PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY
Physical and Mathematical Sciences
2009, № 2, p. 3–7
Mathematics
ON FUNCTIONS SEMI-ANALYTICAL IN THE POLYDISK
A. I. PETROSYAN∗, N. T. GAPOYAN
Chair of Matemathical Analysis, YSU
In the present paper the class of semi-analytical functions in the polydisk
U n ⊂ n is introduced. This class is an extension of the set of holomorphic functions. For n = 1 the concept of semi-analyticity coincides with analyticity. The
Dirichlet problem with values given on the distinguished boundary of the polydisk
always has a solution in the set of real parts of semi-analytical functions. Therefore, to investigate semi-analytical functions one can apply the potential theory
methods, like one does it for the one-dimensional case. In the present paper the
Schwarz type integral representation for the above-mentioned functions is obtained.
Keywords: polydisk, n-harmonic function, pluriharmonic function, the
Schwarz formula.
Introduction. Let function u be pluriharmonic in the domain D ⊂
satisfies the conditions
∂ 2u
∂ 2u
∂ 2u
∂ 2u
+
= 0,
−
=0
∂xi ∂x j ∂yi ∂y j
∂xi ∂y j ∂x j ∂yi
n
, i.e. it
(1)
( i, j = 1,… , n , if i = j the equations of second group are trivial). The function u is
called n -harmonic (doubly harmonic in the case n = 2 ), if only for i = j the conditions (1) are satisfied, i.e.
∂ 2u ∂ 2u
+
= 0, i = 1,… , n .
(2)
∂xi 2 ∂yi 2
The conditions (2) imply harmonicity in each variable zi = xi + iyi .
The pluriharmonic functions are connected with holomorphic functions of
several variables the same way as harmonic functions defined on plane domain
with holomorphic functions of one variable. In the case of one variable this connection enables to apply methods of the potential theory in the complex analysis. In
the multidimensional case the situation is more difficult: the class of pluriharmonic
(unlike harmonic) functions is too narrow in sense that the Dirichlet problem is not
always solvable in this class. This fact makes it difficult to generalize some onedimensional results for higher dimension case.
∗
E-mail: [email protected]
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2009, № 2, p. 3–7.
4
S. Bergman [1] offered the following idea for n = 2 : to put into correspondence with any doubly harmonic function u ( z1 , z2 ) a complex-valued function
f ( z1 , z2 ) = u ( z1 , z2 ) + iv( z1 , z2 ) so that:
1) f ( z1 , z2 ) is holomorphic in z1 for a fixed z2 ;
2) v(0, z2 ) ≡ 0 .
The obtained class of functions was called the extended class of complex
functions. These functions may be studied with the help of potential theory methods, since the Dirichlet's problem with values given on the distinguished boundary is always solvable in the class of all doubly harmonic functions. However this
class is not an extension of the class of holomorphic functions, as holomorphic
functions do not necessarily satisfy the condition 2.
In [2, 3] a modified version of the Bergman class (class of semi-analytical
functions) is introduced, which has the following advantage: in that special case,
when the real part of semi-analytical function is pluriharmonic, the function itself
is holomorphic. Thus, any holomorphic function is also semi-analytical.
In the present paper the concept of semi-analyticity is defined for functions
of arbitrary number of variables (Definition 1). For such functions we obtain an
integral representation, which is an analogue of the well-known Schwarz integral
representation for one variable case.
Notation. We use the following notations:
U n = z = ( z1 ,…, zn ) ∈ n : zk < 1, k = 1,…, n
{
}
is the unit polydisk in n -dimensional complex space
n
{
T = z∈
n
n
,
: zk < 1, k = 1,…, n
is its distinguished boundary, P ( z ) =
}
1− z
is one-dimensional Poisson kernel,
1− z
Q( z ) is the function harmonically conjugate to P ( z ) ,
1− z
S ( z ) = P ( z ) + iQ ( z ) =
1+ z
is the Schwarz kernel.
Definition 1. A function f ( z ) = f ( z1 , z2 ,… , zn ) defined in U n is called
semi-analytical, if
a) the function Re f ( z ) is n -harmonic;
b) for fixed zk +1 ,…, zn the functions f ( 0,…,0, zk , zk +1 ,…, zn ) are holomorphic in the disk zk < 1 , k = 1,… , n .
Note that in the one-dimensional case (i.e. n=1) the n -harmonicity is simply
harmonicity and the semi-analyticity coincides with analyticity.
