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Essentially Adequate Concept of Holomorphic Functions in Quaternionic Analysis

Michael Parfenov

Bashkortostan Branch of Russian Academy of Engineering, Ufa, Russia

Abstract: The so-called essentially adequate concept of quaternionic holomorphic ( -holomorphic) functions defined as functions, whose quaternionic derivatives are independent of "the way of their computation", is developed. It is established that -holomorphic functions form one remarkable class of quaternionic functions whose properties are fully similar (essentially adequate) to complex ones: the quaternionic multiplication of these quaternionic functions behaves as commutative, the left quotient equals the right one, the rules for differentiating sums, products, ratios, inverses, and compositions are the same as in complex analysis. One can just verify these properties, constructing -holomorphic functions from their complex holomorphic counterparts by using the presented constructing rule. Several examples, confirming the theory in question, are considered. When using this concept there are no principal restrictions to build a quaternionic analysis similar to complex one. The elementary source flow and elementary vortex flow, allowing us to construct different 3D steady state fluid flows by superposition, are considered. To automate the processing of -holomorphic functions the pack of Mathematica® Programs is developed, part of which is presented.

Keywords: quaternionic holomorphic functions, quaternionic analysis, quaternionic generalization of Cauchy-Riemann’s equations, rules for quaternionic differentiation, quaternionic potential, 3D steady state fluid flows

Received May 25, 2020; Revised June 26, 2020; Accepted July 05, 2020

Citation: Michael Parfenov, Essentially Adequate Concept of Holomorphic Functions in Quaternionic Analysis. American Journal of Mathematical Analysis, 8 (1). 14-30. January 2020. doi:10.12691/ajma-8-1-3

1. Introduction

Looking for a quaternionic generalization of the complex theory [1] seems to be a natural and important issue in a development of mathematical analysis [2,3].

The known "left" or "right" definition of a quaternionic derivative (see, e.g. references in [4]) is incomplete (or non-essentially adequate), since each of underlying algebras, viz.: with only the "left" or only the "right" multiplication does not represent [8] all arbitrary rotations of vectors in 3D space. Some attempts to unite both approaches were made, however they were more like heuristic rather than systematic (see, e.g. [3,5]).

Unlike these attempts, the so-called essentially adequate concept of the quaternionic differentiability (holomorphy), uniting the left and the right approaches [4,6,7], is complete (or essentially adequate to properties of 3D space).

We denote an independent quaternionic variable by

where are real values, the values

[picture] | (1.1)

are complex constituents of the quaternion representation in the so-called Cayley-Dickson doubling form [8], and are quaternionic basis vectors in quaternion space

The quaternionic functions are, respectively, denoted by

where and are real-valued functions of real variables in the so-called -representation (see, e.g. [7]).

In the Cayley-Dickson doubling form [4,6,8] we have

[picture] | (1.2)

where

For simplicity, we denote the functions and (in the so-called -representation [7]) by and or simpler by and Upon the transition to the complex plane denoted by the Cayley-Dickson doubling form becomes the following [6]:

In accordance with the concept of essentially adequate differentiability [6] the definition of a derivative is based on the main idea, viz.: each point of any real line is at the same time a point of some plane and space as a whole, and therefore any characterization of differentiability at a point must be the same regardless of whether we think of that point as a point on the real axis or a point in the complex plane, or a point in three-dimensional space.

Given this, we have to define a quaternionic derivative as a limit of a difference quotient as tends to 0 [6]. At that the limit is required to be independent not only of directions to approach (as in complex analysis), but also of the manner of quaternionic division: on the left or on the right. Such an independence is called [6] the "independence of the way of computation".

This requirement corresponds to the requirement usual for analysis: the derivative must be unambiguous. At that an adequate representation of a physical 3D field strength, which is usually described by an unambiguous derivative, is achieved [11].

Definition 1.1 A quaternionic function is said to be the essentially adequate quaternionic holomorphic function (- holomorphic or, briefly, -holomorphic function) at a point p, if it has a quaternionic derivative independent of a way of its computation in some open connected neighborhood of a point

This definition leads to the definition of the necessary and sufficient conditions for to be -holomorphic [6].

Definition 1.2 Suppose that the constituents and of a quaternionic function possess the continuous first-order partial derivatives with respect to and in some open connected neighborhood of a point . Then a function is -holomorphic (further denoted by ) at a point p if and only if the functions and satisfy in the following quaternionic generalization of complex Cauchy-Riemann's equations:

[picture] | (1.3)

Here denotes the partial derivative with respect to For details we refer to [6].

The overbars designate the complex (also quaternionic if needed) conjugation. The brackets with the closing vertical bar indicate that the transition has been already performed in expressions enclosed in brackets.

Thus, -holomorphy conditions (1.3) are defined so that during the check of the quaternionic holomorphy of any quaternionic function we have to do the transition in already computed expressions for the partial derivatives of the functions and

However, this doesn’t mean that we deal with triplets in general, since the transition (or ) cannot be initially done for quaternionic variables and functions [6]. Any quaternionic function remains the same 4-dimensional quaternionic function regardless of whether we check its holomorphy or not. This transition is needed only to check the holomorphy of any quaternionic function. It is also used when solving the 3D tasks [7].

Essentially adequate conditions (1.3) of -holomorphy differ from the known left and right Cauchy - Riemann - Fueter conditions [2,3], which we can characterize as non-essentially adequate [6].

Now we recall the following theorems and their corollary needed for the sequel [6].

Theorem 1.3. Let a complex function be -holomorphic everywhere in a connected open set except, possibly, at certain singularities. Then a -holomorphic function of the same kind as can be constructed (without change of a functional dependence form) from by replacing a complex variable (as a single whole) in an expression for by a quaternionic variable where is defined (except, possibly, at certain singularities) by the relation in the sense that exactly follows from upon transition from to

Theorem 1.4. It is assumed that a quaternionic function where and are differentiable with respect to is -holomorphic everywhere in its domain of definition Then its full (uniting the left and right derivatives) quaternionic derivative, defined by

Where

is -holomorphic in as well, except, possibly, at certain singularities. If a quaternion function is once -differentiable in then it possesses derivatives of all orders in each one -holomorphic.

