Chapter 9
The Fractional Meta-Trigonometry

In the previous four chapters, fractional generalizations of the integer-order trigonometric and hyperbolic functions have been developed. In Chapters 5 and 6, we have seen the relationship of the R₁-trigonometry to the R₁-hyperboletry. Such a direct relationship for the R₂-trigonometry, however, has not been found. After some study of this problem, it has become apparent that more complicated relationships exist between the bases for the various trigonometries. This chapter presents a generalization that helps explore these connections. Furthermore, the chapter contains the hyperboletry¹ and all¹ of the trigonometries that we have explored to this point. This chapter also presents new trigonometries (Lorenzo [82, 83]) not previously available and potentially allows the solution of linear fractional differential equations in terms of these special functions. The result then is the master fractional trigonometry based on the R-function.

The bases of the trigonoboletries that we have considered to this point are as follows:

Clearly, trigonometries can be created for, or associated with, all of the indexed forms of the R-function presented in Table 3.2, which is a total of 16 integer-valued bases. Many of these bases will likely have important application in the fractional calculus and in the solution of fractional differential equations. However, a more global approach is considered here. This chapter considers fractional meta-trigonometries based on

With and ranging between 0 and 4, all of the integer-valued bases are considered together with infinitely more noninteger possibilities.

Multiplication of the a or t variables by rotates the variable in the complex plane by the amount radians. This is shown graphically in Figure 9.1. These rotations can have profound effects on the real and imaginary projections, which relate to the generalized functions. This chapter is adapted from Lorenzo [82, 83], with permission of ASME.

Figure 9.1 Graphical display of for in steps of .

Source: Lorenzo 2009a [82]. Reproduced with permission of ASME.

9.1 The Fractional Meta-Trigonometric Functions: Based on Complexity

We start by separating into real and imaginary parts. Thus, we consider

9.1

9.2

Now, for rational q, v, , and , we may write

9.3

where , , , and . Then, we may write

where or and , and M/D is rational and in minimal form.

Therefore, equation (9.2) may be written as

9.4

9.5

where k is included in the R-function argument to reflect its presence in on the right-hand side. Then, similar to the earlier definitions, we define the generalized fractional functions as

9.6

9.7

These definitions generalize those for the trignoboletries that were developed in previous chapters. The notation is changed, and here drops the preceding R and its subscript. These fractional meta-trigonometric functions are discriminated from the traditional trigonometric functions by the capitalization and the subscripted order variables. Continuing from Ref. [82] with permission of ASME:

As with the previous trigonometric functions, we define the principal functions for t > 0 as

9.8

9.9

Combining equations (9.5)–(9.7) gives

9.10

where D is the product of the denominators of in minimal form. This is the generalized fractional Euler equation. A complimentary fractional meta-Euler equation is derived later as equation (9.173).

9.1.1 Alternate Forms

It should be noted that the form of equation (9.1) was chosen for the convenience that it allowed either the “a” and/or the “t” variables to be made complex. The cost of this convenience, however, was to introduce two new variables, , into the defining summation. Complexly proportional forms with one less variable in the summation are possible and are discussed in Appendix E.

9.1.2 Graphical Presentation – Complexity Functions

A complete graphical presentation for the - and -functions is, of course, impossible. There are seven parameters and variables, and the possible number of charts is limitless. In previous chapters, the effects of variations of q and v have been presented with and values of zero and one. Therefore, the emphasis here is on the new variables introduced in this chapter, namely and .

Figures 9.2–9.5 show the effect of varying in the for q = 1.05, 1.00, 0.75, and 0.50, with a = 1.0, , and with variations from 1.0 to 3.0. For the -function in Figure 9.3, we see that over the range shown.

Figure 9.2 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 1.05, a = 1, , v = 0, k = 0.

Figure 9.3 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 1.00, a = 1, , v = 0, k = 0.

Source: Lorenzo 2009a [82]. Adapted with permission of ASME.

Figure 9.4 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 0.75, a = 1, , v = 0, k = 0.

Figure 9.5 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 0.25, a = 1, , v = 0, k = 0.

The study is repeated for in Figures 9.6–9.9. For the -functions, in these figures, it can be seen that the behavior is symmetric around ; that is, for .

Figure 9.6 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 1.05, a = 1, , v = 0, k = 0.

Figure 9.7 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 1.00, a = 1, , v = 0, k = 0.

Source: Lorenzo 2009a [82]. Adapted with permission of ASME.

Figure 9.8 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 0.75, a = 1, , v = 0, k = 0.

