1, then according to (2) and (8) we have

aj,1Z1^1 + ... + a,j,nZn£n

(lm1 + ... + lmn)

cm

Let us denote

< 1, j = 1,. .. ,m.

, j = 1,. .. ,m, q = 1, ... ,n.

If ^^ = 1,..., \£n\ = 1, then according to (10) it follows

\àj,1Z1£1 + ... + a,j,nZn£n\ < 1, j = 1,...,m.

Then

and

Let us denote

J

-1 < Re(aj,1Z1£1 + ... + aj,nZnèn) < 1, j = 1,...,m.

-1 < 0 < Re(aj,1Z1è1 + ... + aj,nZnèn + 1) < 2, j = 1, .. .,m.

C-l • Cm

l £?1+1...

ei1+1... enn+1 n (aj,1Z1^1 +... + ajnZnèn + 1)'j j=1

,

(10) (11)

(12)

(13)

(14)

(15)

X

C

j

C

j

1

1

where the formal residue res{^(£1 j ■ ■ ■ ,£n)} is the coefficient at £ x ... x £n)-1 of the formal power Laurent series A(£1,...,£n) containing only finitely many terms with negative powers (see [4]).

According to the general scheme of the method of coefficients [4] we substitute in (15) each factor under the sign res^ by the known Mellin formula

1 1 fl

— = --- e-azzj-1dz, Re a > 0, (16)

aj (j - 1)Uq ' ' V ^

and according to (16) and (12) we have

1 1

fOO

(àj1Z1£1 + ... + âjnZn£n + 1)j (tj - 1)Uq j j

j = 1,..., m,. Thus

7 _1_ f 1

J = —-;-;-—-;-— • resii,...,i J "S^i-—rr x

1 • ... • cm • (t1- 1)! • ... • (tm - 1)! • £ri + 1...£nn+1

{ f[f0 e-(à>-iZi!ii+...+~a>nzn«n+1)wjwjj-1dwjJJ ' 1 { 1

c1 • ... • cm • (t1 - 1)! •... • (tm - 1)! • resii'-'^ { tf+l ...£n

r^ rl - E (5jiziÇi + ... + àj„z„Ç„+1)wj ( 1) (; _1) }

... e j=i • w( i ) • .. . • wmm 1) • dw1 A .. . A dwm> =

)q jq

l fix - w

c1 • ... • cm • (t1 - 1)! • ... • (tm - 1)! Jq Jq

p

e p=i x

n i 1 zq^< wj 5™1 (t,- 1) (t _1) x [[ res^l ^q+Y e j=i \ • w1 i '• ... • w(T 1) • dw1 A ... A dw„t =

q=1 ^ £q9 '

(resx{eax fxk+1} = ak/k!)

1 f1 fl - E wp (-Z1)si • (E™=1 wj àj,1)si

= ~t-Ï--■ ■ ■ e p=i--j--... x

ct1i • ... • Cm • (t1 - 1)! • ... • (tm - 1)! Jq Jq S1!

x (-Zn)Sn • (E;"—1 wj àj,n)Sn ^ w(;i-1) • ... • wmm-1) • dw1 A ... A dwm =

Q I

n

liq + ...+lmq— sq . _ , , j

(Since (w101q + ... + wmàmq )sq = £ Sq ''W^i^^T T* ^ ' we have)

11 q ,...,lmq £=Nq

(-1)(si+...+s") • ZSi • ... •

-i Sn fl fl

Z1 ^ ^ ^ Zn ' ' e- wp ,w(;i-1) .■■■.w^t-1);

c\i • ■■■ • cm • (t1 - 1)! • ■■■ • (tm - 1)! Jq Jq

hi + ... + lmi—Si (w1à11)1ii^.11^_(wmàm1)ÎTi I11! • ■ ■ ■ • lm1 !

x Z) ' ' "'• ■—'• • ■■■x

iii,...,imi£NQ

lin+...+lmn—Sn )'in . . (w a )Tn

(w1a1n) • ■ ■ ■ • (wmamn) 7 a A 7

> -;-;- • dw1 A ■ ■ ■ A dwm

h ! • • I ! 1

li„,...,lm„eNQ I1n! ■■■ Imn!

(-1)(si + ... + Sn) • ZSsi • ■ ■ ■ • ZSn

c1 ■ ■ ■ cm

(t1 - 1)! •■■■•(tm - 1)!

X

1

l11 + ... + lm1=s1 l1n + ... + lmn = sn ~ln ~lm1 ~hn ~lmn

a 11 • ... • am 1 a 1 n • . . . • amn

X ^ ... L 11^ ... •Im 1 ! '...'l 1 J. • ...•lmn ! X

l11 ,...,lm1€No hn,...,lmn€No 11 m1 ^ ^

x j e 11 ......"dw1 I

E

/ e-Wl w<f1-1+ll1+~+lln) dwi ... ■ e-Wm wmr-1+lm1+-+'mn)dv

Jo Jo

( 00 ■ (j —

(the computation of each factor in the last expression by the formula: f e-azzj-1dz = -:—

V 0 aj )

(-1)(S1+...+Sn) •Zl1 •...•.