Corollary 1.5 All expressions for full derivatives of a -holomorphic function of the same kind as a complex holomorphic (briefly, -holomorphic) function have the same forms as the expressions for corresponding derivatives of a function

For example, if the first derivative of the -holomorphic function is where is a complex variable, then the first derivative of the -holomorphic function is where is a quaternionic variable.

Theorem 1.4 leads [6] to the following general expression for the full quaternionic derivatives of a -holomorphic function of all orders:

[picture] | (1.4)

where a th derivative of is denoted by and the constituents and are expressed by

Here and are constituents of the th derivative of represented in the Cayley-Dickson doubling form as We also denote the full first quaternionic derivative by the prime mark:

Using the following equations obtained in [4]:

[picture] | (1.5)

one can construct other expressions for derivatives of a -holomorphic function of all orders. For example,

They generalize in the quaternionic area the known formula variants for derivatives of -holomorphic functions [1].

The goals of this article are the following: 1) to formalize a rule for constructing -holomorphic functions, 2) to deduce rules for differentiating combinations of -holomorphic functions, 3) to identify properties of a class of -holomorphic functions similar to complex ones, 4) to consider the examples of 3D applications, 5) to automate cumbersome manual processing of the -holomorphic functions.

In the sequel, -holomorphic functions are supposed to be defined in domain if nothing other else is specified.

2. Constructing -holomorphic Functions

Now we formalize a general rule for constructing the functions from their complex counterparts when replacing a complex variable as a single whole by a quaternionic variable without change of a functional dependence form. The ultimate goal is to obtain from the expression for complex holomorphic function the quaternionic expression for its essentially adequate generalization (Theorem 1.3) in the Cayley-Dickson doubling form without change of a functional dependence form.

Constructing rule 2.1. The constructing rule is divided into 2 steps as follows.

Step 1. In an initial expression for -holomorphic function, depending only on complex variables as a single whole, we replace this variable by a quaternionic without change of a functional dependence form. By virtue of Theorem 1.3, the obtained function is -holomorphic. For example, the functions are -holomorphic in their domains of definition. Complex variables can have other designations.

Step 2. We represent the obtained expression for in the Cayley-Dickson doubling form (1.2). In order to obtain dependencies of complex functions and only on complex variables we replace real components and of a quaternion by the equivalent relations:

[picture] | (2.1)

which follow from (1.1).

If in an initial expression for special complex formulae are used, then in order that the nature of the functional dependence does not change we use instead of them their analogues valid in the quaternionic area. For example, we replace the complex Euler formula [1]:

by its quaternionic analog:

This quaternionic expression follows from the quaternion representation where and that is algebraically equivalent to the complex number representation: At that we also replace the imaginary unit by its analog which can be interpreted as a quaternionic generalization of the complex imaginary unit

Example 2.2. To illustrate this rule we construct the quaternionic natural logarithm function from the complex one. The initial complex natural logarithm function is the following [1]:

where (do not confuse here with quaternion’s component ), is an integer, is the principal value of and

We consider the principal branch of (as usual for all multivalued functions): Then we have

[picture] | (2.2)

Step 1. Replacing by we obtain

where

Step 2. Since when getting (2.2) in complex analysis the Euler formula is used, we replace by and have:

Substituting relations (2.1) and simplifying, we obtain finally the following expression for

[picture] | (2.3)

where

[picture] | (2.4)

[picture] | (2.5)

[picture] | (2.6)

and

As shown in [6], the function is -holomorphic, satisfying generalized Cauchy-Riemann's equations (1.3):

1)

2)

3)

4)

where is after setting

3. Differentiation Rules for Combinations of -holomorphic Functions

We consider the quaternionic generalizations of known complex rules and formulae [1] for differentiating powers, sums, products, ratios, inverses, and compositions of -holomorphic functions.

3.1 Constant function. Let a -holomorphic function be a constant where Then the following formula holds true:

Proof. Since the derivative of a constant with respect to any variable is zero, this formula is proved.

3.2 Multiplying by a constant. A -holomorphic function multiplied by an arbitrary constant

[picture] | (3.1)

is -holomorphic as well. The first quaternionic derivative of is the following:

[picture] | (3.2)

Proof. Let a quaternionic function be -holomorphic. Then it satisfies condition (1.3) of -holomorphy as follows:

[picture] | (3.3)

Consider the function Substituting this function into (1.3), we have

i.e. system of equations which is equivalent to (3.3). Thus the function is -holomorphic. The validity of (3.2) follows from (1.4). This rule is proved.

By virtue of rule 3.1 formula (3.2) is also valid for the function where It is not superfluous to note that its special case, when coincides with the result of the theory based on Cauchy-Riemann-Fueter equations [2,3], however this theory is in principle restricted by this result.

3.3 Power rule. The quaternionic derivative of a power function where is the following:

[picture] | (3.4)

Proof. The validity of formula (3.4) follows from Theorem 1.3 and Corollary 1.5.

3.4 Sum and difference rule. A sum of a finite number of the -holomorphic functions ( is an integer) is also holomorphic. The full -holomorphic derivative of the sum is the following:

Proof. Let the functions be -holomorphic. Then each of them satisfies equations (1.3) as follows:

Adding the functions … by component-wise addition, we obtain as follows:

whence

The derivatives which we need to substitute into equation (1.3-1) are the following:

After performing the transition in them, we can rewrite equation (1.3-1) as follows:

Since by virtue of equations (3.3) we have and so on, we can state that this equation is satisfied for the sum of -holomorphic functions. Analogously, we prove the validity of the other equations of system (1.3) for the sum Thus the sum is -holomorphic.

