- Email: [email protected]
Generalization of analytic functions
Applied Mathematics and Computation 218 (2011) 851–855
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Generalization of analytic functions Sinan Kapcak ⇑, Ünal Ufuktepe Izmir University of Economics, Department of Mathematics, Balcova, Izmir, Turkey
a r t i c l e
i n f o
Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Analytic functions Time scale
a b s t r a c t The concept of analyticity for complex functions on time scale complex plane was introduced by Bohner and Guseinov in 2005. They developed completely delta differentiability, delta analytic functions on products of two time scales, and Cauchy–Riemann equations for delta case. In this research paper we study on continuous, discrete and semi-discrete analytic functions and developed completely nabla differentiability, nabla analytic functions on products of two time scales, and Cauchy–Riemann equations for nabla case. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Gusein Sh. Guseinov and Martin Bohner defined the differential calculus, integral calculus and developed line integration along time scale curves [1,2,3]. They also developed the concept of analytic functions on time scales in [2]. The study of analytic functions on Z2 has a history more than sixty years. The pioneer, in this field is Isaacs [4], who introduced two difference equations, both of them are discrete counterparts of the Cauchy–Riemann equation in one complex variable. 2. Discrete analytic functions Definition 2.1. A complex-valued function f which is defined on a subset A of Z þ iZ is called holomorphic in the sense of Isaacs or monodiffric (discrete analytic) of the first kind if the equation
f ðz þ 1Þ f ðzÞ f ðz þ iÞ f ðzÞ ¼ 1 i
ð1Þ
holds for all z 2 A (z + 1 and z + i are also in A). Here, we propose a new kind of monodiffric function which we call backward monodiffric function. To avoid confusion we call the monodiffric function of first kind like forward monodiffric function. Definition 2.2. A complex-valued function f which is defined on a subset A of Z þ iZ is called backward monodiffric if the equation
f ðzÞ f ðz 1Þ f ðzÞ f ðz iÞ ¼ 1 i holds for all z 2 A (z + 1 and z + i are also in A). ⇑ Corresponding author. E-mail addresses: [email protected] (S. Kapcak), [email protected] (Ü. Ufuktepe). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.025
ð2Þ
852
S. Kapcak, Ü. Ufuktepe / Applied Mathematics and Computation 218 (2011) 851–855
3. Semi-discrete analytic functions Semi-discrete analytic functions are single-valued functions of one continuous and one discrete variable defined on a semi-lattice, a uniformly spaced sequence of lines parallel to the real axis. The appropriate semi-discrete analogues of analytic functions are defined in [5] from the classic Cauchy–Riemann equations by replacing the y-derivative with nonsymmetric differences:
@f ðzÞ ¼ ½f ðz þ ihÞ f ðzÞ=ih; @x
z ¼ x þ ikh;
ð3Þ