Figure 9.9 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 0.50, a = 1, , v = 0, k = 0.

In Figure 9.7, the fractional trigonometric functions appear to be quite similar to exponentially damped sinusoids or cosinusoids. The common feature of these figures is the selection of variables, that is, . Continuing from Ref. [84] with permission of ASME:

Relative to Figure 9.7, we have from equation (9.7)

9.11

Now, from Ref. [56], pp. 118–119, #632,

9.12

Thus,

9.13

a closed-form summation. Also, based on Ref. [56], pp. 116–117, #631, we can determine

9.14

Because is simply a constant for any individual curve, we see that for these special cases the functions are indeed exponentially damped sinusoids under the constraints.

The effects of variations in on with = 1.0–3.0 in steps of 0.2, is shown in Figures 9.10a–9.13. Symmetry around , similar to that observed for the variations, is seen in Figure 9.11 for . The effect of variations on the fractional -functions is presented in Figures 9.14–9.17. For values of q < 1, an increase in increases the apparent damping as seen in Figures 9.16 and 9.17.

Figure 9.10 Effect of on with (a) = 1.0–2.0 and (b) = 2.0–3.0 in steps of 0.2, with q = 1.05, a = 1, , v = 0, k = 0.

Figure 9.11 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 1.00, a = 1, , v = 0, k = 0.

Figure 9.12 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 0.75, a = 1, , v = 0, k = 0.

Figure 9.13 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 0.50, a = 1, , v = 0, k = 0.

Figure 9.14 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 1.05, a = 1, , v = 0, k = 0.

Figure 9.15 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 1.05, a = 1, , v = 0, k = 0.

Figure 9.16 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 0.75, a = 1, , v = 0, k = 0.

Figure 9.17 Effect of on with = 1.0–3.0 in steps of 0.2, with q = 0.50, a = 1, , v = 0, k = 0.

Phase plane plots of Figures 9.18–9.21 study the effects of variations in for values of q = 1.05, 1.00, 0.95, and 0.50. A significant change in the nature of these cross plots is seen with small variations of q around q = 1.0 in Figures 9.18–9.20. We notice in Figure 9.21 the loss of oscillation due to the small value of q.

Figure 9.18 Phase plane versus for = 1–3 in steps of 0.2, with a = 1.0, q = 1.00, , k = 0, v = 0. Arrows indicate increasing function.

Source: Lorenzo 2009a [82]. Adapted with permission of ASME.

Figure 9.19 Phase plane versus for = 1–3 in steps of 0.2, with a = 1.0, q = 1.05, k = 0, , v = 0. Arrows indicate increasing function parameter.

Figure 9.20 Phase plane versus for = 1–3 in steps of 0.2, with a = 1.0, q = 0.95, k = 0, , v = 0.

Figure 9.21 Phase plane versus for = 1–3 in steps of 0.2, with a = 1.0, q = 0.50, k = 0, , v = 0. Arrows indicate increasing function parameter.

Figures 9.22 and 9.23 show the effect of on the phase plane behavior of versus with q = 1, , a = 1, and k = 0. By comparing these two figures, the effect of v, , is seen to have a strong effect on the function behavior.

Figure 9.22 Phase plane versus for = 1–3 in steps of 0.2, with v = 1.0, a = 1.0, q = 1.00, k = 0, . Arrows indicate increasing function parameter.

Source: Lorenzo 2009a [82]. Adapted with permission of ASME.

Figure 9.23 Phase plane versus for = 1–3 in steps of 0.2, with v = −1.0, a = 1.0, q = 1.00, k = 0, .

Figures 9.24 and 9.25 show the effect of on the fractional for q = 1.15 and q = 0.95. Note, for q > 1 the functions depart from the origin and for q < 1 the functions arrive from infinity. Figure 9.26 shows the phase plane, versus , for . Here, all the functions arrive from infinity with increasing t. However, for the functions are captured by the origin, while for the functions are repelled by the unit circle.

Figure 9.24 Phase plane versus for = 1–1.5 in steps of 0.1, with q = 1.15, v = 0, a = 1.0, k = 0, .

Figure 9.25 Phase plane versus for = 1–3 in steps of 0.25, with q = 0.95, v = 0, a = 1.0, k = 0, .

Figure 9.26 Phase plane versus for = 1–2 in steps of 0.2, with a = −1.0, q = 0.85, v = 0, k = 0, , t = 0–12.

Because of symmetrical occurrences in some physical processes, Figures 9.27–9.31 present the - versus -functions together with their symmetric partners versus . Figure 9.31 shows an instance of a barred spiral. Such spirals are of interest in astrophysics; see Chapter 18. The various other special cases are self-explanatory.