This rule also holds true when subtracting of an arbitrary number of -holomorphic functions.

Using formula (1.4) for we get the full first quaternionic derivative of as follows:

The sum rule is proved.

It is evident that this rule remains valid when considering any constants instead of When considering where instead of the sum rule represents the property of linearity of the quaternionic derivative.

The following theorem is needed for the sequel.

Theorem 3.5 Suppose the quaternionic functions and are -holomorphic in Then, in hold true: (i) the quaternionic product is also -holomorphic, (ii) the quaternionic multiplication of the functions and behaves as commutative.

Proof. (i) Consider the quaternionic product As shown in [6] (when proving Theorem 1.3), there exist only two ways to go from equations (1.3) of -holomorphy to the Cauchy-Riemann equations, viz.: when and when By the is here denoted a complex variable with imaginary unit or instead of

Both ways mean the same one-to-one correspondence between the set of all -holomorphic functions and the set of all -holomorphic ones. In other words, upon the transition from quaternions to complex numbers each complex holomorphic function follows uniquely from the corresponding -holomorphic function and vice versa.

If and are -holomorphic, then and are -holomorphic. In complex analysis the product of two and more -holomorphic functions is also a -holomorphic function [1]. Then, replacing by in the product without change of a functional dependence form ( and ), we infer by virtue of Theorem 1.3 that the product is -holomorphic.

In the expression the dot "" denotes complex multiplication, however in the expression the dot "" is already associated with quaternionic multiplication. Such a replacement does not lead to the change of a functional dependence form, since the forms of rules for the complex and quaternionic multiplication are the same. We can see this as follows.

For quaternionic multiplication in the Cayley-Dickson doubling form (1.2) we have the following rule [8]:

[picture] | (3.5)

where quaternions and are components are complex, and and are real variables.

Performing the transition from quaternionic variables to complex ones, we put [6]. This leads to two complex variables and where imaginary unit () plays a role of the habitual complex imaginary unit At that the rule of quaternionic multiplication (3.5) reduces to the complex rule of multiplication [1]:

Since the replacement of a complex variable (as a single whole) by a quaternionic one in expressions for -holomorphic functions is a reverse procedure, we have no changing a functional dependence form. Statement (i) of the theorem is proved. This holds always true when we consider quaternionic product instead of complex one.

(ii) As shown in [4], the general expressions for constituents and of a -holomorphic function are the following:

[picture] | (3.6)

[picture] | (3.7)

where or another symmetric form invariant under complex conjugation.

Assume that the functions and are -holomorphic. This means that they satisfy -holomorphy equations (1.3), where instead of functions and we put, respectively, and in the case of the function or put and in the case of the function Then, introducing the designations we have in accordance with (3.6) (3.7) the following valid expressions:

[picture] | (3.8)

[picture] | (3.9)

Usual quaternion multiplication is non-commutative for arbitrary quaternionic functions and However, we now prove that the -holomorphic functions, satisfying equations (1.3), possess such a property that the quaternionic multiplication of these functions behaves as commutative:

Using rule (3.5) for quaternion multiplication of the functions and we obtain the following expressions:

We will now prove that in the case of -holomorphic functions and the following equalities are valid:

[picture] | (3.10)

[picture] | (3.11)

Using (3.8) and (3.9), we get the following expressions:

whence

Thus, we have proved that equality (3.10), i. e. is valid for -holomorphic functions and The only thing left to do is to prove that the equality (3.11) is also valid for -holomorphic functions and

According to (3.11), we have for the constituent of the product the following expression:

[picture] | (3.12)

Its complex conjugation is

[picture] | (3.13)

Since, as proved in (i), the quaternionic product is -holomorphic, we write the and its conjugation, according to (3.7), as follows:

whence

[picture] | (3.14)

Substituting (3.12) and (3.13) into (3.14), we obtain the following expression:

[picture] | (3.15)

Substituting expressions (3.8) and (3.9) as well as their conjugates into (3.15), we have

Whence

[picture] | (3.16)

Further, multiplying both sides of (3.16) by we have

Finally, using (3.8), (3.9) in the last expression, we get the following expression:

which coincides with (3.11). Statement (ii) of the theorem is proved. This completes the proof of the theorem in whole.

It is evident that by virtue of the associativity law of the quaternion multiplication, this theorem can be applied to an arbitrary number of multiplied -holomorphic functions.

Example 3.6 Consider the quaternionic product of the -holomorphic functions and According to constructing rule 2.1, we have the -holomorphic function

where

The function and its constituents and are defined by (2.3), (2.4), and (2.5).

To simplify all expressions we introduce the following notation:

Using the rule of quaternionic multiplication (3.5), we get the following expression for the quaternionic product of the function and

Where

[picture] | (3.17)

[picture] | (3.18)

Interchanging the order of multiplication, we obtain the following expression for quaternionic multiplying by

where

[picture] | (3.19)

[picture] | (3.20)

Comparing expressions (3.17) and (3.19) as well as (3.18) and (3.20), we prove the equalities:

Thus, we see that the quaternionic multiplication of the -holomorphic functions and behaves as commutative.

3.7 Chain rule. Suppose the quaternionic functions and are -holomorphic. Then the composite function is also -holomorphic. The following differentiation formula holds true:

[picture] | (3.21)

where denotes the derivative of with respect to

Proof. The -holomorphy of the function follows from Theorem 1.3. The differentiation formula (3.21) follows from Corollary 1.5.

3.8 Reciprocal Rule. Let a quaternionic function be -holomorphic. Then the multiplicative inverse is also -holomorphic. The following differentiation rule holds true:

[picture] | (3.22)

Proof. The -holomorphy of the quaternionic function follows from Theorem 1.3. Formula (3.22) follows from Corollary 1.5.