Definition 3.1. Semi-discrete functions which satisfy (3) are called the first kind of semi-discrete analytic functions. Since we propose the backward version of semi-discrete analytic functions, to avoid confusion we call the first kind of forward semi-discrete analytic functions. Definition 3.2. The function defined on R þ ihZ which satisfies the following condition
@f ðzÞ ¼ ½f ðzÞ f ðz ihÞ=ih; @x
z ¼ x þ ikh;
ð4Þ
is called backward semi-discrete analytic function. Let f(z) = u(x, y) + iv(x, y) be semi-discrete analytic function on D R þ ihZ, where u and v are real valued semi-discrete functions, respectively equates with the real and imaginary parts of (3) which yields the following semi-discrete Cauchy– Riemann equations:
@uðx; yÞ 1 ¼ ½v ðx; y þ hÞ v ðx; yÞ; @x h @ v ðx; yÞ 1 ¼ ½uðx; yÞ uðx; y þ hÞ: @x h 4. Time scales and complex plane Let T1 and T2 be time scales. Let T1 T2 = {(x, y):x 2 T1, y 2 T2}. The set T1 T2 is a complete metric space with the metric (distance) d which is defined as
dððx; yÞ; ðx0 ; y0 ÞÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x0 Þ2 þ ðy y0 Þ2 for ðx; yÞ; ðx0 ; y0 Þ 2 T 1 T 2 :
For a given d > 0, the d-neighborhood Ud(x0, y0) of a given point (x0, y0) 2 T1 T2 is the set of all points (x, y) 2 T1 T2 such that d((x0, y0), (x, y)) < d. Let q1 and q2 be the backward jump operators for T1 and T2, respectively. Let u : T 1 T 2 ! R be a function. We define nabla partial derivatives of u at a point ðx0 ; y0 Þ 2 T1j T2j as
@uðx0 ; y0 Þ uðx; y0 Þ uðq1 ðx0 Þ; y0 Þ ¼ lim x!x0 ;x–q1 ðx0 Þ x q1 ðx0 Þ r1 x and
@uðx0 ; y0 Þ uðx0 ; yÞ uðx0 ; q2 ðy0 ÞÞ ¼ lim : y!y0 ;y–q2 ðy0 Þ y q2 ðy0 Þ r2 y For given time scales T1 and T2, The set T1 + i T2 = {z = x + iy:x 2 T1, y 2 T2} is called the time scale complex plane and is a comqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi plete metric space with the metric d is defined as dðz; z0 Þ ¼ jz z0 j ¼ ðx x0 Þ2 þ ðy y0 Þ2 where z = x + iy, z0 = x0 + iy0 2 T1 + i T2. Any function f : T 1 þ iT 2 ! C can be represented in the form
f ðzÞ ¼ uðx; yÞ þ iv ðx; yÞ for z ¼ x þ iy 2 T 1 þ iT 2 ; where u : T 1 T 2 ! R is the real part of f and v : T 1 T 2 ! R is the imaginary part of f. Let q1 and q2 be the backward jump operators for T1 and T2, respectively. For z = x + iy 2 T1 + i T2, we set zq1 ¼ q1 ðxÞ þ iy and zq2 ¼ x þ iq2 ðyÞ. 5. Completely nabla differentiable functions Let’s give the definition of completely nabla differentiability for one and two variable:
S. Kapcak, Ü. Ufuktepe / Applied Mathematics and Computation 218 (2011) 851–855
853
Definition 5.1. The function u : T ! R is called completely nabla differentiable at a point x0 2 Tj if there exists a number B such that satisfies the followings:
uðxÞ uðx0 Þ ¼ Bðx x0 Þ þ aðx x0 Þ for all x 2 U d ðx0 Þ
ð5Þ
uðxÞ uðqðx0 ÞÞ ¼ B½x qðx0 Þ þ b½x qðx0 Þ for all x 2 U d ðx0 Þ
ð6Þ
where a = a(x0, x) and b = b(x0, x) are equal zero at x = x0 and
lim aðx0 ; xÞ ¼ 0 and
x!x0
lim bðx0 ; xÞ ¼ 0:
x!x0