C1 . . . Cm

(t1 - 1)! • ... •(tm - 1)!

l11 + ... + lm1=s1 l1n + ...+lmn=sn ~l11 ~lm1 ~hn 7,lmn

a11 • ... • am1 • • a1n • ... • amn

l11,...,lm1€No l1n,...,lmn€No ^ ^ ^ ^ ' ^ ^ ^ ^ ^ ' 1mJ

x(t1 - 1 + I11 + .. . + hn)! • ... • (tm - 1 + 1m1 + ... + lmn)! =

(the change of the variables (11))

(-1)(S1 + ... + Sn) •Zl1 •... •

Z! •... • Znn

C

i1 •...•№ •(U - 1)! •... •(tm - 1)!

/ \ 1ll / \ lm1 / \ lln / \ lmn

ll1+...+lm1=Si lln + ... + lmn = Sn (^H) • • . . Smn. )

^ V CW ' " V y CW ' " y cm J

X ^ ... lnl. ....¡m1\ -...' ¿1n\. ....¡mn\ X

X(t1 - 1 + I11 + .. . + l1n)! ■ ... ■ (tm - 1 + lm1 + .. . + lmn)! = = (-1)(sl + ... + sn) ^ ■...■z'n^

hl + ... + lml=Sl hn + ...+lmn=Sn ¡n lml Iln almn

v a11 ■ . . . ■ am1 a1n ■ ... ■ amn w

X ^ ln!^ ... ■lm1! lmh ...■lmn! X

(t1 — 1 + l11 + ... + l1n)! (tm — 1 + lm1 + ... + lmn)!

c(lll+...+lln) (lml + ... + lmn)

c1 cm

Remark 1. If the conditions a1z1£1 + ... + anzn^n + c = 0 are fulfilled for 1 = 1,..., |£n| = 1 and -|a1||z1| — ... — |an||zn| + |c| > 0; then from (9) we get

1 f f d£1 A ... A d£n

(2ni)nJ | ?l |=1"J | u | = 1 esl + 1 ■ . .. ■ an+1(a1 z1^1 + ... + anznin + c)t

= ( — 1)(sl+...+sn) ■zS1 ■...■znn aj1 ■...■ asnn (t — 1 + (81 + ... + 8n))! = (t — 1)! ■ct s1! ■ ... ■ sn! c(sl+...+Sn)

Corollary. The following combinatorial identity is valid:

1 lll+...^^l=sl (t1 — 1 + (h1 + ... + hn))! x

. (17)

(t1 - 1)! (tm - 1)! ^ 111 !• ...•1m1!

V 1 7 V m ' l11,...,lm1ENo 11 m1

x hn+. -Un~Sn (tm - 1 + (Im1 + ... + lmn))! = (t1 + .. . + tm - 1 + (s1 + ... + Sj))! (18)

11n! • ... • 1mn! (t1 + ... + tm 1)! ^s1! • ... • sn!

l1n,...,lmn£No

ZSn

n

Conclusion

Earlier in [5] the method of coefficients and the Mellin transform were first effectively used in the theory of rational summation. Here we give an original example of application of the same apparatus for computing multiple integrals of a certain type depending on parameters and obtain new non-trivial combinatorial identities.

Authors consider a possibility of application of the Mellin transform in analysis and calculation of integrals of different type of rational functions arising in various areas of mathematics.

References

[1] V.P.Krivokolesko, Integral representations for linearly convex polyhedra and some combinatorial identities, J. Sib. Fed. Univ. Math. Phys, 2(2009), no. 2, 176-188 (in Russian).

[2] M.Forsberg, M.Passare, A.Tsikh, Laurent Determinants and Arrangements of Hyperplane Amoebas, Advances in Mathematics, 151(2000), 45-70.

[3] V.P.Krivokolesko, Method for Obtaining Combinatorial Identities with Polynomial Coefficients by the Means of Integral Representations J. Sib. Fed. Univ. Math. Phys., 9(2016), no. 2, 192-201.

[4] G.P.Egorychev, Integral representation and the computation of combinatorial sums, Transl. of Math. Monogr., vol. 59, Amer. Math. Soc. Providence, RI, 1984, 2-nd Ed. 1989.

[5] G.P.Egorychev, E.V.Zima, Integral representation and algorithms for closed form summation, Handbook of Algebra, Ed. M. Hazewinkel, Elsevier, 459-529.

Вычисление интеграла от рациональной функции по остову единичного полицилиндра в Cn с помощью преобразования Меллина

Георгий П. Егорычев Вячеслав П. Кривоколеско

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

С помощью преобразования Меллина приведено простое вычисление одного кратного интеграла от рациональной функции от нескольких независимых параметров, возникшего в 'работе [2]. Эффективность этого преобразования обусловлена тем, что в вычислениях степень полинома выступает как степень монома. Ранее в [5] преобразование Меллина впервые было успешно использовано в теории рационального суммирования. Рассматривается возможность применения преобразования Меллина при анализе и вычислении интегралов различного типа от рациональной функции.

Ключевые слова: интегральные представления, преобразования Меллина, комбинаторные тождества.