3.9 Product rule. Let quaternionic functions and be -holomorphic. Then the full derivative of their quaternionic product can be calculated by the following formulae:

[picture] | (3.23)

[picture] | (3.24)

Proof. The -holomorphy of the function follows from Theorem 3.5. The differentiation formulae (3.23) and (3.24) follow from Corollary 1.5.

Assertion 3.10 Let the quaternionic functions and be -holomorphic. Then the left quotient of by is equal to the right one.

Proof. By the definition [8] the left and right quotients of by are, respectively, the following:

Since the left and right quotients can be represented as follows:

[picture] | (3.25)

[picture] | (3.26)

According to reciprocal rule 3.8, the function is -holomorphic. Then, by Theorem 3.5, the quaternionic product behave as commutative and we get

The assertion is proved.

Given it, the left or right manner of quotient computation alone may be used in the case of -holomorphic functions.

Example 3.11 We consider the left and right quotients of the function by

According to constructing rule 2.1, we obtain

[picture] | (3.27)

where

[picture] | (3.28)

According to rule 2.1, beginning with Step 2, we get

[picture] | (3.29)

where

[picture] | (3.30)

and is defined by (2.6). It is easy to verify that the function as well as its conjugate are -holomorphic.

There exists the following identity:

Using (3.25) and (3.5), we get the following expressions:

where

[picture] | (3.31)

[picture] | (3.32)

On the other hand, we have as follows:

Where

[picture] | (3.33)

[picture] | (3.34)

Substituting (3.28), (3.30) into (3.31), (3.32), (3.33) and (3.34), we finally get the following identities:

Thus, the left and right quotients of by are equal.

3.12 Quotient rule If the quaternionic functions and are -holomorphic, then the following quotient rule holds true:

Proof. Using formula (3.25) for the left quotient, product rule 3.9, reciprocal rule 3.8 and Theorem 3.5 we obtain

Using formula (3.26) for the right quotient, we have the same result. Obviously that all algebraic operations here are correct, since the quaternionic derivatives and by virtue of Theorem 1.4 are also -holomorphic functions and their multiplication behaves also as commutative. The quotient rule is proved.

Note that the algebraic similarity between -holomorphic and -holomorphic functions allows proving the presented rules in terms of limits as in the complex analysis. We see that all got differentiating rules are similar to complex ones.

Example 3.13 To illustrate the efficiency of the above rules we consider the composite -holomorphic function According to rule 2.1, we obtain

[picture] | (3.35)

where

[picture] | (3.36)

[picture] | (3.37)

[picture] | (3.38)

and is defined by (2.6).

Consider the first derivative of the function Using formula (1.4), we calculate the first derivative as follows:

[picture] | (3.39)

where

[picture] | (3.40)

[picture] | (3.41)

After cumbersome and quite tedious calculation we get

Substituting the last two results into (3.40), we get the following expression for

[picture] | (3.42)

Analogously, we obtain as follows:

Combining these expressions and (3.41), we get

[picture] | (3.43)

Further, substituting (3.42), (3.43) into (3.39) we get the following expression for the first derivative of

Substituting (3.38), uncovering brackets, and then uniting the summands, involving the functions and we obtain after rather cumbersome calculations the following expression:

whence, using (3.36), (3.37) and (3.38), we have

Given the rule of quaternionic multiplication (3.5), we can rewrite this expression as follows:

On the other hand, using chain rule 3.7, we can directly get this expression:

[picture] | (3.44)

Thus, we see that using chain rule 3.7 reduces essentially the volume of calculations.

Consider the second derivative of Using (1.4), we obtain after tedious calculations the following expression:

where

[picture] | (3.45)

[picture] | (3.46)

On the other hand, given (3.44), and using rules 3.2, 3.7, and 3.9, we get

Substituting (3.27) and (3.35), we obtain as follows:

Using (3.36), (3.37), (3.28) and (3.5), we get, after some algebra, expressions for and coinciding with expressions (3.45) and (3.46). Obviously, calculations with using the got rules for differentiating combinations of -holomorphic functions are much simplier. In Appendix we present computing programmes, regarding to this example and essentially simplifying all considered calculations.

4. Class of -holomorphic Functions

The class of the -holomorphic functions includes quaternionic functions that satisfy equations (1.3) of the essentially adequate quaternionic generalization of the complex Cauchy-Riemann equations. Among all quaternionic functions they alone possess one remarkable feature: their algebraic and differential operations are fully identical to complex ones. Each of these functions can be obtained from its complex holomorphic counterpart by using rule 2.1. We can list their properties as follows.

1) The quaternionic multiplication of the -holomorphic functions behaves as commutative one and the left quotient of two -holomorphic functions equals the right one.

2) The differentiation rules for -holomorphic functions are the same as for -holomorphic ones.

3) The constituents of the -holomorphic functions (and their derivatives) in the Cayley-Dickson doubling form (1.2) have the following general representation form [4]:

Such forms are typical of -holomorphic functions and their derivatives. They could serve as gauge for correctness of results obtained by using constructing rule 2.1.

4) The constituents and of the -holomorphic functions (as well as and and of their derivatives, etc) are symmetric in variables and respectively [4,6]. Such a symmetry for derivatives is a consequence of uniting unsymmetrical parts of the left and right derivatives [4], reflecting undoubtedly a symmetry of physical space.

5) The constituents and satisfy also equations (1.5) [4,6].

6) The constituents of -holomorphic functions in -representation are -harmonic functions in the sense that they satisfy the generalizated quaternionic Laplace equations [7].

7) -holomorphic functions have local representations by convergent power series [9], i.e. are analytic functions.

It can be supposed that the commutative behavior of quaternionic multiplication and the equality of the left and right quotients in the case of -holomorphic functions exist in an "objective reality" independently from whether any theory exists or not. One can just verify these properties without even knowing any theory, constructing -holomorphic functions from their complex holomorphic counterparts in accordance with rule of constructing 2.1.

