On Properties of Holomorphic Functions in Quaternionic Analysis

Abstract We draw the conclusions from the earlier presented quaternionic generalization of Cauchy-Riemann’s equations. The general expressions for constituents of -holomorphic functions as well as the relations between them are deduced. The symmetry properties of constituents of -holomorphic functions and their derivatives of all orders are proved. For full derivatives it is a consequence of uniting the left and right derivatives within the framework of the developed theory. Some -holomorphic generalizations of C − holomorphic functions are discussed in detail to demonstrate particularities of constructing H-holomorphic functions. The power functions are considered in detail.

Moreover, they don't enable to construct quaternionic holomorphic functions from complex holomorphic counterparts by direct replacing a complex variable by a quaternionic in expressions for complex holomorphic functions, whilst a similar way exists in complex analysis. We develop further the "essentially adequate" theory [11] that unites both approaches and solves this problem.
We denote a quaternionic variable by = + + + = + ∈ ℍ , where = + , = + ( , , , are real values, , , are quaternionic basis vectors, ℍ denotes a quaternionic space), and a quaternion-valued (briefly, quaternionic) function by ( ) = 1 + 2 , where 1 = 1 ( , ) = 1 + 2 and 2 = 2 ( , ) = 3 + 4 ( 1 , 2 , 3 , 4 are real-valued functions of , , , ). In accordance with [11] we define a ℍ-holomorphic function as follows. Definition 1.1. It is assumed that the constituents 1 Here and in the sequel, the complex conjugation is denoted as usual (for example, 2 ); the brackets (. . | with the closing vertical bar indicate that the transition = = has been already performed in expressions enclosed in brackets. Thus, the ℍ-holomorphicity condition is defined so that during the check of quaternionic holomorphicity of any quaternionic function we have to do the transition = = in already computed expressions for partial derivatives of functions 1 and 2 to be used in the equations (1). However, this does not mean that we deal with triplets in general; this transition cannot be initially done for quaternionic variables and functions [11]. Any quaternionic function of a quaternion variable remains the same 4-dimensional quaternionic function regardless of whether we check its quaternionic holomorphicity or not.
Equations 2) and (1)(2)(3)4) represent left and right ℍ -holomorphicity conditions, respectively. The additional condition = � = , reflecting the requirement of an uniqueness of a derivative in space, unites them and allows us to construct ℍ-holomorphic functions from the complex holomorphic (briefly, ℂ-holomorphic) counterparts by replacing variables, as noted above. We will establish some properties of ℍ -holomorphic functions, defined by equations (1).
The purpose of this article is to obtain the general expressions for constituents 1 ( , ) and 2 ( , ) of ℍholomorphic functions as well as some important relations between them.
To represent all expressions in the Cayley-Dickson doubling form [6], which plays a principal role further, we use the identity for any .