Figure 9.27 Phase plane versus and versus for = 0.2, with a = 1.0, q = 1.10, v = −1, k = 0, , t = 0–18.

Figure 9.28 Phase plane versus and for versus = 0, with a = 1.0, q = 1.05, v = 0.2, k = 0, , t = 0–20.

Figure 9.29 Phase plane versus and for versus , = 0.1, with a = 1.0, q = 1.05, v = 0, k = 0, , t = 0–20.

Figure 9.30 Phase plane versus and for versus = 0.1, with a = 1.0, q = 1.05, v = 0.2, k = 0, , t = 0–20.

Figure 9.31 Phase plane versus and for versus , = 0, with a = 1.0, q = 1.05, v = 0.05, k = 0, , t = 0–10.

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity

Continuing from Ref. [82] with permission of ASME:

We now consider based on parity of the exponent of a. Then, equation (9.1) is written as

9.15

9.16

The summation becomes

9.17

Forming two summations by separating the even and odd powers of a, we have

9.18

9.19

This also may be expressed as

9.20

The summations of equation (9.19) contain the even and odd powers of a, respectively. In parallel with the previous development of the R-trigonometric functions, we define the generalized or meta-Corotation and Rotation functions as

9.21

9.22

where the nomenclature means the terms with even powers of a in .

Similarly,

9.23

9.24

and where the nomenclature means the terms with odd powers of a in .

As in the previous trigonometries, the generalized Corotation and Rotation functions are also, in general, complex. Clearly, we may also write

9.25

The real and imaginary parts of these functions are now used to define the four new real fractional meta-trigonometric functions. The -function is given in equation (9.21) as

9.26

Now, applying equation (3.123) to with rational, we have

9.27

with M/D a rational number in minimal form. Thus, the may be written as

9.28

where . The real part of the is now defined as the generalized or meta-Covibration function

9.29

with t > 0, and .

The meta-Vibration function, , is defined as the imaginary part of the -function; thus,

9.30

with t > 0, and . Then, we have

9.31

The generalized or meta-Rotation function similarly defines two new functions based on its real and imaginary parts; these are the meta-Flutter and meta-Coflutter functions, that is,

9.32

and

9.33

and where is the product of the denominators .

In parallel with equation (9.31), we also have

9.34

Table 9.1 presents special values for the fractional meta-trigonometric functions when and . The row can apply to the R₁-hyperboletry, while the row can apply to the R₁- and R₂-trigonometries. The R₃-trigonometry is a special case of the row. Of course, any of these rows may also be applied to noninteger values of and . Continuing from Ref. [82] with permission of ASME:

Some observations may be made from the table. For example, for the -function column all terms are of the form . This may be shown as follows:

9.35

9.36

Similarly, for the column, all terms are of the form , since

9.37

9.38

Thus, we see that these functions are backward compatible with the hyperbolic functions with imaginary arguments. Other relationships are observed for the remaining columns.

Table 9.1 Special values of the generalized trignobolic functions

Source: Lorenzo 2009a [82]. Reproduced with permission of ASME.

For this table , also , – indicates only series description found.

9.3 Commutative Properties of the Complexity and Parity Operations

In this section, we demonstrate that the operations of determining the real/imaginary parts and even/odd parts of the parity functions may be interchanged. We start with the real part of from equation (9.6); then, for , we have

9.39

where . Expanding the summation yields

9.40

Collecting even and odd powers of a gives

9.41

However, we also observe that

9.42

and that

9.43

Now, from equations (9.29) and (9.42), we have

9.44

and from equations (9.32) and (9.43)

9.45

proving the assertion for these functions.

For the remaining functions, we begin with the imaginary part of , from equation (9.7)

9.46

where and . Expanding this summation yields

9.47

Again, collecting even and odd powers of a gives

9.48

Here, we observe

9.49

and

9.50

Now, using equations (9.30) with (9.49)

9.51

and from equations (9.33) and (9.50),

9.52

completing the demonstration of the complexity–parity commutivity properties of . These properties are summarized for the meta-trigonometric functions with t > 0 [83]:

9.53

9.54

9.55

9.56

9.57

9.58

9.59

9.60

where and refer to terms containing the odd and even powers of a, respectively.

9.3.1 Graphical Presentation – Parity Functions

Once again, it is not possible to show a representative display of the parity functions. The problem is exacerbated because the number of functions is doubled. Thus, the focus is on the effect of the generalizing variables and .