When using this class there are no principal restrictions to build a quaternionic differential analysis similar to complex one.

Given that the -holomorphic functions are infinitely differentiable, one can denote this class by the symbol

Note that due to identity of properties of - holomorphic and - holomorphic functions all proofs of limit theorems in quaternionic area can be the same as in complex one.

5. Some Applications

We follow the theory of quaternionic potential and the notation presented in [7]. Two elementary steady state fluid flows from which it is possible to construct more flows by superposition are here considered.

5.1. Elementary Source Flow

The elementary source flow is described in complex analysis [11] by the -holomorphic function or complex potential The quaternionic potential of this flow is correspondingly the -holomorphic function [7]. We rewrite expressions (2.3) - (2.6) as follows:

where

[picture] | (5.1)

[picture] | (5.2)

Substituting (1.1) and grouping terms with imaginary units and we get the following expression for the quaternionic potential in -representation:

where

After transition (to 3D space), we obtain

where

As shown in [6], we have Using (1.1), we obtain the first quaternionic derivative of the function in -representation as follows:

where

According to the quaternionic potential theory [7], we have for the quaternionic potential the following flow velocity vector (field vector) in 3D space:

[picture] | (5.3)

where

[picture] | (5.4)

[picture] | (5.5)

[picture] | (5.6)

On the other hand, we can directly apply the 3D gradient operator [7]:

[picture] | (5.7)

to the function Then, we obtain the following result:

where

The last expressions coincide with expressions (5.4), (5.5) and (5.6), respectively. Hence the steady state fluid flow, corresponding to quaternionic potential is a potential one in except at the singularity

The 3D flow velocities calculated in accordance with (5.3) are plotted in Figure 1.

Figure 1. The overall picture of the velocity vectors for quaternionic potential Logp. The vectors near the origin are highlighted by orange color. The number of depicted flow lines is 6

The flow velocity vectors are directed radially away from the origin and indicated with arrows. The sizes of depicted arrows depend on the absolute values of the flow velocity:

Note that this dependence corresponds to the analogous dependence for complex potential [11].

Now by analogy to the example of 3D flow modeling considered in [7], we find equations of 3D equipotential surfaces and stream surfaces for the quaternionic potential

For 3D equipotential surfaces we obtain the equation

whence it follows that 3D equipotential surfaces can be represented as spheres with centres at

[picture] | (5.8)

where is an arbitrary constant.

According to [7], the 3D stream surfaces are to be defined by the following generating function:

[picture] | (5.9)

where are constants.

Equations when and when give the following traces of the desired stream surfaces in the planes and respectively:

Leaving designations for constants unchanged, we reduce these equations for traces to the following equations:

,

.

Since and lie in the range of values of the function we have the restrictions

Putting and rotating counterclockwise the plane around the -axis by the angle we can obtain the trace from Hence the desired stream surfaces for quaternionic potential in the simplest case are the surfaces of revolution [12] around the -axis.

To obtain the surfaces of revolution from we use the known method of replacing the variable by [12] in the expression for and get the following equation for the stream surfaces

[picture] | (5.10)

where

It easy to see that equation (5.10) is the equation of 3D conical surfaces [12] that have the axis and vertexes at the origin. The streamlines are situated on these imagined 3D stream surfaces

From equations (5.8) and (5.10), it follows that the equation of a curve of intersection of the surfaces and is the following:

where and is the circle of radius with the center on the -axis.

Assuming, for definiteness, that we get the following values of constants: and for the surfaces and Then the required equations of the surfaces reduce, respectively, to the following ones:

[picture] | (5.11)

[picture] | (5.12)

Figure 2 demonstrates two orthogonal surfaces and depicted, according to equations (5.11) and (5.12).

Figure 2. The quaternionic potential Logp. Orthogonal stream and potential surfaces: S for K²=0,69 and φ₁ for K₁²=13. The curves of intersection are highlighted by blue color

Verifying the orthogonality of and at the points is essentially the same as in the example of 3D flow modeling considered in [7]. We shall not dwell on this here.

To retain an analogy with a "planar" flow in the complex plane, we can regard the described 3D flow to the flow with rate of mass transport equal to 1, however a further study of this analogy is beyond the scope of the present paper.

5.2. Elementary Vortex Flow

In the complex analysis the elementary vortex flow is represented by the principal value of the function where denotes the habitual imaginary unit [11]. This function satisfies known complex Cauchy-Riemann's equations [1,11]. However the quaternionic function doesn't satisfy the quaternionic system of generalized Cauchy-Riemann's equations (1.3).

As mentioned above (constructing rule 2.1), we can replace the by the quaternionic "imaginary vector" defined by (3.29). The function satisfies equations (1.3):

1)

2)

3)

4)

Hence the function is -holomorphic.

Since the functions and are -holomorphic, the function by virtue of Theorem 3.5 is also -holomorphic. Considering this function as the quaternionic potential of elementary vortex flow, rewrite it as follows:

where in accordance with multiplication rule (3.5) we have

Substituting expressions (3.30), (5.1) and (5.2) into the last two expressions, we finally obtain as follows:

It is easy to verify that these expressions are equal, respectively, to expressions for and that is, the quaternionic product is commutative.