Definition 5.2. We say that the function u : T 1 T 2 ! R is completely nabla differentiable at a point ðx0 ; y0 Þ 2 T1j T2j if the numbers B1 and B2 which do not depend on (x, y) 2 T1 T2 exist (but, in general, depend on (x0, y0)) such that satisfy the following conditions:
uðx; yÞ uðx0 ; y0 Þ ¼ B1 ðx x0 Þ þ B2 ðy y0 Þ þ a1 ðx x0 Þ þ a2 ðy y0 Þ;
ð7Þ
uðx; yÞ uðq1 ðx0 Þ; y0 Þ ¼ B1 ½x q1 ðx0 Þ þ B2 ðy y0 Þ þ b11 ½x q1 ðx0 Þ þ b12 ðy y0 Þ;
ð8Þ
uðx; yÞ uðx0 ; q2 ðy0 ÞÞ ¼ B1 ðx x0 Þ þ B2 ½y q2 ðy0 Þ þ b21 ½x x0 þ b22 ½y q2 ðy0 Þ
ð9Þ
for all (x, y) 2 Ud(x0, y0), where d > 0 is sufficiently small, aj = aj(x0, y0; x, y) and bjk = bjk(x0, y0; x, y) are defined on Ud(x0, y0) such that they are equal to zero at (x,y) = (x0, y0) and
lim
ðx;yÞ!ðx0 ;y0 Þ
aj ðx0 ; y0 ; x; yÞ ¼
lim
ðx;yÞ!ðx0 ;y0 Þ
bjk ðx0 ; y0 ; x; yÞ ¼ 0 for j; k 2 f1; 2g
If we take T 1 ¼ T 2 ¼ Z, then we have
@uðx0 ; y0 Þ r1 x @uðx0 ; y0 Þ B2 ¼ uðx0 ; y0 Þ uðx0 ; y0 1Þ ¼ : r2 y
B1 ¼ uðx0 ; y0 Þ uðx0 1; y0 Þ ¼
Lemma 5.3. Let the function u : T 1 T 2 ! R be completely nabla differentiable at the point ðx0 ; y0 Þ 2 T1j T2j , then it is continuous at (x0, y0) and has the first order partial nabla derivatives
@uðx0 ; y0 Þ ¼ B1 r1 x
and
@uðx0 ; y0 Þ ¼ B2 : r2 y
Theorem 5.4. Suppose that a and b are two arbitrary points in T; a ¼ minfa; bg; b ¼ maxfa; bg, and f is a continuous function on [a, b] which has a nabla derivative at (a, b]. Then there exist n, n0 2 (a, b] such that
f r ðnÞða bÞ 6 f ðaÞ f ðbÞ 6 f r ðn0 Þða bÞ: Theorem 5.5. If the function u : T 1 T 2 ! R is continuous, has the first order continuous partial nabla derivatives @uðx;yÞ r1 x ; some d-neighborhood Ud(x0, y0) of the point ðx0 ; y0 Þ 2 T1j T2j , then u is completely r-differentiable at (x0, y0).
@uðx;yÞ r2 y
in
6. Nabla analytic functions Definition 6.1. The complex-valued function f : T1 þ iT2 ! C is nabla differentiable (or nabla analytic) at the point z0 ¼ x0 þ iy0 2 T1j þ iT2j if there exists a complex number B (generally depends on z0) such that
f ðzÞ f ðz0 Þ ¼ Bðz z0 Þ þ aðz z0 Þ
ð10Þ
q q q f ðzÞ f ðz0 1 Þ ¼ B z z0 1 þ b z z0 1
ð11Þ
q q q f ðzÞ f ðz0 2 Þ ¼ B z z0 2 þ c z z0 2
ð12Þ
for all z 2 Ud(z0), where Ud(z0) is a d-neighborhood of z0 in T1 þ iT2 ; a ¼ aðz0 ; zÞ; b ¼ bðz0 ; zÞ and c = c(z0, z) are defined for z 2 Ud(z0), they are equal to zero at z = z0, and
lim aðz0 ; zÞ ¼ lim bðz0 ; zÞ ¼ lim cðz0 ; zÞ ¼ 0:
z!z0
z!z0
z!z0
Then the number B is called the nabla derivative (or r-derivative) of f at z0 and is denoted by fr(z0).
854
S. Kapcak, Ü. Ufuktepe / Applied Mathematics and Computation 218 (2011) 851–855
Theorem 6.2. The function f : T1 þ iT2 ! C where
f ðzÞ ¼ uðx; yÞ þ iv ðx; yÞ for z ¼ x þ iy 2 T1 þ iT2 : is r-differentiable at the point z0 ¼ x0 þ iy0 2 T1j þ iT2j if and only if the functions u and v are completely r-differentiable at the point (x0, y0) and satisfied the Cauchy–Riemann equations
@u
r1 x
¼
@v
@u
and
r2 y
¼
r2 y
@v
ð13Þ
r1 x
at (x0, y0). fr(z0) can be represented in any of the following forms:
f r ðz0 Þ ¼
@u
r1 x
þi
@v
r1 x
¼
@v
r2 y
i
@u
r2 y
¼
@u
r1 x
i
@u
r2 y
¼
@v
r2 y
þi
@v
r1 x
ð14Þ
;