Figures 9.32–9.35 show the effect of the primary order variable, q, on the parity functions, , , , and as a function of t time, with = 0.5 and = 0.5. Because = 1, = 0 corresponds to the R₁-trigonometry and = 0, = 1 corresponds to the R₂-trigonometry, the choice of = 0.5 and = 0.5 examines a new trigonometry sharing aspects of both R₁- and R₂-trigonometries. An observed feature of these figures is that increasing the order q increases the oscillatory behavior of the function in most cases. Note that the reversal of the first peak amplitude for for the - and -functions.

Figure 9.32 The effect of q for , q = 0.1–1.0 in steps of 0.1, with a = 1, v = 0, k = 0, = 0.5, = 0.5.

Figure 9.33 The effect of q for , q = 0.1–1.0 in steps of 0.1, with a = 1, v = 0, k = 0, = 0.5, = 0.5.

Figure 9.34 The effect of q for , q = 0.1–1.0 in steps of 0.1, with a = 1, v = 0, k = 0, = 0.5, = 0.5.

Figure 9.35 The effect of q for , q = 0.1–1.0 in steps of 0.1, with a = 1, v = 0, k = 0, = 0.5, = 0.5.

Figure 9.36 The effect of and , , for , with = 0.0–2.0 in steps of 0.2, and with q = 0.5, a = 1, v = 0, k = 0.

Figure 9.37 The effect of and , , for , with = 0.0–2.0 in steps of 0.2, and with q = 0.5, a = 1, v = 0, k = 0.

Figure 9.38 The effect of and , , for , with = 0.0–2.0 in steps of 0.2, and with q = 0.5, a = 1, v = 0, k = 0.

The effect of and , where , for , , , and functions is shown in Figures 9.36–9.39. For these figures, = 0.0–2.0 in steps of 0.2, and q = 0.5, a = 1, v = 0, k = 0. Note that the responses for the ,- and -functions are symmetric around , and the and functions overlay themselves for , where .

Figure 9.39 The effect of and , , for , with = 0.0–2.0 in steps of 0.2, and with q = 0.5, a = 1, v = 0, k = 0.

Figures 9.40 and 9.41 study the effect of a for - and -functions with a = 0.25– 2.0 in steps of 0.25, and with q = 0.5, = 0.5, = 0.5, v = 0, k = 0.

Figure 9.40 The effect of a for , with a = 0.25–2.0 in steps of 0.25, and with q = 0.5, = 0.5, = 0.5, v = 0, k = 0.

Figure 9.41 The effect of a for , with a = 0.25–2.0 in steps of 0.25, and with q = 0.5, = 0.5, = 0.5, v = 0, k = 0.

Figures 9.42 and 9.43 study the effect of the differintegration variable, v, for the - and -functions with v = −1.0 to 1.0 in steps of 0.25, and with q = 0.5, a = 1, = 0.5, = 0.5, k = 0.

Figure 9.42 The effect of v for , with v = −1.0–1.0 in steps of 0.25, and with q = 0.5, a = 1, = 0.5, = 0.5, k = 0.

Figure 9.43 The effect of v for , with v = −1.0–1.0 in steps of 0.25, and with q = 0.5, a = 1, = 0.5, = 0.5, k = 0.

Figure 9.44 considers the effect of the index variable k, for k = 0–4, with q = 3/5, a = 1, = 0.5, = 0.5, v = 0. Note again the untypical symmetric responses resulting from the particular choice of q.

Figure 9.44 Effect of k, for k = 0–4, with q = 3/5, a = 1, = 0.5, = 0.5, v = 0.

Figure 9.45 Phase plane showing the effect of and , , for versus with = 0.0–1.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, k = 0, t = 0–9.

The remaining figures (Figures 9.45–9.52) are phase plane plots chosen to expose a few of the many varied possibilities. Figures 9.45 and 9.46 are phase planes showing the effect of and , where , for versus . For Figure 9.45, = 0.0–1.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, k = 0. For Figure 9.46, = 1.0 to 2.0 in steps of 0.1, also with q = 0.5, a = 1, v = 0, k = 0. All cases arrive from infinity and spiral into the origin except = 2.0, which is attracted to the unit circle. From Lorenzo [83]:

A particularly interesting, and important, pair of plots is found in Figures 9.47 and 9.48 for the versus phase plane. For both cases, = 1.0–2.0 and q = 0.5, a = 1, v = 0, k = 0. While it is not obvious because of the change of scale between the pair, in Figure 9.47 as is varied over the range , with , the responses start from the unit circle and diverge to infinity. In Figure 9.48 with and all other variables the same, the responses start from the same identical points as in Figure 9.47 but are attracted to the origin. Furthermore, the slopes across the unit circle are preserved for each response. This behavior and its interpretation are discussed in more detail in Section 9.12.