Using (1.1) and grouping terms with imaginary units and we get the following expression for the quaternionic potential in -representation:

where

and is defined by (2.6),

After transition (to 3D space), we have

where

[picture] | (5.13)

Combining expression (1.4) for and (3.30), we calculate the first derivative of as follows:

[picture] | (5.14)

By virtue of product rule 3.9 and zero result (5.14) the first derivative of is the following:

Given the rule of multiplication (3.5), we have as follows:

where

[picture] | (5.15)

[picture] | (5.16)

As shown in [6], the constituents of the function are the following:

[picture] | (5.17)

Substituting expressions (3.30) and (5.17) into (5.15) and (5.16), we obtain the following expressions:

Substituting (1.1) into the last expressions and grouping terms with imaginary units and we get the following expression for the quaternionic derivative of the function in -representation:

where

According to the quaternionic potential theory [7], we get the following expression for the flow velocity vector (field vector) in 3D space:

[picture] | (5.18)

where

[picture] | (5.19)

[picture] | (5.20)

[picture] | (5.21)

Applying 3D gradient operator (5.7) directly to the function defined by (5.13), we obtain the following expression:

where

The obtained expressions for coincide with expressions (5.19), (5.20), (5.21), respectively. Thus, the steady state fluid flow, corresponding to the quaternionic potential is a potential one in except at the singularity point

According to [7], we get equations of 3D equipotential surfaces and stream surfaces for the quaternionic potential

For 3D equipotential surfaces we have the equation

where is a constant. Leaving the designations for constants unchanged, we reduce it to the following:

[picture] | (5.22)

The constant is restricted to since it lies in the range of values of the function

It easy to see that equation (5.22) is the equation of 3D conical surfaces that have the axis and the vertexes at the origin.

To define the stream surfaces we use the generating function (5.9). From the equations when and when it follows that the desired stream surface has in the planes and the following equations of traces, respectively:

where are constants.

Leaving the designations for constants unchanged, we reduce these equations to the following ones:

which represent the circles with centres at the origin.

Putting and rotating counterclockwise the plane around the -axis by the angle we can obtain the trace from Hence the desired stream surfaces when considering the quaternionic potential in the simplest case, are the surfaces of revolution [12] around the -axis. Analogously to the previous subsection, we get these from the trace by replacing the variable by

[picture] | (5.23)

The surfaces (5.23) are 3D spheres with centres at the origin. The streamlines are situated on these imagined 3D stream surfaces

Comparing expressions (5.8), (5.10) with expressions (5.22), (5.23), we see that the equations of and for quaternionic potentials and switch places with each other, just as in complex analysis [11]. At that the coefficients and is changed with each other.

Analogously to the previous subsection, we obtain the equation of a curve of intersection of surfaces and from equations (5.22) and (5.23) as follows:

Assuming, just as in the case of the quaternionic potential that we get the following values of constants: and for the surfaces and in the case of quaternionic potential Then equations (5.22) and (5.23) reduce to the following:

[picture] | (5.24)

[picture] | (5.25)

In complex analysis the velocity vector of the elementary vortex flow is represented [11] by the expression Moving of the flow in a circle in the complex plane is due to the fact that the flow velocity vector in accordance with this expression becomes opposite in sign (remaining equal in absolute value), if signs of the variables and are changed simultaneously.

Since components (5.19), (5.20), and (5.21) of the velocity do not change their signs to opposite ones due to the simultaneous change in the signs of expression (5.18) is not good enough for modeling an elementary 3D vortex flow. To correct this we introduce the piecewise smooth function as follows:

[picture] | (5.26)

It is evident that this function, according to rule 3.2 (at ), is piecewise -holomorphic. The equations of equipotential surfaces and stream surfaces are the same for both parts of expression (5.26).

The corresponding function is the following:

Ultimately, we can write the flow velocity vector as follows:

[picture] | (5.27)

where the vector components in braces are defined by expressions (5.19), (5.20), (5.21).

Figure 3 demonstrates two orthogonal surfaces and computed by using formulae (5.24) and (5.25).

Figure 3. The quaternionic potential rLogp. Example of orthogonal stream and potential surfaces: S for K²=13 and φ₁ for K₁²=0,69. The intersection curves are highlighted by blue color

The flow velocity vectors indicated with arrows on the surface are computed by using formula (5.27).

They have the same length, since have the same absolute value on the sphere, depending only on the radius of this sphere:

Such a vector picture could be imagined as rotating a complex plane (with depicted moving of the flow in a circle) counterclockwise around the -axis by the angles from 0 to radians. We do not present an overall picture of the velocity vectors for quaternionic potential since it is the same as on the depicted sphere distributed at the whole space.

Just as in the previous subsection we shall not dwell on verifying the orthogonality of and at the points

By creating Figure 1, Figure 2, Figure 3 the computing system Wolfram Mathematica® [10] was used.

The presented results give a reason for building other segments of quaternionic analysis similar to complex ones.

To avoid an error-prone cumbersome and tedious manual procedure of calculations with -holomorphic functions we have developed the special software pack written in the programming language Wolfram®. By using this pack all calculations can be immediately carried out.

This pack allows us to test the holomorphy of any quaternionic function, the multiplicative commutativity of -holomorphic functions, calculate expressions for them and their derivatives, including the quaternionic potentials and expressions for 3D potential fields, field divergence and vortex density as well as get pictures of 3D potential fields such as in Figure 1, Figure 2, Figure 3. A lot of examples of -holomorphic functions is considered in this pack. The part of developed programmes is presented in Appendix below.

References

Appendix

We present here the only five computing programs to illustrate processing of the -holomorphic function (see Example 3.13).

Program 1 computes the functions f1 (= or for the derivative) and f2 ( or ) in the case when the function in question (or its derivative) does not have operations of the quaternionic multiplication explicitly. The function is such a case. The got functions f1 and f2 for are further used as input data for Program 5, which tests whether a quaternionic function is -holomorphic. The variables are denoted by a, ac, b, bc, respectively. The conjugate functions have extra sign "c".

