Chapter 6
The R1-Fractional Trigonometry

6.1 R₁-Trigonometric Functions

The R₁-fractional trigonometry [73–75] is defined by taking the parameter a in the argument of the R-function to be imaginary. This is in contrast to Chapter 5 in which both the a and t parameters were taken to be real. Then, the R₁-trigonometry is based on , where R-function is expanded into even and odd series, that is, series with terms that are even and odd powers of a. This yields

6.1

6.2

6.3

We define to be the even part of , that is, the terms in equation (6.1) with even powers of a, and to be the odd part of , that is, the terms in equation (6.1) with odd powers of a. Then, we have and from equation (6.3). Therefore, -function is defined as the real part of equation (6.1) and the -function as the imaginary part. Because t is raised to a fractional power in these series, the series are multivalued or indexed. To elucidate the indexed behavior of these functions, we consider the roots of t. Then, from equation (3.122),

where for the exponent of t is

where and are assumed rational and irreducible, , and M/D is rational and in minimal form. From Lorenzo and Hartley [74], with permission of Springer:

Then, the multivalued, or indexed, definition for the is given by

6.4

with , , with common factors in M/D removed. The and are defined in a similar manner

6.5

with , and

6.6

The principal -generalized trigonometric functions are real functions of the real variable t. They are obtained by taking , in equations ((6.4)–(6.6). Thus,

6.7

6.8

and

6.9

where the simplified notation introduced earlier is used for the principal, k = 0, functions. These functions, equations (6.4)–(6.9), are generalizations of the circular functions (also known as harmonic functions) of the classical integer-order trigonometry. The indexed behavior seen in equations (6.4)–(6.6) has no parallel in the classical trigonometry. We note, when q = 1 and v = 0, the -trigonometric functions revert back to the circular functions for t > 0.

The principal -trigonometric functions are shown in Figures 6.1a, b and 6.2 for various values of q. As the value of q decreases from 1 to 0.1, the oscillatory nature of the - and -functions is also seen to decrease. For q < 1, it can also be seen that the curves tend to zero as t increases. The functions grow at an increasing rate, as q increases for values of q > 1 (not shown). For smaller values of q in the to range, the lack of zero crossings of the -function strongly affects the behavior of the -function.

Figure 6.1 (a) versus t-Time, for q = 0.2–1.2 in steps of 0.1, v = 0, a = 1. (b) versus t-Time, for q = 0.2–1.2 in steps of 0.1, v = 0, a = 1. (a, b)

Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

Figure 6.2 versus t-Time, for q = 0.2–1.2 in steps of 0.2, v = 0, a = 1.

Figures 6.3–6.8 show the effects of the a parameter on the R₁Cos-, R₁Sin-, and the -functions, respectively. Increasing the a parameter increases the response rate of the function. In other words, increasing a decreases the apparent period and increases the overshoot. The reader should also consider the effect of the order variable q in these figures. For the and the , it also increases the response rate. For the , it changes the character of the response.

Figure 6.3 Effect of a on for a = 0.25–1.0 in steps of 0.25, with q = 0.25, v = 0 and k = 0.

Figure 6.4 Effect of a on for a = 0.25–1.0 in steps of 0.25, with q = 0.75, v = 0 and k = 0.

Figure 6.5 Effect of a on for a = 0.25–1.0 in steps of 0.25, with q = 0.25, v = 0 and k = 0.

Figure 6.6 Effect of a on for a = 0.25–1.0 in steps of 0.25, with q = 0.75, v = 0 and k = 0.

Figure 6.7 Effect of a on for a = 0.25–1.0 in steps of 0.25, with q = 0.25, v = 0 and k = 0.

Figure 6.8 Effect of a on for a = 0.25–1.0 in steps of 0.25, with q = 0.75, v = 0 and k = 0.

Figures 6.9–6.14 show the effects of secondary order parameter v on the R₁Cos-, R₁Sin-_, and -functions, respectively. In general, increasing v tends to increase the rate of response of the functions and reduce the period. From the Laplace transforms of the functions (Section 6.5), it is clear that v > 0 fractionally differentiates the v = 0 function, while v < 0 integrates it. The effect of v reduces the period of the -function; this is shown clearly in Figure 6.14.

