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 - holomorphic functions are discussed in detail to demonstrate particularities of constructing H-holomorphic functions. The power functions are considered in detail.

1. Introduction

There exist various versions of quaternionic analysis. They imply, explicitly or implicitly, the “left“and “right” definitions of a quaternionic derivative (e. g., ^{1, 2, 3, 4, 5, 6, 8, 9, 12, 13}), which are regarded as equivalent. Each of them by itself is incomplete (may be called "inessentially adequate"), since only the left or right version considered separately doesn’t represent all arbitrary rotations of vectors in space ^{7, 11} by the underlying algebra.

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 , where , ( are real values, are quaternionic basis vectors, denotes a quaternionic space), and a quaternion-valued (briefly, quaternionic) function by where and ( are real-valued functions of ). In accordance with ¹¹ we define a - holomorphic function as follows.

Definition 1.1. It is assumed that the constituents and of a quaternionic function possess continuous first-order partial derivatives with respect to and in some open connected neighborhood of a point . Then a function is said to be -holomorphic and denoted by at a point , if and only if the functions and satisfy in the following quaternionic generalization of Cauchy-Riemann's equations:

(1)

Here and in the sequel, the complex conjugation is denoted as usual (for example, ); the brackets with the closing vertical bar indicate that the transition has been already performed in expressions enclosed in brackets.

Thus, the -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 and 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 (1-1,2) and (1-3,4) represent left and right -holomorphicity conditions, respectively. The additional condition , reflecting the requirement of an uniqueness of a derivative in space, unites them and allows us to construct -holomorphic functions from the complex holomorphic (briefly, -holomorphic) counterparts by replacing variables, as noted above. We will establish some properties of -holomorphic functions, defined by equations (1).

The purpose of this article is to obtain the general expressions for constituents and of -holomorphic functions as well as some important relations between them.

Denote a th derivative of a -holomorphic function by

(2)

where the constituents and are expressed ¹¹ by

(2a)

(2b)

and are the constituents of the th derivative of , represented in the Cayley–Dickson doubling form as , ; and , . As we will see later, these formulae generalize expressions for derivatives of -holomorphic functions.

To represent all expressions in the Cayley–Dickson doubling form ⁶, which plays a principal role further, we use the identity

(3)

2. Expressions for Constituents

When formulating the essentially adequate quaternionic generalization of the complex Cauchy-Riemann equations it has been shown ¹ that the following equation:

(4)

for the constituent of any -holomorphic function holds. Indeed, according to (1-1) and (1-3), the derivative is simultaneously equal to the derivatives and ; hence (4) holds.

The equation (4) can only be satisfied if the constituent has the following general form:

(5)

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

(6)

where is any non-negative integer. The number of summands in (6) is equal to . If , then we can state that .

Since the full derivatives of -holomorphic functions defined by (2) are -holomorphic too ¹¹, we can state that the same dependence of on symmetric invariant forms exists for derivatives of all orders, that is, for , , 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.

Assertion 1.1 The symmetric (in variables ,) form of the complex constituents of quaternionic -holomorphic derivatives is a consequence of uniting unsymmetrical constituents and of the left and right derivatives.

Proof. It suffices to prove this assertion for , since every th derivative is constructed from a previous -holomorphic th derivative. Differentiating , we obtain from (5) the following expressions:

(7)

(8)

The derivatives and are the constituents of the left () and right () derivatives, defined ¹¹ respectively by the left (1-1,2) and right (1-3,4) holomorphicity equations.

Differentiating with respect to and , we get

(9)

(10)

Combining (2b) for , (7), and (8), we obtain

(11)

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 unites left and right versions of quaternionic analysis, forming its symmetric expression. Q.e.d.

As regards the constituent , it can be expressed in the following general form:

Indeed, the equations (1-2) and (1-4) can be only satisfied if is symmetric in variables and .

Example 1.2 Consider the quaternionic function , which is -holomorphic because it is constructed from the -holomorphic function by replacing (as a single whole) by (see Theorem 4.4 in ¹¹). Straightforward computing yields upon application of (3) the representation of in the Cayley–Dickson doubling form, whence further we have

The expression for is symmetric in variables and .

The expression for is symmetric in variables and .

The derivatives and are

Summing these, we obtain the following expression:

which is symmetric in variables and too.