In the case when a quaternionic function does contain explicitly the quaternionic multiplication operations Program 4 is used. Program 4 calculates f1 and f2 to be further used in Program 5 and also tests whether the quaternionic multiplication in this case behaves as commutative. The input data for Program 4 are the components inia1, inia2 of the first factor of the quaternionic product (inia1+ inia2)( inib1+inib2) and also the components inib1, inib2 of the second factor, calculated by Program 2 for the first derivative of or Program 3 for the second derivative of

All the programs are written each in a separated cell. If needed the programmes can be directly copied cell by cell from this text or its PDF into input cells (only one program into one cell!) of the opened Notebook blank .nb. The programmes are ready for use immediately after copying. The functions without semicolon at the end of them must be in separate lines. They can be also used with initial data of other functions. To launch each program it is needed to click on it's cell, and then holding down the Shift key while press the Enter key. For the sake of brevity, the results of only calculating for the second derivative of are below represented. First the cell with Program 3 is evaluated, then the cell with Program 4, and finally the cell with Program 5.

(*Program 1*)

(*The function e^p^2, input data for Program 5*)

Clear[f1,f2,f1c,f2c];

quatfunctiontested:=Evaluate[e^p^2];

v=(1/2)*Sqrt[4*(a*ac+b*bc)-(a+ac)^2];

β1=E^((a^2+ac^2-2 b bc)/2);θ=(a+ac)*v;

f1[a_,ac_,b_,bc_]:=β1*(Cos[θ]+((a-ac)Sin[θ])/(2*v));

f2[a_,ac_,b_,bc_]:=(β1*Sin[θ]*b)/v;

f1c[a_,ac_,b_,bc_]:=β1*(Cos[θ]+((ac-a)Sin[θ])/(2*v));

f2c[a_,ac_,b_,bc_]:=(β1*Sin[θ]*bc)/v;

Print["Constituents f1,f2:"];

f1[a,ac,b,bc]

f2[a,ac,b,bc]

(*Program 2*)

(*The first derivative (e^p^2)´= e^p^2* 2p, initial data

for Program 4*)

Clear[f1,f2,f1c,f2c,inia1,inia1c,inia2,inia2c,inib1,inib1c,inib2,inib2c];

quatfunctiontested:=Evaluate[2e^p^2*p];

v=(1/2)*Sqrt[4*(a*ac+b*bc)-(a+ac)^2];

β1=E^((a^2+ac^2-2 b bc)/2);θ=(a+ac)*v;

inia1[a_,ac_,b_,bc_]:=β1*(Cos[θ]+((a-ac)Sin[θ])/(2*v));

inia1c[a_,ac_,b_,bc_]:=β1*(Cos[θ]+((ac-a)Sin[θ])/(2*v));

inia2[a_,ac_,b_,bc_]:=(β1*Sin[θ]*b)/v;

inia2c[a_,ac_,b_,bc_]:=(β1*Sin[θ]*bc)/v ;

inib1[a_,ac_,b_,bc_]:=2*( a^2-b*bc); inib1c[a_,ac_,b_,bc_]:=2*( ac^2-b*bc);

inib2 [a_,ac_,b_,bc_]:= 2*(a+ac)*b;

inib2c [a_,ac_,b_,bc_]:= 2*(a+ac)*bc;

Print["Constituents inia1,inia2:"];

inia1[a,ac,b,bc]

inia2[a,ac,b,bc]

Print["Constituents inib1,inib2:"];

inib1[a,ac,b,bc]

inib2[a,ac,b,bc]

(*Program 3*)

(*The second derivative (e^p^2)´´= 2e^p^2*(2p^2+1), initial data for Program 4*)

Clear[inia1,inia1c,inia2,inia2c,inib1,inib1c,inib2,inib2c];

v=(1/2)*Sqrt[4*(a*ac+b*bc)-(a+ac)^2];θ=(a+ac)*v;

mu:=Cos[θ]+((a-ac)*Sin[θ])/(2*v); muc:=Cos[θ]+((ac-a) *Sin[θ])/(2*v); beta:=E^((a^2+ac^2-2 b bc)/2);

quatfunctiontested:=Evaluate[2*e^p^2*(2p^2+1)];

inia1[a_,ac_,b_,bc_]:=2*beta*mu;

inia1c[a_,ac_,b_,bc_]:=2*beta*muc;

inia2[a_,ac_,b_,bc_]:=(2*beta*Sin[θ]*b)/v;

inia2c[a_,ac_,b_,bc_]:=(2*beta*Sin[θ]*bc)/v ;

inib1[a_,ac_,b_,bc_]:= 1+2*(a^2-b*bc);

inib1c[a_,ac_,b_,bc_]:=1+2*(ac^2-b*bc);

inib2 [a_,ac_,b_,bc_]:= 2*b*(a+ac);

inib2c [a_,ac_,b_,bc_]:=2*bc*(a+ac) ;

Print["Constituents inia1, inia2 :"];

inia1[a,ac,b,bc]

inia2[a,ac,b,bc]

Print["Constituents inib1, inib2 :"];

inib1[a,ac,b,bc]

inib2[a,ac,b,bc]

Outputs:

Constituents inia1, inia2:

Constituents inib1, inib2:

Remark. The results for inia1, inia2 coincide, respectively, with above expressions (3.36), (3.37) for and multiplied by 2. The results for inib1, inib2 coincide, respectively, with above expressions (3.28) for and multiplied by 2, with addition of +1 to

(*Program 4*)

(*Input data for Program 5 and commutativity testing*)

vorf1[a_,ac_,b_,bc_]:=inia1[a,ac,b,bc]*inib1[a,ac,b,bc]-inia2[a,ac,b,bc]*inib2c[a,ac,b,bc];

vorf1c[a_,ac_,b_,bc_]:=inia1c[a,ac,b,bc]*inib1c[a,ac,b,bc]-inia2c[a,ac,b,bc]*inib2[a,ac,b,bc];

f1[a_,ac_,b_,bc_]:=FullSimplify[vorf1[a,ac,b,bc]];

f1c[a_,ac_,b_,bc_]:=FullSimplify[vorf1c[a,ac,b,bc]];

q1q2f1:=f1[a,ac,b,bc]; q1q2f1c:=f1c[a,ac,b,bc];

vorf2[a_,ac_,b_,bc_]:=inia1[a,ac,b,bc]*inib2[a,ac,b,bc]+

inia2[a,ac,b,bc]*inib1c[a,ac,b,bc];

vorf2c[a_,ac_,b_,bc_]:=inia1c[a,ac,b,bc]*inib2c[a,ac,b,bc]+

inia2c[a,ac,b,bc]*inib1[a,ac,b,bc];

f2[a_,ac_,b_,bc_]:=FullSimplify[vorf2[a,ac,b,bc]];

f2c[a_,ac_,b_,bc_]:=FullSimplify[vorf2c[a,ac,b,bc]];