Figure 6.9 Effect of v on for v = −0.20–0.20 in steps of 0.10, with q = 0.25, a = 1.0 and k = 0.

Figure 6.10 Effect of v on for v = −0.20–0.20 in steps of 0.10, with q = 0.75, a = 1.0 and k = 0.

Figure 6.11 Effect of v on for v = −0.20–0.20 in steps of 0.10, with q = 0.25, a = 1.0 and k = 0.

Figure 6.12 Effect of v on for v = −0.20–0.20 in steps of 0.10, with q = 0.75, a = 1.0 and k = 0.

Figure 6.13 Effect of v on for v = −0.20–0.20 in steps of 0.10, with q = 0.25, a = 1.0 and k = 0.

Figure 6.14 Effect of v on for v = −0.20–0.20 in steps of 0.10, with q = 0.75, a = 1.0 and k = 0.

Figure 6.15 shows the phase plane for variations in the order q, with a = 1.0, v = 0.0, and k = 0. Note that for q < 1 the functions are attracted to the origin from infinity, while for q > 1, the functions start at the origin and spiral out to infinity.

Figure 6.15 Phase plane versus for q = 1.1 and q = 0.25–1.75 in steps of 0.25, a = 1.0, v = 0.0.

6.1.1 R₁-Trigonometric Properties

From the definitions (6.4) and (6.5), we have that

6.10

and

6.11

These results substituted into equation (6.6) yield

6.12

Using the definitions (6.4) and (6.5) with equation (6.3) gives

6.13

which is a fractional Euler equation for the R₁-trigonometry. Expanding in the same manner as done for equation (6.3) yields

6.14

a complementary fractional Euler equation. The addition of equations (6.13) and (6.14) gives

6.15

If we subtract equation (6.15) from (6.14), we have

6.16

Both equations (6.15) and (6.16) are the fractional generalizations of the classical equations (1.18) and (1.19), respectively. Based on the definitions of the -trigonometric functions, we have

6.17

another R-function expression. Combining this with equation (6.15) gives

6.18

Similarly, we have

6.19

6.20

taken with equation (6.16) yields

6.21

A similar relation for the may be obtained from the ratio of equations (6.18) and (6.21), that is,

6.22

The addition of the squares equations (6.15) and (6.16) gives

6.23

a fractional-order generalization of the classical identity . Because the right-hand side of this identity is a real number, this complex R product behaves like the multiplication of complex conjugates.

In a similar manner, the subtraction of the square of equation (6.16) from the square of equation (6.15) gives

6.24

Figure 6.16 shows the value of the product for a range of q and t values. This is a generalization of the identity . Figure 6.17 shows the value of the conjugate sum for a range of q and t values. The substitution of equations (6.18) and (6.21) into (6.13) gives

6.25

the fractional Euler equation in an exponential form. Note that for q = 1/2 and v = 0,

Furthermore, with a = 1, we have

Figure 6.16 Conjugate product , q = 0.2–1.4 in steps of 0.2, k = 0.

Figure 6.17 Conjugate sum , q = 0.2–1.2 in steps of 0.2, k = 0, a = 1.0.

6.2 R₁-Trigonometric Function Interrelationship

A relationship between the -function and the -function is easily determined. Based on equation (6.4)

6.26

Thus,

6.27

Conversely,

The substitution into the defining series for the fractional trigonometric functions gives

6.28

6.29

and

6.30

demonstrating backward compatibility to common trigonometry.

6.3 Relationships to R₁-Hyperbolic Functions

The relationship between the R₁-hyperbolic and the R₁-trigonometric functions may be determined by appropriate substitutions into the defining series. It is clear from equations (5.21) and (6.17) that

6.31

and conversely

6.32

From equations (5.22) and (6.21), we have that

6.33

and conversely

6.34

These relationships between R₁-hyperbolic and R₁-trigonometric functions exactly parallel and generalize those between the integer-order trigonometric functions (q = 1, v = 0) and classical hyperbolic functions.

