J. Rajasthan Acad. Phy. Sci., Vol. 5, No. 1, June 2006, pp.131-142 © Printed in India
BICOMPLEX GAMMA AND BETA FUNCTION S.P. GOYAL*, TRILOK MATHUR** and RITU GOYAL*** Department of Mathematics, University of Rajasthan, Jaipur-302004 (India) e-mail:
[email protected] e-mail:
[email protected] e-mail:
[email protected] Abstract: The aim of this paper is to define bicomplex Gamma and Beta functions. We also discuss conditions of validity and T-holomorphicity for these functions. Various properties including Legendre duplication formula, Gauss multiplication theorem and Binomial theorem are established. These functions which are believed to be new will provide a fundamental tool to develop the theory of bicomplex special functions. Key words: bicomplex number, Idempotent basis, Gamma and Beta functions.
1. Introduction, Definitions and Preliminaries In search for development of special algebras, Segre1 published a paper in 1892, in which he treated an infinite set of algebras whose elements, he called bicomplex numbers, tricomplex,…, n-complex numbers. The algebras of quarternions and bicomplex numbers are developed by making use of so called complex pairs. One 2
3
can also refer to the recent works by Price and Rönn for other details of bicomplex 4
algebras and function theory. Rochon has developed a bicomplex Riemann-Zeta 5
function while Goyal and Goyal have recently developed bicomplex Hurwitz-Zeta function. Motivated by these works, in this paper we extend the domain of the well known Gamma and Beta functions from the set of complex numbers to the set of bicomplex numbers. We also discuss various properties connected with these functions. (a) Bicomplex Number 3
The bicomplex number (see, e.g. Rönn ) is defined as a i b i c jd (a, b, c, d R) 1
2
2
2
(1)
2
where i1 i 2 1 j 1 i 2 j j i 2 i1 i1 j j i1 i 2 and i1i 2 i 2i1 j can also be written as ( a i1b) i 2 (c i1d ) ,
(a, b, c, d, R)
(2)
or
z1 i 2 z 2
(z1 , z 2 C1)
(3)
S.P. Goyal, Trilok Mathur and Ritu Goyal
132
where z a i b , z c i and 1 1 2 1 2 C1 x i1y : i1 1 x, y R} is the set of complex numbers.
Let C2 be the set of bicomplex numbers, then 2 2 2 C2 a i1b i 2 c j d : i1 i 2 1 j 1 a, b, c, d R }
(4)
can be viewed as complexification of C1 and a bicomplex number can be seen as an 2
element of C 1 . Moreover C2 is commutative unitary ring with the following characterization for the non invertible elements.
z1 i 2 z 2 C2 is non invertible iff
2
2
z1 z 2 0
(b) Bicomplex Algebra The algebraic operations for bicomplex numbers are defined as follows, Let C2 such that 1 z1 i 2 z 2 and 2 x1 i 2 x 2 , z1 , z 2 , x1 , x 2 C1. Then 1
2
(i)
1 2 z1 x1 i 2 z 2 x 2 C2
(5)
(ii)
1 2 z1x1 z 2 x 2 i 2 z 2 x1 z1 x 2
(6)
where represent bicomplex multiplication and ‘.’ represent complex multiplication. z z 2 2 1 (iii) 1 2 1 2 i 2 2 2 2 ( z1 z 2 0 ) (7) z1 z 2 z1 z 2
(iv)
1
2
1 21
z1x1 z 2 x 2 x12 x 22
i2
z 2 x1 z1x 2 x12 x 22
(8)
x12 x 22 0 . (v) (vi)
10 1 and
n
n 1
1 1 1
z1 i 2 z 2
n Z
R
(9) (10)
Remark: The bicomplex multiplication is associative, distributive over bicomplex addition + and most significantly is commutative. For other properties one 2
can refer to the book by Price .