Print["Result of computing: Constituents f1, f2 of ",quatfunctiontested, ":"];

f1[a,ac,b,bc]

f2[a,ac,b,bc]

q1q2f2:=f2[a,ac,b,bc]; q1q2f2c:=f2c[a,ac,b,bc];

vorϕf1[a_,ac_,b_,bc_]:=inia1[a,ac,b,bc]*inib1[a,ac,b,bc]-inia2c[a,ac,b,bc]*inib2[a,ac,b,bc];

vorϕf1c[a_,ac_,b_,bc_]:=inia1c[a,ac,b,bc]*inib1c[a,ac,b,bc]-inia2[a,ac,b,bc]*inib2c[a,ac,b,bc];

ϕf1[a_,ac_,b_,bc_]:=FullSimplify[vorϕf1[a,ac,b,bc]];

ϕf1c[a_,ac_,b_,bc_]:=FullSimplify[vorϕf1c[a,ac,b,bc]];

q2q1f1:=ϕf1[a,ac,b,bc]; q2q1f1c:=ϕf1c[a,ac,b,bc];

vorϕf2[a_,ac_,b_,bc_]:=inia1c[a,ac,b,bc]*inib2[a,ac,b,bc]+

inia2[a,ac,b,bc]*inib1[a,ac,b,bc];

vorϕf2c[a_,ac_,b_,bc_]:=inia1[a,ac,b,bc]*inib2c[a,ac,b,bc]+

inia2c[a,ac,b,bc]*inib1c[a,ac,b,bc];

ϕf2[a_,ac_,b_,bc_]:=FullSimplify[vorϕf2[a,ac,b,bc]];

ϕf2c[a_,ac_,b_,bc_]:=FullSimplify[vorϕf2c[a,ac,b,bc]];

q2q1f2:=ϕf2[a,ac,b,bc]; q2q1f2c:=ϕf2c[a,ac,b,bc];

coins1:=True===FullSimplify[q1q2f1==q2q1f1];

coins2:=True===FullSimplify[q1q2f2==q2q1f2];

coins:=coins1&&coins2;

If[coins,Print["Result of testing: Quaternionic multiplication in the case of ",quatfunctiontested," is commutative"],Print["Result of testing: Quaternionic multiplication in the case of ",quatfunctiontested," is not commutative"]]

Outputs:

Result of computing: Constituents f1, f2 of

Result of testing: Quaternionic multiplication in the case of is commutative.

Remark. The results for f1 and f2 (constituents of the second derivative ) coincide with above expressions (3.45) and (3.46) for and respectively.

(*Program 5*)

(*Testing H-holomorphy of quaternionic functions by equations (1.3)*)

df1:=D[f1[a,ac,b,bc],{{a,bc}}];

trdf1:=df1/.{a->x,ac->x};

sftrdf1:=FullSimplify[trdf1];

df1c:=D[f1c[a,ac,b,bc],{bc}];

trdf1c:=df1c/.{a->x,ac->x};

sftrdf1c:=FullSimplify[trdf1c];

df2:=D[f2[a,ac,b,bc],{{a,ac,b}}];

trdf2:=df2/.{a->x,ac->x};

sftrdf2:=FullSimplify[trdf2];

df2c:=D[f2c[a,ac,b,bc],{bc}];

trdf2c:=df2c/.{a->x,ac->x};

sftrdf2c:=FullSimplify[trdf2c];

trmdf1da:=PowerExpand[sftrdf1[ [ 1 ] ] ]//Expand;

trmdf2cdbc:=PowerExpand[sftrdf2c]//Expand;

eq1:=True===FullSimplify[trmdf1da==trmdf2cdbc];

trmdf2da:=PowerExpand[sftrdf2[ [ 1 ] ] ]//Expand;

trmdf1cdbc:=PowerExpand[-sftrdf1c]//Expand;

eq2:=True===FullSimplify[trmdf2da==trmdf1cdbc];

trmdf2db:=PowerExpand[sftrdf2[ [ 3 ] ] ]//Expand;

eq3:=True===FullSimplify[trmdf1da==trmdf2db];

trmdf2dac:=PowerExpand[sftrdf2[ [ 2 ] ] ]//Expand;

trmdf1dbc:=PowerExpand[-sftrdf1[ [ 2 ] ] ]//Expand;

eq4:=True===FullSimplify[trmdf2dac==trmdf1dbc];

If[eq1∧eq2∧eq3∧eq4,Print["Result of testing: The quaternionic function ",quatfunctiontested," is H-holomorphic"],Print["Result of testing: The quaternionic function ",quatfunctiontested,"

is not H-holomorphic"]];

If[eq1,,Print["The equation 1) is not satisfied"]]

If[eq2,,Print["The equation 2) is not satisfied"]]

If[eq3,,Print["The equation 3) is not satisfied"]]

If[eq4,,Print["The equation 4) is not satisfied"]]

Output:

Result of testing: The quaternionic function is H- holomorphic.

We see that the second derivative of is -holomorphic.

Remark. We can also verify the -holomorphy of the function by launching Program 1 and than Program 5 as well as the -holomorphy of the first derivative by launching Program 2, then Program 4 and finally Program 5.

The complete pack of programmes entitled "Processing of the -holomorphic functions" is scheduled to be shortly published.

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