where the partial derivatives are evaluated at (x0, y0). Remark 6.3 (See Theorem 5.5). If the functions u; v : T1 T2 ! R are continuous and have the first order continuous partial nabla derivatives
@uðx;yÞ @uðx;yÞ @ v ðx;yÞ @ v ðx;yÞ r1 x ; r2 y ; r1 x ; r2 y
in some d-neighborhood Ud(x0, y0) of the point ðx0 ; y0 Þ 2 T1j T2j , then u and
v
are completely r-differentiable at (x0, y0). In addition, if the Cauchy–Riemann Eq. (13) are satisfied, then f(z) = u(x, y) + iv(x, y) is r-differentiable at z0 = x0 + iy0. Example 6.4 (i) The function f(z) = x2 y2 + i(2xy x y) is r-analytic everywhere on Z þ iZ. Since, Cauchy–Riemann equations are satisfied:
@uðx; yÞ ¼ x þ q1 ðxÞ ¼ 2x 1; r1 x
@uðx; yÞ ¼ y q2 ðyÞ ¼ 2y þ 1; r2 y
@ v ðx; yÞ ¼ 2y 1; r1 x
@ v ðx; yÞ ¼ 2x 1: D2 y
Note that this function is not delta analytic on Z þ iZ. (ii) The function f(z) = x2 y2 + i3xy is not r-analytic on T þ iT where T ¼ f2n : n 2 Zg [ f0g. Since, Cauchy–Riemann equations are not satisfied:
@uðx; yÞ 3x @ v ðx; yÞ – 3x ¼ ¼ x þ q1 ðxÞ ¼ ; 2 r1 x r2 y
@uðx; yÞ 3y @ v ðx; yÞ – 3y ¼ ¼ y q2 ðyÞ ¼ : 2 r2 y r2 y
Note that this function is delta analytic everywhere on T þ iT with the same T. Remark 6.5 (i) If T1 ¼ T2 ¼ R, then T1 þ iT2 ¼ R þ iR ¼ C is the usual complex plane and the three condition (10)–(12) of Definition 6.1 coincide and reduce to the classical definition of analyticity (differentiability) of functions of a complex variable. (ii) Let T1 ¼ T2 ¼ Z. Then T1 þ iT2 ¼ Z þ iZ ¼ Z½i is the set of Gaussian integers. The neighborhood Ud(z0) of z0 contains the single point z0 for d < 1. In this case, the condition (10) disappears, while the conditions (11) and (12) reduce to the single condition
f ðz0 Þ f ðz0 1Þ f ðz0 Þ f ðz0 iÞ ¼ 1 i
ð15Þ
with fr(z0) equal to the left hand side of (15) (and also to the right hand side). The condition (15) coincides with the definition of backward monodiffric functions we defined in (2). (iii) If T1 ¼ R and T2 ¼ hZ ¼ fhk : k 2 Zg where h > 0, then (10) and (11) coincide and in any of them, dividing both sides by (z z0) where z = x + ikh and z0 = x0 + ik0h, and taking limit as z ? z0 (which just means x ? x0), with k = k0 we have
lim
x!x0
f ðx0 þ ik0 hÞ f ðx þ ik0 hÞ ¼ B: x0 x
Similarly by (12) we get
f ðzÞ f ðz ihÞ ¼ B: ih Equating this two results gives the condition of backward semi-discrete analytic functions defined in (4).
S. Kapcak, Ü. Ufuktepe / Applied Mathematics and Computation 218 (2011) 851–855
855
Remark 6.6. Then we can unify Cauchy–Riemann equations for either discrete or continuous or semi-discrete cases in one equation as follow:
@f
r1 x
¼
1 @f i r2 y
ð16Þ
(i) Suppose that T1 ¼ T2 ¼ R, then we have @u þ i @@xv ¼ 1i @u þ i @@yv . We can get the usual Cauchy–Riemann equations by @x @y equating real and imaginary parts. (ii) Suppose that T1 ¼ T2 ¼ Z then from the Eq. (16) we get
1 i
rx f ¼ ry f which is the condition for backward monodiffric functions. (iii) Suppose that T1 ¼ R and T2 ¼ hZ, then from the Eq. (16) we get
@f 1 f ðzÞ f ðz ihÞ ¼ @x i h which is the equation for backward semi-discrete analytic functions.