Integral representation. The following theorem is an analogue of the
Schwarz formula for semi-analytical functions.
T h e o r e m 1 . Let the function f ( z ) be semi-analytical in the unit polydisk
U n , and
ρ1 ,…, ρ n
be arbitrary numbers from
(0,1) . Then for any
z ∈ { z = ( z1 ,… , zn ) : zk < ρ k , k = 1,… , n} the following formula
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2009, № 2, p. 3–7.
f ( z1 ,…, zn ) = iv ( 0,…,0) +
⎛ z1
1
−iθ1
∫ Sn ⎜ e
(2π ) T ⎝ ρ1
n
,…,
n
5
⎞
e−iθn ⎟ u ρ1e−iθ1 ,…, ρne−iθn dθ1
ρn
⎠
(
zn
)
dθn
is true, where
n
n
( )
n
( ) ∏ P(z )
S n ( z1 ,… , zn ) = ∏ P z j + i ∑ Q z j
j =1
j =1
(3)
k
k = j +1
is the n -dimensional analogue of the Schwarz kernel. Note that for n = 1 the function S n ( z1 ,… , zn ) coincides with the usual Schwarz kernel: S n = S .
Proof. For fixed zk , | zk |< 1, k = 2,… , n , the function f ( z1 , z2 ,… , zn ) is
analytical in the disk | z1 |< 1 . By the Schwarz formula (see, for instance, [3]) we
obtain
1 2π ⎛ z1 −iθ1 ⎞
f ( z1 ,… , zn ) = iv ( 0, z2 ,… , zn ) +
S ⎜ e ⎟ u ρ1eiθ1 , z2 ,… , zn dθ1 . (4)
∫
2π 0 ⎝ ρ1
⎠
(
)
Then for a fixed zk , | zk |< 1, k = 3,… , n , the function f (0, z2 ,… , zn ) is analytical in the disk | z2 |< 1 . Hence
1 2π ⎛ z2 −iθ2 ⎞
iθ
∫ S ⎜ e ⎟ u 0, ρ2 e 2 , z3 ,…, zn dθ 2 . (5)
2π 0 ⎝ ρ 2
⎠
Reasoning by analogy, on the j -th step we obtain
(
f ( 0, z2 ,… , zn ) = iv ( 0,0, z3 ,… , zn ) +
(
)
)
(
)
f 0,… , 0, z j ,… , z n = iv 0,… ,0, z j +1 ,… , z n +
⎞
iθ
⎟⎟ u 0,… , 0, ρ j e j , z j +1 ,… , z n dθ j , j = 1, 2,… , n.
0
⎝ j
⎠
Since the function u is n -harmonic, hence for the fixed ρ1eiθ1 we have
+
⎛ zj
2π
1
2π
∫ S ⎜⎜ ρ
e
− iθ j
(
)
⎛z
⎞
⎛z
⎞
1
P ⎜ 2 e−iθ2 ⎟ P ⎜ n e−iθn ⎟ u ρ1eiθ1 ,…, ρneiθn dθ2
n −1 ∫
(2π ) T n−1 ⎝ ρ2
⎠
⎝ ρn
⎠
The obtained result and (4) imply
f ( z1 ,… , z n ) = iv ( 0, z 2 ,… , z n ) +
(
(
)
u ρ1eiθ1 , z2 ,…, zn =
+i
1
2π
⎛ z1
⎞ ⎛z
⎞
e − iθ1 ⎟ P ⎜ 2 e − iθ 2 ⎟
⎝ 1
⎠ ⎝ ρ2
⎠
∫ Q⎜ ρ
Tn
⎛z
P ⎜ n e − iθ n
⎝ ρn
)
⎞
iθ
iθ
⎟ u ρ1e 1 ,… , ρ n e n d θ1
⎠
(
)
dθn .
dθ n + (6)
⎛z
⎞
⎞
e − iθ1 ⎟ P ⎜ n e − iθ n ⎟ u ρ1e iθ1 ,… , ρ n eiθ n d θ1 d θ n .