Bicomplex Gamma And Beta Function
133
(c) Differentiability of a bicomplex function Differentiability of a bicomplex function (see, e.g. Rönn3) at a point of C2 can be defined as: Let U be an open set of C2 and 0 U, then f : U C2 C2 is said to be C2differentiable at 0 with derivative equal to f ' (0) C2, given by following equation f ( f(0 lim f ' 0 (11) 0 0 provided that the limit exists and – 0 is invertible. We shall say that the function f is C2 - holomorphic on an open set U iff f is C2 differentiable at each point of U. A function f ( f (z i z f z , z i f z z of C2 can be seen 1
2 2
1
1
2
2 2
1
2
as a mapping f (z z f z , z f z z of C2. 1
2
1
1
2
2
1
2
Lemma 1: Let U be an open set of C, and f : U C2 C2. Also f (z1 + i2z) = f1 (z1, z2) + i2 f2 (z1, z2), then f is C2 -holomorphic iff f1 and f2 are holomorphic in U and f1 f 2 f 2 f 1 on U (12) z1 z 2 z1 z 2 Aforementioned equations are known as complexified Cauchy-Riemann (C – R) equations. The class of T-holomorphic mappings on an open set U C2 is defined as:
f f f f T H(U) f : U C 2 C 2 f H (U) and 1 2 2 1 on U z1 z 2 z1 z 2
(13)
where H(U) is the class of C2-holomorphic functions on U C2. (d) Bicomplex Integration The bicomplex integration of a bicomplex function 1 z1 z 2 i 2 2 z1 z 2 is defined as a line integral, that is evaluated with respect to some four-dimensional curve H in C2. More specifically, the bicomplex integration (see, e.g. Rönn 3 ) is defined as
d
d dz1 dz 2
(14)
S.P. Goyal, Trilok Mathur and Ritu Goyal
134
where H is a piecewise continuously differentiable curve in C2 and has the parametric equation H : P= P (t) , P (t) = (a (t), b (t) ), for r t s and H can be taken as a curve made up of two component curves 1 and 2 in C i.e. H = (1, 2) thus
I
s
r
t) ] ' t) dt
(15)
(e) Idempotent Basis Every bicomplex number = z1 + i2z2 has the following unique idempotent representation (16) z1 i 2 z 2 z1 i1z 2 e1 z1 i1z 2 e 2 where e 1 j)/2, e 1 j)/2 and 1 2
j i1 i 2
This representation is very useful because addition, multiplication and division can be done term-by-term. Also an element is non invertible iff
z12 z 22 0 z1 i1 z 2 0 or z1 i1z 2 0 . Hence we can say that A C2 is a C2-cartesian set determined by A1 C1 and C1 if A A1 e A 2
A2
{ z1 i 2 z 2 C2 : z1 i 2 z 2 1e1 2e 2 and (1 , 2 )A1 A 2 } . Thus if C2 is in the form = 1e1 + 2e2 and also if = z1 + i2z2 , then 1 z1 i1z 2 2 z1 i1z 2 (17)
z1 1 2 2
z2
i1
2 (18) 2 1 The set B = {e1, e2} is called the idempotent basis of C2. Lemma 2: If f A C1 and f A C1 are holomorphic (analytic) functions e 1 e 2 1
2
of C1 on the domain A1 and A2 respectively. Then the function f : A1 ×e A2 C2 is defined as (19) f (z i z f z i z e f z i z e 1
2 2
e1
1
1 2
1
e2
1
1 2
2
z1 + i2z2 A1 ×e A2 is T-holomorphic on the domain A1 ×e A2 C2.
Bicomplex Gamma And Beta Function
Lemma 3: If f
e1
135
A1 C1 and f e A1 C1 are holomorphic functions of C1 on 2
the domain A1 and A2 respectively. Also let f : A1 ×e A2 C2 is defined by (20) f ( f ( ) e f e e1
1
1
e2
2
2
where e e d d e d e 1 1 2 2 1 1 2 2
f( ) d f ( ) d e f ( ) d e1 1 1 1 e2 2 2 e2 1 2 where H t) , (t) 1 t) e1 2 t) e 2 r t s then
(21) (22)
Remark: If u e u e and v e v e are two elements of C2, then 1 1 1 2 2 2 1 1 2 2 (i)
1 2 iff u1 v1 and u 2 v 2
(23)
(ii)
1 2 u1 v1 e1 u 2 v 2 e 2
(24)
(iii)
1 2 u1v1 e1 u 2 v 2 e 2
(25)
(iv)
1 1 1 1 e1 e 2 u1 u 2 0 u u 1 2
(26)
u u (27) 1 e1 2 e 2 v1 v 2 0 v 2 v1 2 2. Bicomplex Gamma Function (A) We define the bicomplex Gamma function in the Euler product form as (v)
1
1 e 2
n 1
1 exp n n
(28)
where z1 i 2 z 2 z1 i1z 2 e1 z1 i1z 2 e 2 , and 0 1 is the Euler constant (see, e.g. Rainville6 ) defined as
lim H n log n)
(29)
1 k
(30)
n n
Hn
k 1
This definition is justified by the following
S.P. Goyal, Trilok Mathur and Ritu Goyal
Theorem 1: Let z i z C2 with z 1 2 2 1
m ) and z 2 1 m) where 2 2
m,n N {0}, then 1 1 1 e1 e z i z z i z 2 2
1
1 2
1
136
(31)
1 2
Proof. Consider
e
n
z i z e z i z e (z i z 2 1 2 2 exp 1 2 2 exp 1 1 2 1 n n
z i z z i z exp 1 1 1 e1 exp 1 1 2 e 2 n n e 1e1 2 e 2 e 1 )e e 2 e 1 2
z i z 1 exp 1 1 1 2 n n n
z i z exp 1 1 2 e n 1
z i z 1 1 1 2 n
z i z exp 1 1 2 e n 2
e
n 1
1 n exp n
z i z z1 i1z 2 z i z z1 i1z 2 e 1 1 n 1 2 exp 1 n 1 2 e1 n 1 z i z z i z z i z z1 i1z 2 e 1 1 2 1 1 1 2 exp 1 1 2 e 2 n n n 1 1 1 1 (using Lemma 2) e1 e 2 z1 i1z 2 z 2 i1z1 2 provided that z1 i1z 2 0 1 2 and z 2 i1z1 0 1 2
m m z1 and z 2 i1 2 2
m, N {0}.