Expressions for Constituents and
When formulating the essentially adequate quaternionic generalization of the complex Cauchy-Riemann equations it has been shown [1] that the following equation: for the constituent 2 of any ℍ -holomorphic function ( ) = 1 + 2 holds. Indeed, according to (1-1) and (1-3), the derivative 1 is simultaneously equal to the derivatives 2 and 2 ; hence (4) holds. The equation (4) can only be satisfied if the constituent 2 ( , ) has the following general form: where the function �( ), ( ) , � �� is dependent on symmetric (in variables , and , ) basic forms , ( ) , and invariant under complex conjugation. Then �( ), ( ) , � �� is equal to its conjugate and (4) is valid. In accordance with [11] the general expression for the symmetric form ( ) = ( ) can be represented as where is any non-negative integer. The number of summands in (6) is equal to ( + 1). If = 0, then we can state that ( ) 0 = 0 0 = 1.
Since the full derivatives of ℍ-holomorphic functions defined by (2) are ℍ-holomorphic too [11], we can state that the same dependence of 2 on symmetric invariant forms exists for derivatives of all orders, that is, for 2 (1) , 2 (2) , and so on. The obtained general expressions (5) and (6) now allow us to prove the following assertion, which was discussed earlier in [11] without a proof. Proof. It suffices to prove this assertion for = 1 , since every ′th derivative is constructed from a previous ℍ -holomorphic ( − 1)′th derivative. Differentiating 2 , we obtain from (5) the following expressions: The derivatives 2 and 2 are the constituents of the left ( ′ ( ) = 1 + 2 ) and right ( ′ ( ) = 1 + 2 ) derivatives, defined [11] respectively by the left (1-1,2) and right (1-3,4) holomorphicity equations.
Differentiating ( ) with respect to and , we get Finally, substituting (9) and (10) into (11), we have ( ) Since the derivatives ( ) and ( ) are symmetric in and , this expression is symmetric in and too. Thus the constituent 2 ( ) unites left and right versions of quaternionic analysis, forming its symmetric expression. Q.e.d.
As regards the constituent 1 ( , ), it can be expressed in the following general form: A a a bb φ = Indeed, the equations (1-2) and (1-4) can be only satisfied if 1 ( , ) is symmetric in variables and .
Example 1.2 Consider the quaternionic function 4 = ( + ) 4 = 1 + 2 , which is ℍ -holomorphic because it is constructed from the ℂ-holomorphic function 4 by replacing (as a single whole) by (see Theorem 4.4 in [11]). Straightforward computing yields upon application of (3) the representation of 4 in the Cayley-Dickson doubling form, whence further we have ( ) The expression for 1 is symmetric in variables and .
The expression for 2 is symmetric in variables and . The derivatives 2 and 2 are Summing these, we obtain the following expression: which is symmetric in variables and too.

Relations between Constituents ,
Recall that we deal with quaternionic holomorphic functions constructed from the corresponding ℂ -holomorphic functions by a direct replacement of a complex variable by a quaternionic in expressions for ℂ-holomorphic functions without change of a functional dependence form. We begin by the theorem: Assume that 1 ( , ) and 2 ( , ) have continuous mixed second order partial derivatives with respect to , , and , which do not vanish at ∈ . Then, in hold true: Proof. The proof is by reductio ad absurdum. Obviously if the equations (12) and (13) were valid, then we would, by differentiating both sides of the equation (12) with respect to , and by differentiating both sides of the complex conjugate of the equation (13) with respect to � , have the following equations for mixed derivatives: where interchanging the order of taking partial derivatives is possible by the condition of the theorem (second-order mixed partial derivatives commute, if they are continuous [10]). (We use here the notation of a type instead of more usual ). Subtracting the second equation from the first, we would further have the following relation: Next, assume that any one of equations (12) and (13) at all ∈ are not valid. Then the expression (14) becomes According to the assumption of the theorem, the second partial derivatives are not to be equal to zero here. By changing the order of taking partial derivatives, we get from (15) the following inequality: However this inequality contradicts the true expression which follows from the equation (4) when differentiating its both sides with respect to . Thus, the assumption (15) is not true; hence the validity of equations (12) and (13) for ℍ -holomorphic functions is proved by reductio ad absurdum. Q.e.d. We now have proved the validity of the equations (12) and (13) introduced without a proof in [11]. Based on (12) and (13), it is easy to prove one more equation also previously-stated in [11]: Corollary 3.2 The constituents 1 ( , ) of the ℍ-holomorphic functions satisfies the following equation: Proof. Differentiating both sides of (12) with respect to , and both sides of (13) with respect to , we get Taking into consideration that the second partial derivatives commute, and subtracting the second equation from the first, we obtain If the equation (16) were not valid, then we would have an immediate contradiction to the true equation (17), since by differentiating the inequality 1 ≠ 1 with respect to � we would get 1 ≠ 1 . Hence the equation (16) is valid. Q.e.d.
To illustrate that such a way generalizes expressions for complex derivatives we consider in detail two quaternionic generalizations: ( ) ( ) ( ) ( ) and then the transition from them to complex counterparts. According to [11], for such a transition it suffices to rule out the dimensions with quaternionic imaginary units and . This can be achieved by the following replacements: 2  1  3   1  1  2 1  1  2  1   2  3  4  2  3  4  3   ,   , , Performing these in (19) and (20) we get, respectively, , .
According to the Cayley-Dickson doubling [6] formula: = + , where and are real values when denotes complex numbers, and and are complex values when represents quaternions, while the remains the same, these replacements represent the transition reverse to the replacement as a single whole by . We see that a representation of initial complex functions in the complex plane = + (with the instead of the ) must take place when constructing ℍ -holomorphic functions from complex counterparts.