6.4 Fractional Calculus Operations on the R₁-Trigonometric Functions

This section determines fractional differintegrals of the principal fractional R-trigonometric functions. This section is from Lorenzo and Hartley [74], with permission of Springer:

The -order differintegral of is determined as follows: Differintegrating term-by-term (see Section 3.16) gives

6.35

Then applying equation (5.37), we have

Thus,

6.36

Applying equation (6.27), this may be written as

6.37

For , this becomes

6.38

The -order differintegral of , is determined as follows: We may differintegrate (6.26) term-by-term based on Section 3.16. Then, using equation (5.37)

6.39

Thus,

6.40

Applying equation (6.27), this may be written as

6.41

and for , we have the important case

6.42

6.5 Laplace Transforms of the R₁-Trigonometric Functions

Because the series are uniformly convergent (see Chapter 14), the Laplace transforms for the principal -trigonometric functions are readily determined by transforming the defining infinite series term-by-term.

Laplace Transform of R₁Cos_{q. v}(a, k, t)

Then, the transform of equation (6.4) is

6.43

Inverse transforming term-by-term (see Section 3.16),

6.44

where and . Thus,

Replacing cos and sin with exponential forms,

6.45

6.46

Now,

6.47

therefore, we have

6.48

where and . For the principal function,

6.49

6.5.2 Laplace Transform of R₁Sin_{q. v}(a, k, t)

Determination of this transform proceeds in a similar manner as that for the ; thus,

6.50

From Section 3.16, we may transform term-by-term; thus,

6.51

where and . Thus,

Replacing cos and sin with exponential forms,

6.52

Now applying equation (6.47), we have

6.53

where and . For the principal function,

6.54

The principal -trigonometric and -hyperbolic functions, their defining series, Laplace transforms, and R-function relationships are presented in Table 6.1 along with those of the classical circular functions.

Table 6.1 Summary of the principal R₁-trigonometric, R₁-hyperbolic, and the traditional functions

Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.

6.6 Complexity-Based R₁-Trigonometric Functions

In Section 6.1, fractional trigonometric functions that parallel the familiar classical trigonometric functions were derived based on the parity-based series (equation (6.2). In this section, we shall consider complexity-based functions starting from equation (6.1). Equation (6.1) can be written as

6.55

6.56

Furthermore, we have that

6.57

a complex plane version of the R-function. Because the results are R-functions which have been studied earlier, we do not pursue this area further.

6.7 Fractional Differential Equations

Continuing from Ref. [74], with permission of Springer:

The -trigonometric functions may also be solutions to fractional differential equations. This transform (equation (6.49) indicates that the uninitialized fractional differential equation

6.58

has a solution of the form

6.59

when , that is, an impulse function. As with the -hyperbolic functions for arbitrary , the solution is given by the convolution

6.60

Assuming composition equation (6.58) may also be written in the form of a fractional integro-differential equation, namely

In a similar manner, equation (6.54) indicates that the uninitialized fractional differential equation

6.61

has a solution of the form

6.62

when , that is, an impulse function. For arbitrary , the solution is obtained by the convolution similar to equation (6.60). Equation (6.61) may also be written as a fractional integro-differential equation, namely

6.63

Linear combinations of fractional trigonometric functions may also be used to infer solutions for fractional differential forms of noncommensurate order. For example, from the linear combination

6.64

or using the relationships (6.17) and (6.20)

we may infer the Laplace transformed equation

6.65

or

From this, it is seen that the uninitialized fractional differential equation

6.66

where

will have a solution of the form given by equation (6.64). Various specializations of the orders and parameters lead to other useful results. Still more varied fractional differential equations may be obtained by considering linear combinations of the both the R₁-trigonometric functions and the -hyperbolic functions, using, for example, forms such as

Chapter 7 explores the effect of imaginary t in the definition of a fractional trigonometry.