Bicomplex Gamma And Beta Function
137
Remarks (i) (ii)
m m 2 () is T-holomorphic if z1 z 2 i1 2 2 2 () has simple pole at m m z1 and z 2 i1 , m, n N {0} 2 2 p, q N {0} or z i z p and z1 i1z 2 q , 1 1 2
(iii)
2 has essential singularities at z1 = or z2 =
(iv)
2 is never zero since
1 has no poles.
(B) Integral form: For z1 i 2 z 2 C2 , p p1e1 p 2 e 2 p1 p 2 R+ we define
2
e
p
p
1
dp
(32)
L
where H (1 2
and 1 1 p1 and 2 2 p 2
This definition is justified since
1 1 e p p 1 dp e p1 p1 1 dp1 e1 e p 2 p 2 2 dp 2 e 2 H 0 0 where 1 z1 i1z 2 , 2 z1 i1z 2
2 ( ) [ (1 )]e1 [ (2 )]e 2
{using Rainville6 [p.15, Theorem 6]} (33)
Bicomplex Beta Function: For 1 2 C2 p p1e1 p 2 e 2 C2, p1p 2 01 we define B2 (1 2 where
p
1 1 1 1 p) 2 dp
(34)
L
H ( 1 , 2 ) and 1 1(p1 ) and 2 2 ( p 2 )
Theorem 2: Let 1 u1 i 2 u 2 , 2 v1 i 2 v 2 , with Re (u1 ) | Im (u 2 ) | and Re (v1 Im (v2 then B2 (1 2 B (u1 i1u 2 v1 i1v 2 e1 B (u1 i1u 2 v1 i1v 2 e 2
(35)
S.P. Goyal, Trilok Mathur and Ritu Goyal
138
where B (x,y) is ordinary Beta function. Theorem 3: Let u i u v i v e e C2, 1
1
2 2
2
1
2 2
1 1
2 2
1
2
with Re (u Im (u and Re (v Im (v then 1 2 1 2 B2 ( 2 1 2
sin
21 1
cos
2 2 1
d
(36)
H
where and 1 1 1 2 2 2
0 1 2
Proof of above theorem is straight forward by using the substitution p1 2 = sin 1 and p2 = sin 2. Theorem 4. Let 1 u1 i 2 u 2 and 2 v1 i 2 v 2 valid under the conditions 2
mentioned in Theorem 3. Then 2 2 B2 1 2 2 1 2 1 2
(37)
Proof: Consider
2 (1 ) 2 (2 ) e p p1 1 dp ⊙ e q q 2 1 dq 1 2 z1 z 2 C1 ; 2 y1e1 y 2e 2 y1 y 2 C1 , where 1 z1e1 z 2e 2 z1 u1 i u 2 z 2 u1 i1u 2 , y1 v1 i1v 2 p p1e1 p 2e 2
y 2 v1 i1v 2
p1 p 2 R+ and q q1e1 q 2e 2
q1 q 2 R+
p z 1 q y 1 Thus 2 1 2 2 e 1 p1 1 dp1 e 1 q1 1 dq1 e1 0 0 p 2 z 2 1 q 2 y 2 1 e p 2 dp 2 e q 2 dq 2 e 2 0 0 I1 e1 I 2 e 2 (38) p z 1 q y 1 I1 e 1 p1 1 dp1 e 1 q1 1 dq1 0 0 I1 z1 y1 z1 y1 B (z1 y1
(39)
Bicomplex Gamma And Beta Function
139
p z 1 q y 1 similarly I 2 e 2 p 2 2 dp 2 e 2 q 2 2 dq 2 z 2 y 2 B (z 2 y 2 0 0 (40) Therefore 2 1 2 2
z1 y1 B (z1 y1 e1 z 2 y 2 B (z 2 y 2 e 2 z1 y1 e1 z 2 y 2 e 2 B (z1 y1 e1 B (z 2 y 2 e 2 z1e1 z 2e 2 y1e1 y 2e 2 B (z1 y1 e1 B (z 2 y 2 e 2
2 1 2 2 2 1 2 B2 1 2 which gives the result (37). We can also prove that
H