3. Relations between Constituents

Recall that we deal with quaternionic holomorphic functions constructed from the corresponding -holomorphic functions by a direct replacement of a complex variable by a quaternionic in expressions for -holomorphic functions without change of a functional dependence form. We begin by the theorem:

Theorem 3.1 Let a quaternionic function be -holomorphic throughout a connected open set . Assume that and have continuous mixed second order partial derivatives with respect to , , and , which do not vanish at . Then, in hold true:

(12)

(13)

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 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:

(14)

Next, assume that any one of equations (12) and (13) at all are not valid. Then the expression (14) becomes

(15)

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 , 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-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 of the -holomorphic functions satisfies the following equation:

(16)

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

and further

(17)

If the equation (16) were not valid, then we would have an immediate contradiction to the true equation (17), since by differentiating the inequality with respect to we would get . Hence the equation (16) is valid. Q.e.d.

Putting together equations (4), (12), (13), and (16), we get the following system of equations:

(18)

which holds if the function is -holomorphic, that is, and satisfy the system (1). Equations (18-2,3,4) are valid if the second mixed partial derivatives don't vanish. These equations enable to deduce various formulae for -holomorphic derivatives from the basic expressions (2), (2a), and (2b).

To illustrate that such a way generalizes expressions for complex derivatives we consider in detail two quaternionic generalizations:

(19)

(20)

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:

Performing these in (19) and (20) we get, respectively,

These are the expressions for derivatives of complex functions in the complex plane with the imaginary unit , where satisfies corresponding Cauchy-Riemann's equations

(21)

It is easy to see also that the complex Cauchy-Riemann equations (21) follow from the quaternionic Cauchy-Riemann equations (1) by the same replacements.

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 -holomorphic functions from complex counterparts.

4. Particularities of Constructing

As has been said above, we consider the -holomorphic functions constructed from the -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 -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 and , allowing us to be sure that "a functional dependence form" does not change upon such a transition.

1) Computation of constituents and when an initial complex counterpart, depending only on as a single whole, involves the imaginary unit as a multiplicative factor. For example, the -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 -holomorphic function. Next the replacement of as a single whole by gives the -holomorphic functions and .

Consider the function . We obtain , whence , as well as , . We can see that this function is -holomorphic, since the equations (1) hold:

Note that there is no need to use the requirement . The full derivative of this function is in accordance with (19) the following:

This result corresponds to the complex case . The formula (20) yields the same result.

Consider the function . We obtain , whence , as well as , . This function is -holomorphic too, since the equations (1) hold:

There is also no need to do the transition , since the derivatives are independent of variables. The derivative of this function in accordance with (19) is as follows:

This corresponds to the complex case too: .

Consider the function . Since the complex function is -holomorphic, the function must be -holomorphic too. We have , whence , as well as , . In fact, this function is -holomorphic too:

According to (19), the derivative of this function is

It generalizes the complex case too: .

2) Computation of constituents and when some additional formulae are used in an initial expression for the -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 and the next step is to represent this formula in the Cayley–Dickson doubling form . One evident way is to use Euler's formula , 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 is a real value, is a purely imaginary unit quaternion, so its square is −1.

Then Euler's formula for quaternions is , and the expression for the quaternionic exponential function is

whence . The -holomorphicity of this function is explored in detail in ¹¹. We shall not dwell on this here.

5. 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 , where is a non-negative integer. We denote further and simply by and . For the power functions it is possible to separate variables , and , and we can represent and by polynomials (sums of monomials) in , as follows:

(22)

(23)

where

(24)

(25)

and , .

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 and cannot be greater than , and hence the values of and become zero, respectively if and , we omit the upper limit of summation over in (22) and (23) and use the simple symbol It is easy to see that in the above Example 1.2 .

For now we need the multiplication rule

represented in the Cayley–Dickson doubling form ⁶ for arbitrary quaternions and . Setting and we get

where

(26)

(27)

Substituting expressions (22) - (25) into the formulae (26) and (27), and grouping the terms with identical exponents of variables and , we get the following expressions:

(28)

(29)

Representing (28) and (29) in the form similar to (22) and (23), we obtain

where

(30)

(31)

(32)

Since in accordance with the associativity of quaternion multiplication the equality holds too, we get

whence, arguing as above, we deduce the expressions

(33)

(34)

(35)

equivalent to (30) - (32).

Theorem 5.1 Let a quaternion power function , where is any non-negative integer and and are differentiable with respect to , , , , be represented by polynomials in , . Then the factors and satisfy the following relations:

(36)

(37)

(38)

(39)

Proof. The proof will be divided into 2 steps.