References [1] [2] [3] [4] [5]
M. Bohner, G. Sh Guseinov, Partial differentiation on time scales, Dyn. Syst. Appl. 13 (2004) 351–379. M. Bohner, G. Sh Guseinov, An introduction to complex functions on product of two time scales, J. Diff. Equat. Appl. (2005). G.Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003) 107–127. R.P. Isaacs, A finite difference function theory, Univ. Nac. Tucuman Rev. 2 (1941) 177–201. G.J. Kurowski, Semi-discrete analytic functions, Trans. American Soc. 106 (1) (1963) 1–18.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Generalization of analytic functions Sinan Kapcak ⇑, Ünal Ufuktepe Izmir University of Economics, Department of Mathematics, Balcova, Izmir, Turkey
a r t i c l e
i n f o
Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Analytic functions Time scale
a b s t r a c t The concept of analyticity for complex functions on time scale complex plane was introduced by Bohner and Guseinov in 2005. They developed completely delta differentiability, delta analytic functions on products of two time scales, and Cauchy–Riemann equations for delta case. In this research paper we study on continuous, discrete and semi-discrete analytic functions and developed completely nabla differentiability, nabla analytic functions on products of two time scales, and Cauchy–Riemann equations for nabla case. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Gusein Sh. Guseinov and Martin Bohner defined the differential calculus, integral calculus and developed line integration along time scale curves [1,2,3]. They also developed the concept of analytic functions on time scales in [2]. The study of analytic functions on Z2 has a history more than sixty years. The pioneer, in this field is Isaacs [4], who introduced two difference equations, both of them are discrete counterparts of the Cauchy–Riemann equation in one complex variable. 2. Discrete analytic functions Definition 2.1. A complex-valued function f which is defined on a subset A of Z þ iZ is called holomorphic in the sense of Isaacs or monodiffric (discrete analytic) of the first kind if the equation
f ðz þ 1Þ f ðzÞ f ðz þ iÞ f ðzÞ ¼ 1 i
ð1Þ
holds for all z 2 A (z + 1 and z + i are also in A). Here, we propose a new kind of monodiffric function which we call backward monodiffric function. To avoid confusion we call the monodiffric function of first kind like forward monodiffric function. Definition 2.2. A complex-valued function f which is defined on a subset A of Z þ iZ is called backward monodiffric if the equation
f ðzÞ f ðz 1Þ f ðzÞ f ðz iÞ ¼ 1 i holds for all z 2 A (z + 1 and z + i are also in A). ⇑ Corresponding author. E-mail addresses: [email protected] (S. Kapcak), [email protected] (Ü. Ufuktepe). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.025
ð2Þ
852
S. Kapcak, Ü. Ufuktepe / Applied Mathematics and Computation 218 (2011) 851–855
3. Semi-discrete analytic functions Semi-discrete analytic functions are single-valued functions of one continuous and one discrete variable defined on a semi-lattice, a uniformly spaced sequence of lines parallel to the real axis. The appropriate semi-discrete analogues of analytic functions are defined in [5] from the classic Cauchy–Riemann equations by replacing the y-derivative with nonsymmetric differences:
@f ðzÞ ¼ ½f ðz þ ihÞ f ðzÞ=ih; @x
z ¼ x þ ikh;
ð3Þ
Definition 3.1. Semi-discrete functions which satisfy (3) are called the first kind of semi-discrete analytic functions. Since we propose the backward version of semi-discrete analytic functions, to avoid confusion we call the first kind of forward semi-discrete analytic functions. Definition 3.2. The function defined on R þ ihZ which satisfies the following condition
@f ðzÞ ¼ ½f ðzÞ f ðz ihÞ=ih; @x
z ¼ x þ ikh;
ð4Þ
is called backward semi-discrete analytic function. Let f(z) = u(x, y) + iv(x, y) be semi-discrete analytic function on D R þ ihZ, where u and v are real valued semi-discrete functions, respectively equates with the real and imaginary parts of (3) which yields the following semi-discrete Cauchy– Riemann equations:
@uðx; yÞ 1 ¼ ½v ðx; y þ hÞ v ðx; yÞ; @x h @ v ðx; yÞ 1 ¼ ½uðx; yÞ uðx; y þ hÞ: @x h 4. Time scales and complex plane Let T1 and T2 be time scales. Let T1 T2 = {(x, y):x 2 T1, y 2 T2}. The set T1 T2 is a complete metric space with the metric (distance) d which is defined as
dððx; yÞ; ðx0 ; y0 ÞÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x0 Þ2 þ ðy y0 Þ2 for ðx; yÞ; ðx0 ; y0 Þ 2 T 1 T 2 :
For a given d > 0, the d-neighborhood Ud(x0, y0) of a given point (x0, y0) 2 T1 T2 is the set of all points (x, y) 2 T1 T2 such that d((x0, y0), (x, y)) < d. Let q1 and q2 be the backward jump operators for T1 and T2, respectively. Let u : T 1 T 2 ! R be a function. We define nabla partial derivatives of u at a point ðx0 ; y0 Þ 2 T1j T2j as
@uðx0 ; y0 Þ uðx; y0 Þ uðq1 ðx0 Þ; y0 Þ ¼ lim x!x0 ;x–q1 ðx0 Þ x q1 ðx0 Þ r1 x and
@uðx0 ; y0 Þ uðx0 ; yÞ uðx0 ; q2 ðy0 ÞÞ ¼ lim : y!y0 ;y–q2 ðy0 Þ y q2 ðy0 Þ r2 y For given time scales T1 and T2, The set T1 + i T2 = {z = x + iy:x 2 T1, y 2 T2} is called the time scale complex plane and is a comqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi plete metric space with the metric d is defined as dðz; z0 Þ ¼ jz z0 j ¼ ðx x0 Þ2 þ ðy y0 Þ2 where z = x + iy, z0 = x0 + iy0 2 T1 + i T2. Any function f : T 1 þ iT 2 ! C can be represented in the form
f ðzÞ ¼ uðx; yÞ þ iv ðx; yÞ for z ¼ x þ iy 2 T 1 þ iT 2 ; where u : T 1 T 2 ! R is the real part of f and v : T 1 T 2 ! R is the imaginary part of f. Let q1 and q2 be the backward jump operators for T1 and T2, respectively. For z = x + iy 2 T1 + i T2, we set zq1 ¼ q1 ðxÞ þ iy and zq2 ¼ x þ iq2 ðyÞ. 5. Completely nabla differentiable functions Let’s give the definition of completely nabla differentiability for one and two variable:
S. Kapcak, Ü. Ufuktepe / Applied Mathematics and Computation 218 (2011) 851–855
853
Definition 5.1. The function u : T ! R is called completely nabla differentiable at a point x0 2 Tj if there exists a number B such that satisfies the followings:
uðxÞ uðx0 Þ ¼ Bðx x0 Þ þ aðx x0 Þ for all x 2 U d ðx0 Þ
ð5Þ
uðxÞ uðqðx0 ÞÞ ¼ B½x qðx0 Þ þ b½x qðx0 Þ for all x 2 U d ðx0 Þ
ð6Þ
where a = a(x0, x) and b = b(x0, x) are equal zero at x = x0 and
lim aðx0 ; xÞ ¼ 0 and
x!x0
lim bðx0 ; xÞ ¼ 0:
x!x0
Definition 5.2. We say that the function u : T 1 T 2 ! R is completely nabla differentiable at a point ðx0 ; y0 Þ 2 T1j T2j if the numbers B1 and B2 which do not depend on (x, y) 2 T1 T2 exist (but, in general, depend on (x0, y0)) such that satisfy the following conditions:
uðx; yÞ uðx0 ; y0 Þ ¼ B1 ðx x0 Þ þ B2 ðy y0 Þ þ a1 ðx x0 Þ þ a2 ðy y0 Þ;
ð7Þ
uðx; yÞ uðq1 ðx0 Þ; y0 Þ ¼ B1 ½x q1 ðx0 Þ þ B2 ðy y0 Þ þ b11 ½x q1 ðx0 Þ þ b12 ðy y0 Þ;
ð8Þ
uðx; yÞ uðx0 ; q2 ðy0 ÞÞ ¼ B1 ðx x0 Þ þ B2 ½y q2 ðy0 Þ þ b21 ½x x0 þ b22 ½y q2 ðy0 Þ
ð9Þ
for all (x, y) 2 Ud(x0, y0), where d > 0 is sufficiently small, aj = aj(x0, y0; x, y) and bjk = bjk(x0, y0; x, y) are defined on Ud(x0, y0) such that they are equal to zero at (x,y) = (x0, y0) and
lim
ðx;yÞ!ðx0 ;y0 Þ
aj ðx0 ; y0 ; x; yÞ ¼
lim
ðx;yÞ!ðx0 ;y0 Þ
bjk ðx0 ; y0 ; x; yÞ ¼ 0 for j; k 2 f1; 2g
If we take T 1 ¼ T 2 ¼ Z, then we have
@uðx0 ; y0 Þ r1 x @uðx0 ; y0 Þ B2 ¼ uðx0 ; y0 Þ uðx0 ; y0 1Þ ¼ : r2 y
B1 ¼ uðx0 ; y0 Þ uðx0 1; y0 Þ ¼
Lemma 5.3. Let the function u : T 1 T 2 ! R be completely nabla differentiable at the point ðx0 ; y0 Þ 2 T1j T2j , then it is continuous at (x0, y0) and has the first order partial nabla derivatives
@uðx0 ; y0 Þ ¼ B1 r1 x
and
@uðx0 ; y0 Þ ¼ B2 : r2 y
Theorem 5.4. Suppose that a and b are two arbitrary points in T; a ¼ minfa; bg; b ¼ maxfa; bg, and f is a continuous function on [a, b] which has a nabla derivative at (a, b]. Then there exist n, n0 2 (a, b] such that
f r ðnÞða bÞ 6 f ðaÞ f ðbÞ 6 f r ðn0 Þða bÞ: Theorem 5.5. If the function u : T 1 T 2 ! R is continuous, has the first order continuous partial nabla derivatives @uðx;yÞ r1 x ; some d-neighborhood Ud(x0, y0) of the point ðx0 ; y0 Þ 2 T1j T2j , then u is completely r-differentiable at (x0, y0).
@uðx;yÞ r2 y
in
6. Nabla analytic functions Definition 6.1. The complex-valued function f : T1 þ iT2 ! C is nabla differentiable (or nabla analytic) at the point z0 ¼ x0 þ iy0 2 T1j þ iT2j if there exists a complex number B (generally depends on z0) such that
f ðzÞ f ðz0 Þ ¼ Bðz z0 Þ þ aðz z0 Þ
ð10Þ
q q q f ðzÞ f ðz0 1 Þ ¼ B z z0 1 þ b z z0 1
ð11Þ
q q q f ðzÞ f ðz0 2 Þ ¼ B z z0 2 þ c z z0 2
ð12Þ
for all z 2 Ud(z0), where Ud(z0) is a d-neighborhood of z0 in T1 þ iT2 ; a ¼ aðz0 ; zÞ; b ¼ bðz0 ; zÞ and c = c(z0, z) are defined for z 2 Ud(z0), they are equal to zero at z = z0, and
lim aðz0 ; zÞ ¼ lim bðz0 ; zÞ ¼ lim cðz0 ; zÞ ¼ 0:
z!z0
z!z0
z!z0
Then the number B is called the nabla derivative (or r-derivative) of f at z0 and is denoted by fr(z0).
854
S. Kapcak, Ü. Ufuktepe / Applied Mathematics and Computation 218 (2011) 851–855
Theorem 6.2. The function f : T1 þ iT2 ! C where
f ðzÞ ¼ uðx; yÞ þ iv ðx; yÞ for z ¼ x þ iy 2 T1 þ iT2 : is r-differentiable at the point z0 ¼ x0 þ iy0 2 T1j þ iT2j if and only if the functions u and v are completely r-differentiable at the point (x0, y0) and satisfied the Cauchy–Riemann equations
@u
r1 x
¼
@v
@u
and
r2 y
¼
r2 y
@v
ð13Þ
r1 x
at (x0, y0). fr(z0) can be represented in any of the following forms:
f r ðz0 Þ ¼
@u
r1 x
þi
@v
r1 x
¼
@v
r2 y
i
@u
r2 y
¼
@u
r1 x
i
@u
r2 y
¼
@v
r2 y
þi
@v
r1 x
ð14Þ
;
where the partial derivatives are evaluated at (x0, y0). Remark 6.3 (See Theorem 5.5). If the functions u; v : T1 T2 ! R are continuous and have the first order continuous partial nabla derivatives
@uðx;yÞ @uðx;yÞ @ v ðx;yÞ @ v ðx;yÞ r1 x ; r2 y ; r1 x ; r2 y
in some d-neighborhood Ud(x0, y0) of the point ðx0 ; y0 Þ 2 T1j T2j , then u and
v