(2π ) T n ⎝ 1
⎠
⎝ ρn
⎠
Equating the imaginary parts in (5), we get
1 2π ⎛ z2 −iθ2 ⎞
v ( 0, z2 ,… , zn ) = v ( 0,0, z3 ,… , zn ) +
Q ⎜ e ⎟ u 0, ρ 2 eiθ2 , z3 ,… , zn dθ 2 . (7)
∫
2π 0 ⎝ ρ 2
⎠
+
⎛ z1
1
n
(
∫ P⎜ ρ
)
(
)
Using the integral representation of the function f ( 0, z2 ,…, zn ) (which is
holomorphic in z2 ) and separating again the imaginary parts, we obtain
v ( 0,0, z3 ,… , zn ) = v ( 0,0,0, z4 ,… , zn ) +
1
2π
2π
⎛ z3
⎞
e − iθ3 ⎟ u 0,0, ρ 3eiθ3 , z4 ,… , zn dθ 3 .
⎝ 3
⎠
∫ Q⎜ ρ
0
(
)
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2009, № 2, p. 3–7.
6
Continuing in a similar way and substituting the obtained formulas consecutively in (7), we get
1 n 2π ⎛ zk −iθk ⎞
v ( 0, z2 ,…, zn ) = v ( 0,0,…,0 ) +
∑ ∫ Q ⎜ e ⎟ u 0,…,0, ρk eiθk , zk +1 ,…, zn dθk .
2π k = 2 0 ⎝ ρk
⎠
(
)
(8)
Taking into account the n -harmonicity of u , we receive
(
)
u 0,…,0, ρ k eiθk , zk +1 ,…, zn =
⎛z
⎞
⎛z
⎞
1
P ⎜ k +1 e −iθk +1 ⎟ P ⎜ n e−iθn ⎟ u 0,…,0, ρ k eiθk ,…, ρ n eiθn dθ k +1
n−k ∫
(2π ) T n−k ⎝ ρ k +1
⎠
⎝ ρn
⎠
Due to the mean value theorem
(
=
(
)
u 0,…,0, ρ k eiθk ,…, ρ n eiθn =
)
(
)
1
u ρ1eiθ1 ,… , ρ n eiθn dθ1
k −1 ∫
(2π ) T k −1
From the last two equalities it follows that
(
dθ n .
dθ k −1.
)
u 0,…,0, ρk eiθk , zk +1 ,…, zn =
⎛z
⎞
⎛z
⎞
1
P ⎜ k +1 e−iθk +1 ⎟ P ⎜ n e−iθn ⎟ u ρ1eiθ1 ,…, ρn eiθn dθ1
n −1 ∫
(2π ) T n−1 ⎝ ρk +1
⎠
⎝ ρn
⎠
Substituting this equality into (8), we obtain
(
=
1
k = 2 2π
n
v ( 0, z2 ,… , zn ) = v ( 0,0,… ,0 ) + ∑
2π
⎛ zk
∫ Q⎜ ρ
0
)
⎝
k
(
+
dθn .
⎞
⎛z
⎞
1
e −iθk ⎟
P ⎜ k +1 e −iθk +1 ⎟
n −1 ∫
⎠ (2π ) T n−1 ⎝ ρ k +1
⎠
⎛z
⎞
P ⎜ n e −iθn ⎟ u ρ1eiθ1 ,… , ρ n eiθn dθ1
⎝ ρn
⎠
or
dθk −1dθk +1
)
dθ n ,
v ( 0, z2 ,…, zn ) = v ( 0,0,…,0 ) +
1
⎛z
n
(2π )
n
∑ ∫ Q ⎜ ρk
k =2 T n
⎝
k
⎞ ⎛z
⎞
e−iθk ⎟ P ⎜ k +1 e−iθk +1 ⎟
⎠ ⎝ ρk +1
⎠
⎛z
⎞
P ⎜ n e−iθn ⎟ u ρ1eiθ1 ,…, ρn eiθn dθ1
⎝ ρn
⎠
(
)
dθ n .
From here and from (6) we have
f ( z1 ,…, zn ) = iv ( 0,0,…,0) +
+i
+i
+
1
⎛z
n
(2π )
Q⎜ k
n ∑ ∫
ρ
k =2 T n
⎝
⎛ z1
1
k
∫ Q⎜ e
(2π ) T ⎝ ρ1
n
⎛ z1
∫ P⎜ e
(2π ) T ⎝ ρ1
n
−iθ1
n
1
n
⎞ ⎛z
⎞
e−iθk ⎟ P ⎜ k +1 e−iθk +1 ⎟
⎠ ⎝ ρk +1
⎠
−iθ1
⎞ ⎛ z2 −iθ2 ⎞
⎟ P⎜ e ⎟
⎠ ⎝ ρ2
⎠
⎞
⎟
⎠
⎛z
⎞
P ⎜ n e−iθn ⎟ u ρ1eiθ1 ,…, ρn eiθn dθ1
⎝ ρn
⎠
(
)
⎛z
⎞
P ⎜ n e−iθn ⎟ u ρ1eiθ1 ,…, ρn eiθn dθ1
⎝ ρn
⎠
(
)
⎛z
⎞
P ⎜ n e−iθn ⎟ u ρ1eiθ1 ,…, ρn eiθn dθ1
⎝ ρn
⎠
(
)
dθn +
dθn +
dθn .