Step 1. First we prove the formulae (36), and then (37) for :

(40)

Relabeling as in the recursive relation (32), we get

(41)

To eliminate recursion from the function we must use (41) repeatedly until obtaining an explicit close-form expression, that is, until obtaining the obvious base case:

(42)

(It is easy to see that from we have and as well as and .) Using the relation (41) two times, we have

Further, using the relation (41) times, we obtain the following recursive relation:

Taking into account (42), we see that the iteration must be terminated after steps when . Thus we have the following explicit close-form expression:

Comparing this expression with (6), we finally get:

The statement (36) of Theorem 5.1 is proved.

The formula (6) can be also represented as

then, differentiating it with respect to , we get

(43)

Combining (36) and (43) we obtain further the recursive formula for :

Substituting (43) into this formula one more time, that is, using (43) in the expression for two times, we get the following expression:

Using the relation (43) times, we obtain:

The iteration terminates after steps when , , . Given this, we obtain the following expression for the derivative:

Finally, making use of (36), we have

(44)

It remains to get an analogous expression for . We use the formula (30) for , relabelling as :

(45)

Using (45) times (with respect to ) like above, we get the following recursive formula for :

The iteration terminates after steps, since becomes 1, and we achieve the base case of the recurrence: and . Finally, we get for :

(46)

Comparing (46) with (44), we immediately prove that (40), that is, (37) for is valid.

Step 2. We prove the formulae (37) and (38) for and by using mathematical induction.

The base case , , whence, given (22) - (25), we have the sequences : and : , . Applying (37) to elements of , we obtain , and so on, which coincide with the corresponding elements of . Applying (38) to elements of , we obtain and so on, which coincide with the corresponding elements of . The formulae (37) and (38) are true in this case.

The base case , . By direct multiplication and using (3), we get in the Cayley–Dickson doubling form , whence and Given (22) - (25), we have the sequences : and : Applying (37) to elements of , we obtain , and so on, which coincide with the corresponding elements of Applying (38) to elements of , we obtain and so on, which coincide with the corresponding elements of . The formulae (37) and (38) are true in this case too. We see that the values of and become zero in both base cases automatically when values of increasing as noted above.

The inductive step. Assuming that the equations (37) and (38) are valid for any , we will prove the validity of these equations for :

(47)

(48)

Substituting (30) and (34) into (47), we obtain the following expression:

Performing the differentiation on the right-hand side of this expression and simplifying, we get as follows:

Using (37), we now come to the expression

that is valid by the hypothesis of induction (38). Thus, the relation (47) is valid if the relations (37) and (38) are supposed to be valid. It remains to prove the validity of the expression (48) if the validity of the expression (37) and (38) is supposed.

By complex conjugating (33), and relabelling as , we get

Differentiating this expression with respect to , we have

Further, multiplying (31) by yields

Given the last two results, the expression (48) becomes

(49)

Relabeling as in the formula (37), we have

(50)

Finally, substituting (50) into (49), we get the following expression:

which is valid by the hypothesis of induction (38). Thus, the validity of the expression (48) is also proved if the validity of (37) and (38) is supposed. In sum, we have proved by induction on that the statements (37) and (38) of Theorem 5.1 are valid.

The validity of the statement (39) follows from the relations (37) and (38). Indeed, substituting the conjugate of the expression (50) into (38) and taking into account that

we obtain the expression (39). Thus, the statement (39) of Theorem 5.1 is proved too. This completes the proof of theorem 5.1 in whole. Q.e.d.

By starting with the expression (36), the recursive relation (39) allows us to compute step by step the factors for all . The formulae (37) and (38) represent the relations between the factors and , hence between the constituents and . It is not difficult to show that, computing the derivatives of functions represented by (22), (23) to be used in (12) and (13), and combining the obtained results with (37), we get the equations (12) and (13) for power functions.

To illustrate how the relations (37) - (39) work we continue the consideration of Example 1.2. Recall that the function represented in the Cayley–Dickson doubling form as has

Given (22) - (25), we obtain from these expressions the sequences : , and : , Applying (37) to the corresponding elements of the sequence , we obtain the expressions , The formula (37) gives the values of , coinciding with the elements of the above sequence .

On the other hand, applying the formula (38) to the corresponding elements of the sequence , we get

The formula (38) gives the values, coinciding with values of elements of the above sequence .

The initial value satisfies (36). The formula (39) provides consistently the following :

coinciding with the values of the initial sequence . This example demonstrates the validity of relations (36)-(39).

Acknowledgements

I am indebted to Irina Levina for the continued support and Ephraim Gurevich for useful discussions.

References