Figure 9.46 Phase plane showing the effect of and , , for versus with = 1.0–2.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, k = 0, t = 0–9.

Figure 9.47 Phase plane showing the effect of for versus with = 1.0–2.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, = 1, k = 0.

Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

Figure 9.48 . Phase plane showing the effect of for versus with = 1.0–2.0 in steps of 0.1, and with q = 0.5, a = 1, v = 0, = 3, k = 0.

Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

Figure 9.49 Phase plane for versus with = 0.16, = 1.25, and with q = 0.80, a = 0.8, v = 0, k = 0, t = 0–40.

Figure 9.50 Phase plane for versus with = 0.5, and with q = 1.09, a = 1.0, v = 0, k = 0, t = 0–19.

Figure 9.51 Phase plane for versus and versus with = 0.5–0.8, = 0.6, and with q = 1.04, v = −0.3, a = 1.0, k = 0, t = 0–4.8.

Figure 9.52 Phase plane for versus and versus with = 0.5–0.8, = 0.6, and with q = 1.04, v = −0.3, a = 1.0, k = 0, t = 0–4.8.

Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

Figure 9.49 shows an interesting spiral converging to the origin that is similar to some of the classical spirals [64], pp. 188, 189. Figures 9.50–9.53 are phase plane plots of several barred spirals based on the parity functions. Figure 9.51 presents a phase plane for versus and versus with = 0.5–0.8. The spirals start at the origin and spiral out to infinity with increasing rate as increases.

Figure 9.53 Taxonomy of the fractional meta-trigonometric functions.

Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

Figure 9.52 is a very interesting phase plane plot for versus and versus with = 0.5–0.8. The unusual feature here is that the departure is from the end of the spiral bar as increases. Such barred spiral behavior is of particular interest in the astrophysics of the galaxies. More such behaviors are studied in Chapter 18.

9.4 Laplace Transforms of the Fractional Meta-Trigonometric Functions

The development of the Laplace transform for the generalized functions parallels that of the previous trigonometries.

9.61

where , , and where D is the product of the denominators , in equation (9.3), in minimal form. Because the series converges uniformly (see Sections 3.16 and 14.3), we may transform term-by-term. Thus,

9.62

Continuing from Ref. [83]:

9.63

Recognizing the summations using equation (7.52) gives

9.64

9.65

9.66

where , , and . The two forms, those of equations (9.65) and (9.66), are particularly useful.

The derivation for the Laplace transform of proceeds in a similar manner:

9.67

with , and giving the results

9.68

and

9.69

where and .

The Laplace transform for the follows.

Continuing from Ref. [83]:

9.70

9.71

9.72

where , and is the product of the denominators , in minimal form and where and .

Similarly, for the , we have

9.73

9.74

where , and .

The Laplace transform for the follows:

9.75

9.76

where , and .

The derivation for the is similar to that for and yields

9.77

9.78

where , and .

This collection of Laplace transforms generalizes those derived for the previous trigonometries. Comparison of these transforms with the parallel results for the R₃-trigonometric functions shows that the transforms are structurally the same and the R₃ results may be had by the substitutions . Table 9.2 summarizes the Laplace transforms of the meta-trigonometric functions.

Table 9.2 Summary of the meta-trigonometric functions

Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

, , ; D is the product of the denominators of with repeated multipliers removed.

9.5 R-Function Representation of the Fractional Meta-Trigonometric Functions

The fractional exponential or R-function representation of the trigonometric functions is useful for analysis and numerical computation. Continuing from Ref. [83]:

Now, from the definition of , we have for t > 0

9.79

Now,

Applying equation (3.124), namely , yields

9.80

Then,

The summations are recognized as R-functions giving the result

9.81

Applying equation (3.121) simplifies the result to

9.82

The remaining functions are determined in a similar manner and are listed in two forms, t real and complex; thus, we have

9.83

or

9.84

9.85

or

9.86

9.87

or

9.88

9.89

or

9.90

9.91

or

9.92

Finally, summarizing the results of equations (9.22) and (9.24)

9.93

9.94

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions

For the fractional differintegrations that follow, it is assumed that the integrated function, and all of its derivatives, are identically zero for all t < 0. Continuing from Ref. [83]:

9.6.1 Cos_q,v(a, α, β, k, t)

The u-order differintegral of is determined for t > 0 as

9.95

We may differintegrate term-by-term (see Section 3.16); thus,

9.96

Applying equation (5.37), that is,

9.97

Thus equation (9.96) becomes

9.98

From the sum and difference formulas for the integer-order trigonometry, the following identities may be derived:

9.99

and

9.100

Now, in equation (9.98), let , and , with ; then applying equation (9.99) gives

9.101

The summations are recognized as and , respectively, yielding the final result

9.102

where . Taking , we have and giving

9.103

Now, taking

9.104

An alternative development of the result of equation (9.103) is obtained as follows:

9.105

by the differentiation equation (3.114)

9.106

When , we have

9.107

which is recognized as

9.108

which is identically equation (9.104) when .

9.6.2 Sin_q,v(a, α, β, k, t)

Determination of the differintegral for the -function proceeds in a manner similar to that of the . Then, the u-order differintegral of is determined as

9.109

Application of equation (9.97) to this equation gives

9.110

In equation (9.110), let and , ; then applying equation (9.100), we have

9.111

The summations are seen to be and , respectively, yielding the final result

9.112

where . Taking , we have and , giving

9.113

Furthermore, taking gives

9.114

The R-function-based development of the result of equation (9.113) is obtained based on equation (9.84) as

9.115

by the differentiation equation (3.114)

9.116

When and k = 0, this yields the same result as equation (9.113).

9.6.3 Cor_q,v(a, α, β, t)

The u-order differintegral for the -function is determined, using definition (9.2.8), as

9.117

Application of equation (9.97) to this equation gives

9.118

9.119

with , and where the summation has been recognized as an -function, to yield the final result. In terms of the R-function, we have

9.120

or

9.121

9.6.4 Rot_q,v(a, α, β, t)

Determination of the differintegral for the -function proceeds in a manner similar to that of . Then, the u-order differintegral of is determined as

9.122

9.123

Application of equation (9.97) to this equation gives

9.124

The summation is seen to be , yielding

9.125

In terms of the R-function representation, we have

9.126

9.6.5 Coflut_q,v(a, α, β, k, t)

In this section, we determine the differintegral of the Coflutter function. The u-order differintegral of is determined as

9.127

Application of equation (9.97) to this equation gives

9.128

with . Now, in equation (9.128), let , , also, let ; then, applying equation (9.99), we have

9.129

The summations are recognized as and , yielding the desired differintegral form

9.130

where . For the special case with , we have and , giving

9.131

Taking in equation (9.130) gives the general form for the differintegral of the Coflutter function as

9.132

Furthermore, with , we have

9.133

The alternative R-function-based development of the results of equation (9.90) is obtained as

9.134

by the differentiation equation (3.114)

9.135

For the case ,

9.136

9.137

9.6.6 Flut_q,v(a, α, β, k, t)

The -order differintegral of is determined as

9.138

where is in minimal form.

Applying equation (9.97) to this equation gives

9.139

Now, in equation (9.139), let , and let ; then, applying equation (9.100), we have

9.140

with . The summations are recognized as and , respectively, yielding the final result

9.141

where . For the special case where , we have and , giving

9.142

The alternative R-function-based development of the results of equation (9.92) is obtained as follows:

9.143

Application of the differentiation equation (3.114) gives the result

With , we have

9.144

or

9.145

9.6.7 Covib_q,v(a, α, β, k, t)

The -order differintegral of the is determined as follows:

9.146

Application of equation (9.97) to this equation gives

9.147

Now let , in equation (9.147) and let , ; then, applying equation (9.99), we have

9.148

The summations are seen to be and , respectively, yielding the key result

9.149

where . When we have and ; thus,

9.150

The R-function-based differintegral is obtained as

9.151

By the differentiation equation (3.114)

When , we have

9.152

The right-hand side is seen to be , thus validating equation (9.150).

9.6.8 Vib_q,v(a, α, β, k, t)

The -order differintegral of the generalized vibration function, , is derived as follows:

9.153

Application of equation (9.97) to this equation gives

9.154

with . Now, let , , , in equation (9.154). Then, applying equation (9.100), we have

9.155

The summations are recognized as and , respectively, yielding the key result

9.156

where . When , we have and , giving

9.157

Now, taking in equation (9.156),

9.158

With both and with , we have

9.159

However, if additionally , ; therefore, we have

9.160

where . The alternative R-function-based development of the results of equation (9.88) is obtained as

9.161

by the differintegration equation (3.114)

9.162

When , we have

9.163

Tables 9.2 and 9.3 summarize the various properties of the meta-trigonometric functions.