Particularities of Constructing ,
As has been said above, we consider the ℍ-holomorphic functions constructed from the ℂ-holomorphic functions by the direct replacement of variables without change of a functional dependence form. The examples considered in [1] are mostly those, in which a complex variable is replaced as a single whole by a quaternionic variable .
However, if we use additional formulae of complex analysis to represent some ℂ-holomorphic functions, then we must be sure that analogous formulae are valid upon the transition to a quaternion case. Now, we would like to discuss some particularities of constructing the constituents 1 and 2 , allowing us to be sure that "a functional dependence form" does not change upon such a transition. 1) Computation of constituents 1 and 2 when an initial complex counterpart, depending only on as a single whole, involves the imaginary unit as a multiplicative factor. For example, the ℂ -holomorphic function = = , where and = + denote, respectively, the imaginary complex unit and a complex variable written in the familiar form. Then, as noted above, the must be replaced by ; by = + , and we get the correct initial expression = = to construct the ℍ-holomorphic function. Next the replacement of as a single whole by gives the ℍ-holomorphic functions ( ) = and ( ) = .
2) Computation of constituents 1 and 2 when some additional formulae are used in an initial expression for the ℂ -holomorphic function. For example, by direct replacing by in the expression for the complex exponential function ( ) = we get the quaternionic exponential function ( ) = . To obtain the constituents 1 and 2 the next step is to represent this formula in the Cayley-Dickson doubling form ( ) = = 1 + 2 • . One evident way is to use Euler's formula = cos + sin , but first we must establish, in which form this formula is valid in the quaternion area.
The simple replacement of the by would be doubtful, because the is not a multiplicative factor in the function expression depending only on complex variable as a single whole. We can only use the representation formula for quaternions algebraically equivalent to the complex formula = + . Such a formula is = + , where = � 2 + 2 + 2 is a real value, 2 2 2 yi zj uk r y z u is a purely imaginary unit quaternion, so its square is −1. Then Euler's formula for quaternions is = cos + sin , and the expression for the quaternionic exponential function ( ) = is The ℍ -holomorphicity of this function is explored in detail in [11]. We shall not dwell on this here.

Expressions for Power Functions
Taking into account a very important role of computing algorithms in practice, we work out in detail here the expressions for the power functions = ( + ) = 1 ( , , ) + 2 ( , , ) , where is a non-negative integer. We denote further 1 ( , , ) and 2 ( , , ) simply by 1 ( ) and 2 ( ). For the power functions it is possible to separate variables , and , � and we can represent 1 ( ) and 2 ( ) by polynomials (sums of monomials) in , � as follows: The term in (22) is evident, since is the base = + raised to the power of , hence the variable too. Since the sum of exponents of factors included in each monomial 1, ( ) and 2, ( ) cannot be greater than , and hence the values of ( ) and ( ) become zero, respectively if 2 + 2 > and 2 + 1 > , we omit the upper limit of summation over in (22) and (23) and use the simple symbol ∑ .
≥0 It is easy to see that in the above Example 1.2 = 0,1.
For now we need the multiplication rule Representing (28)  Taking into account (42), we see that the iteration must be terminated after λ = (n -1) steps when − = 1. Thus we have the following explicit close-form expression: The formula (6)   Using (45) times (with respect to 0 ( )) like above, we get the following recursive formula for 0 ( ): Comparing (46) with (44), we immediately prove that (40), that is, (37) for = 0 is valid.