are completely r-differentiable at (x0, y0). In addition, if the Cauchy–Riemann Eq. (13) are satisfied, then f(z) = u(x, y) + iv(x, y) is r-differentiable at z0 = x0 + iy0. Example 6.4 (i) The function f(z) = x2 y2 + i(2xy x y) is r-analytic everywhere on Z þ iZ. Since, Cauchy–Riemann equations are satisfied:
@uðx; yÞ ¼ x þ q1 ðxÞ ¼ 2x 1; r1 x
@uðx; yÞ ¼ y q2 ðyÞ ¼ 2y þ 1; r2 y
@ v ðx; yÞ ¼ 2y 1; r1 x
@ v ðx; yÞ ¼ 2x 1: D2 y
Note that this function is not delta analytic on Z þ iZ. (ii) The function f(z) = x2 y2 + i3xy is not r-analytic on T þ iT where T ¼ f2n : n 2 Zg [ f0g. Since, Cauchy–Riemann equations are not satisfied:
@uðx; yÞ 3x @ v ðx; yÞ – 3x ¼ ¼ x þ q1 ðxÞ ¼ ; 2 r1 x r2 y
@uðx; yÞ 3y @ v ðx; yÞ – 3y ¼ ¼ y q2 ðyÞ ¼ : 2 r2 y r2 y
Note that this function is delta analytic everywhere on T þ iT with the same T. Remark 6.5 (i) If T1 ¼ T2 ¼ R, then T1 þ iT2 ¼ R þ iR ¼ C is the usual complex plane and the three condition (10)–(12) of Definition 6.1 coincide and reduce to the classical definition of analyticity (differentiability) of functions of a complex variable. (ii) Let T1 ¼ T2 ¼ Z. Then T1 þ iT2 ¼ Z þ iZ ¼ Z½i is the set of Gaussian integers. The neighborhood Ud(z0) of z0 contains the single point z0 for d < 1. In this case, the condition (10) disappears, while the conditions (11) and (12) reduce to the single condition
f ðz0 Þ f ðz0 1Þ f ðz0 Þ f ðz0 iÞ ¼ 1 i
ð15Þ
with fr(z0) equal to the left hand side of (15) (and also to the right hand side). The condition (15) coincides with the definition of backward monodiffric functions we defined in (2). (iii) If T1 ¼ R and T2 ¼ hZ ¼ fhk : k 2 Zg where h > 0, then (10) and (11) coincide and in any of them, dividing both sides by (z z0) where z = x + ikh and z0 = x0 + ik0h, and taking limit as z ? z0 (which just means x ? x0), with k = k0 we have
lim
x!x0
f ðx0 þ ik0 hÞ f ðx þ ik0 hÞ ¼ B: x0 x
Similarly by (12) we get
f ðzÞ f ðz ihÞ ¼ B: ih Equating this two results gives the condition of backward semi-discrete analytic functions defined in (4).
S. Kapcak, Ü. Ufuktepe / Applied Mathematics and Computation 218 (2011) 851–855
855
Remark 6.6. Then we can unify Cauchy–Riemann equations for either discrete or continuous or semi-discrete cases in one equation as follow:
@f
r1 x
¼
1 @f i r2 y
ð16Þ
(i) Suppose that T1 ¼ T2 ¼ R, then we have @u þ i @@xv ¼ 1i @u þ i @@yv . We can get the usual Cauchy–Riemann equations by @x @y equating real and imaginary parts. (ii) Suppose that T1 ¼ T2 ¼ Z then from the Eq. (16) we get
1 i
rx f ¼ ry f which is the condition for backward monodiffric functions. (iii) Suppose that T1 ¼ R and T2 ¼ hZ, then from the Eq. (16) we get
@f 1 f ðzÞ f ðz ihÞ ¼ @x i h which is the equation for backward semi-discrete analytic functions.
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M. Bohner, G. Sh Guseinov, Partial differentiation on time scales, Dyn. Syst. Appl. 13 (2004) 351–379. M. Bohner, G. Sh Guseinov, An introduction to complex functions on product of two time scales, J. Diff. Equat. Appl. (2005). G.Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003) 107–127. R.P. Isaacs, A finite difference function theory, Univ. Nac. Tucuman Rev. 2 (1941) 177–201. G.J. Kurowski, Semi-discrete analytic functions, Trans. American Soc. 106 (1) (1963) 1–18.