Table 9.3 Summary of the meta-trigonometric functions

Source: Lorenzo 2009b [83]. Reproduced with permission of ASME.

, ; D is the product of the denominators of in minimum form.

9.6.9 Summary of Fractional Calculus Operations on the Meta-Trigonometric Functions

For ease of reference the fractional calculus operations are summarized here. The derivations are for t > 0, and and :

9.164

9.165

9.166

9.167

9.168

9.169

9.170

9.171

9.7 Special Topics in Fractional Differintegration

In this, and previous chapters, fractional differintegrals of the fractional trigonometric functions have been derived. In all cases, the fractional differintegration has started from t = 0. Because the functions are zero for t < 0, initialization or history effects have not been an issue. However, when a function does not start from zero or is segmented, such as

and preceded by another different function, fractional differintegrations must consider the initialization effects caused by the preceding function. Clearly, there will be times when it is desired to start a fractional differintegration at times other than t = 0. The reader is cautioned that fractional differintegration differs from integer-order integration in this important regard. Detailed analysis of these important differences is addressed in Appendix D.

9.8 Meta-Trigonometric Function Relationships

This section determines generalized identities for the fractional meta-trigonometric functions in terms of the fractional exponential functions. These are parallel to the very basic identities presented in equations (1.18) and (1.19). The results will, of course, apply to all particular fractional trigonometries that may be derived by specializing the results of this chapter.

9.8.1 Cos_q,v(a, α, β, t) and Sin_q,v(a, α, β, t) Relationships

The generalized fractional Euler equation, with k = 0, for the - and -functions was determined in Section 9.1 as

9.172

The complementary generalized fractional Euler equation, with k = 0, can be determined by expanding , in a manner similar to the expansion from equation (9.1) to (9.5), or by subtracting equation (9.84) from (9.82), which gives

9.173

Continuing from Ref. [83]:

Squaring equations (9.82) and (9.84) yields

9.174

9.175

Adding and subtracting the resultant equations provides the following two identities:

9.176

9.177

From the result of equation (9.176), it is clear that because and are real, the product must also be real. Thus, we note that is the complex conjugate of . These identities (9.176) and (9.177) are broad generalizations of and from the ordinary trigonometry. This may be shown by taking in equations (9.176) and (9.177).

9.8.2 Cor_q,v(a, α, β, t) and Rot_q,v(a, α, β, t) Relationships

Consider, from equation (9.124) with

9.178

The summation is recognized as yielding

9.179

From equation (9.125), we have

9.180

Therefore,

9.181

Continuing from Ref. [83]:

9.8.3 Covib_q,v(a, α, β, t) and Vib_q,v(a, α, β, t) Relationships

Equations (9.86) and (9.88) are rewritten in the following form:

9.182

9.183

Adding these equations gives the generalized fractional semi-Euler equation for the Vibration functions as

9.184

or

9.185

While subtracting yields the generalized complementary semi-Euler equation

9.186

or

9.187

The terminology semi-Euler is used because of the q to q/2 relationship between the right- and left-hand sides of the equations. Squaring equations (9.86) and (9.88) yields

9.188

9.189

Adding and subtracting these equations give

9.190

9.191

meta-identities relating the and -functions.

9.8.4 Cofl_q,v(a, α, β, t) and Flut_q,v(a, α, β, t) Relationships

The - and -functions can also be related. We rewrite equations (9.90) and (9.92) as follows:

9.192

and

9.193

Adding equations (9.192) and (9.193) gives

9.194

Alternatively,

9.195

the generalized fractional semi-Euler equation for the Flutter functions. Subtracting equation (9.193) from (9.192) yields

9.196

or

9.197

the generalized complementary fractional semi-Euler equation for the Flutter functions. The functions as represented in equations (9.90) and (9.92) are squared to give

9.198

9.199

The sum and difference of these equations yield

9.200

9.201

generalized identities relating the and -functions.

9.8.5 Cofl_q,v(a, α, β, t) and Vib_q,v(a, α, β, t) Relationships

Taking and k = 0 in equation (9.139) gives

9.202

Now,

9.203

Let and with

9.204

Then, we may write equation (9.203) as

9.205

More succinctly,

9.206

Taking , and k = 0 in equation (9.204) gives

9.207

Now,

9.208

Using the same substitutions and with into equation (9.208) gives

9.209

Adding equations (9.206) and (9.209) yields the interesting result

9.210

where . Subtracting equation (9.206) from (9.209) gives the complementary result, which may be combined into the single identity

9.211

where again .