Taking into account (3) from the last equality, we obtain the required integral representation. The Theorem 1 is proved.
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2009, № 2, p. 3–7.
7
T h e o r e m 2 . The imaginary part of an arbitrary semi-analytical function
f = u + iv is a n -harmonic function.
Proof. Separating the imaginary parts of the left-hand and right-hand sides in
the formula of Theorem 1, we obtain
⎛z
⎞
z
1
v ( z1,…, zn ) = v ( 0,…,0) +
Im Sn ⎜ 1 e−iθ1 ,…, n e−iθn ⎟ u ρ1e−iθ1 ,…, ρne−iθn dθ1 dθn .
n ∫
ρn
(2π ) T n
⎝ ρ1
⎠
(9)
According to the definition (3) of the kernel S n , we have
(
)
⎛z
⎞ n ⎛ z j −iθ ⎞ n
⎛z
⎞
z
Im S n ⎜ 1 e −iθ1 ,… , n e −iθn ⎟ = ∑ Q ⎜ e j ⎟ ∏ P ⎜ k e −iθk ⎟ ,
(10)
⎟ k = j +1 ⎝ ρ k
ρn
⎝ ρ1
⎠ j =1 ⎜⎝ ρ j
⎠
⎠
whence it is clear that the left-hand side of this formula is n -harmonic in the polydisk { z = ( z1 ,… , zn ) : zk < ρ k , k = 1,… , n} . Taking into account that ρ1 ,… , ρ n are
arbitrary numbers from (0, 1), we obtain the assertion of the Theorem from (9) and
(10).
Received 24.02.2009
REFERENCES
1.
2.
3.
4.
Bergman S. The Kernel Function and Conformal Mappings, Mathematical Surveys. Amer.
Math. Soc., 1970, p. 214–218.
Petrosyan A.I. Izv. AN Arm. SSR. Mat., 1974, v. 9, № 1, p. 3–13 (in Russian).
Petrosyan A.I. Mathematics in Higher School, 2007, v. 3, № 2, p. 37–43 (in Armenian).
Shabat B.V. Introduction to Complex Analysis. Part 2. M.: Nauka, 1985 (in Russian).
8
Proc. of the Yerevan State Univ. Phys. and Mathem. Sci., 2009, № 2, p. 3–7.
Պոլիդիսկում կիսաանալիտիկ ֆունկցիաների մասին
Աշխատանքում ներմուծվում է U n ⊂ n միավոր պոլիդիսկում կիսաանալիտիկ ֆունկցիաների դաս, որը հոլոմորֆ ֆունկցիաների դասի ընդլայնում է:
դեպքում կիսաանալիտիկության հասկացությունը համընկնում է
n =1
անալիտիկության հետ: Պոլիդիսկի հենքի վրա տրված արժեքների համար
Դիրիխլեի խնդիրը միշտ ունի լուծում կիսաանալիտիկ ֆունկցիաների իրական
մասերի բազմության մեջ: Ուստի, ինչպես և միաչափ դեպքում, կիսաանալիտիկ
ֆունկցիաները հետազոտելիս մենք կարող ենք կիրառել պոտենցիալի
տեսության մեթոդները: Հոդվածում այդ ֆունկցիաների համար ստացված է
Շվարցի ինտեգրալային բանաձևը:
О функциях, полуаналитических в полидиске
В работе вводится класс полуаналитических в полидиске U n ⊂ n функций,
являющийся расширением множества аналитических функций. В случае n = 1 понятие полуаналитичности совпадает с аналитичностью. Проблема Дирихле с заданными на основе полидиска значениями всегда имеет решение в классе вещественных
частей полуаналитических функций. Поэтому, как и в одномерном случае, при исследовании полуаналитических функций можно применять методы теории потенциалов.
Для этих функций в работе получено интегральное представление Шварца.