1 1 2 1 2 2 2 2 1 2 sin cos d 2 2 22 1 2
(41)
3. Properties of Gamma Function 2 1 2 (i) Proof: Let z1 i 2 z 2 C2
2 z1 i1z 2 e1 z1 i 2 z 2 e 2
2 1 z1 1 i1z 2 e1 z1 1 i 2 z 2 e 2 z1 i1z 2 z1 i1z 2 e1 z1 i1z 2 z1 i1z 2 e 2 z1 i1z 2 e1 z1 i1z 2 e 2 z1 i1z 2 e1 z1 i1z 2 e 2 2 1 2 sin Proof: Using Theorem 4, we have (ii)
2 2 1
2 2 1 B2 (1
z1 i 2 z 2 C2 p 1 1 p) dp H
(42)
S.P. Goyal, Trilok Mathur and Ritu Goyal
140
where p = p1e1 + p2e2 , p1, p2 [0,1] z1 i1z 2 1 dp1 p dp 1 p1 e1 0 p p1 H 1 p 0 1 p1 p p Now put 1 q1 2 q 2 q1 q 2 R+ 1 - p1 1- p2
p 2 1 p 2
z1 i1z 2
dp 2 e p2 2
q z1 i1z 2 q z1 i1z 2 1 2 2 2 1 dq1 e1 dq 2 e 2 0 0 1 q1 1 q2 e1 e2 sin z1 i1z 2 sin z1 i1z 2 sin z1 i1z 2 e1 z1 i1z 2 e 2
sin (iii) Pochammer Symbol
6
(Using Rainville [p. 21, Eq.(4) ] )
n
n 1 n 1 i 1
n 1
(43)
i 1
It is easy to show that n
2 n) 2 ()
(44)
(iv) Gauss Multiplication Theorem For z1 i 2 z 2 C2 we have
s 1 1 2 k 1 2 k1/2 k 2 k 2 k s 1 Proof. Consider s 1 s 1 s 1 2 z1 i1z 2 e1 z1 i1z 2 e k k k 2 k
k
s 1
s 1 s 1 z1 i1z 2 e1 z1 i1z 2 e2 k k s 1 k
(45)
Bicomplex Gamma And Beta Function
141
1/2 k(z1 i1z 2 21 2 k 1 k k(z1 i1z 2 e1 1/2 k(z1 i1z 2 21 2 k 1 k k(z1 i1z 2 e 2 (Using Gauss Multiplication Theorem for Gamma function)
k(z1 i1z 2 k(z1 i1z 2 21 2 k 1 k1/2 k e1 k e 2 k (z1 i1z 2 e1 k (z1 i1z 2 e 2
21 2 k 1 k1/2 k 2 k (v) Legendre Duplication Formula For z1 i 2 z 2 C2, we have
2 2n 2 2n n 1 2 n Proof. We have 2 2 2n)
2 2
Since
(46)
2 z1 i1z 2 n) }e1 2(z1 i1z 2 n)}e 2 2 z1 i1z 2 ) }e1 2(z1 i1z 2 )}e 2
1 1 1 e1 e 2 , therefore ae1 be 2 a b
Thus 2 2 2n)
2 2
2 z1 i1z 2 n)} 2 z1 i1z 2 )}
e1
2(z1 i1z 2 n)} 2(z1 i1z 2 )}
e2
22n z1 i1z 2 n z1 i1z 2 1 2 n e1 22n z1 i1z 2 n z1 i1z 2 1 2 n e 2 (Using Legendre Duplication Formula for Gamma function)
22n z1 i1z 2 n e1 z1 i1z 2 n e 2 z1 i1z 2 1 2 n e1 z1 i1z 2 1 2 n e 2 2 2n 22n n 1 2 n (vi) Binomial Theorem Let a C2 be a bicomplex constant and z1 i 2 z 2 C2 ,then
S.P. Goyal, Trilok Mathur and Ritu Goyal
a
1
n 0
2 a 1)
n 2 a n 1) n!
L.H.S. (1 a 1 =
a1e1 a 2 e 2
where
142
(47)
a 1 a 2 C2
(1 z1 i1z 2 )a1 e1 (1 z1 i1z 2 )a 2 e 2
(a1 1) ( z1 i1z 2 ) n (a 2 1) ( z1 i1z 2 ) n e1 e2 ( a n 1) n ! ( a n 1) n ! 1 2 n 0 n 0
a1 1 e1 a 2 1 e 2 z1 i1z 2 n e1 z1 i1z 2 n e 2 n 