9.8.6 Cos_q,v(a, α, β, t) and Sin_q,v(a, α, β, t) Relationships to Other Functions

The following relationships are presented without proof. They may be easily shown by substitution into the defining series and with the use of equations (9.99) and (9.100):

9.212

9.213

For the principal functions, we have

9.214

where .

9.215

where .

These are but a few of the many relationships possible for the meta-trigonometric functions. Many other relations paralleling the multiple- and fractional-angle formulas from the integer-order trigonometry and more are yet to be derived.

9.8.7 Meta-Identities Based on the Integer-order Trigonometric Identities

Because the series definitions of the meta-trigonometric functions contain the periodic cosine and sine functions, inter-relationships may be determined for these functions. This section determines meta-identities based on these relationships for the - and -functions. Identities such as and are possible bases, as are the following identities provided by Spanier and Oldham [116], pp. 298–299 from the integer-order trigonometry:

9.216

9.217

Identities for the -functions are considered first.

9.8.7.1 The cos(−x) = cos(x)-Based Identity for Cos_q,v(a, α, β, t)

The question to be answered is: For given values of q and v what values for and will provide an identical function with parameters and ? In other words, under what conditions will

9.218

From the definition of ,

9.219

This will occur when for all n

9.220

Now, from the classical trigonometry using , this requires

9.221

This equality must also hold for n = 0; thus,

This result substituted into equation (9.221) yields

9.222

from which

Thus, the final result is

9.223

9.8.7.2 The sin(−x) = − sin(x)-Based Identity for Sin_q,v(a, α, β, t)

Here, we wish to find conditions under which

9.224

From the definition of ,

9.225

Considering the sine arguments

9.226

This equality must also hold for n = 0; thus,

This result is now substituted into equation (9.226) giving

Combing the argument result with the sign change for yields the final result

9.227

9.8.7.3 The Cos_q,v(a, α, β, t) ⇔ Sin_q,v(a, α, β, t) Identity

Selected arguments of the classical cosine and sine functions in the - and -functions also lead to identities. For this study, we consider

9.228

and

9.229

Then when

9.230

From the classical trigonometry (equation (9.217), we have , where . Applying the identity to equation (9.230) gives

9.231

Comparing arguments

9.232

Again, the relationship must hold for n = 0; thus,

This result substituted into equation (9.232) yields

9.233

Thus, the resulting meta-trigonometric identity is given by

9.234

9.8.7.4 The sin(x) = sin(x ± mπ/2)-Based Identity for Sin_q,v(a, α, β, t)

For , identities will occur when

9.235

Based on the definition of ,

9.236

We require

9.237

Here, from equation (9.217), we use the identity , where ; thus,

9.238

Comparing arguments

9.239

Since the relationship must also hold when n = 0,

9.240

Substituting equation (9.240) into (9.239) gives

9.241

Thus, we have the identity

242

Clearly other, possibly stronger, identities of this type are possible. Importantly, because the series defining the parity functions also contain integer-order sine and cosine terms, similar identities may be created for them. Continuing from Ref. [83]:

9.9 Fractional Poles: Structure of the Laplace Transforms

Consider the Laplace transform of the Covibration function. From equations (9.72) and (9.71), we have

243

This can be further resolved to

244

From the transform pair (equation (244), it may be seen that the may be composed by a weighted sum of four R-functions or four complex fractional poles. The transforms of all of the fractional meta-trigonometric functions may be similarly decomposed.

9.10 Comments and Issues Relative to the Meta-Trigonometric Functions

The previous sections of this chapter have presented the definitions, Laplace transforms, special properties, and identities for the meta-trigonometry based on the fractional exponential function, . From these meta-definitions and properties, the analyst may specialize the variables q, v, , and to obtain particular fractional trigonometries or hyperboletries. The same variables substituted into the meta-transforms, meta-identities, and so on will yield those transforms and identities specialized to the particular fractional trigonometry.

The mathematical process that was used to generate the meta-functions is shown graphically in Figure 9.53. The a and t variables of fractional exponential function, -function expressed in series form, are generalized to the complex forms shown in the level 2 box. Two paths are shown from level 2 to level 3. The left path takes the real and imaginary parts of the series while the right path takes the even and odd powered terms of the series. From level 3 to 4, following the left path, four new functions are generated by taking the even and odd powered terms of the cosine and sine series. Following the right path, the same four functions may be generated by taking the real and imaginary parts of the complex corotation and rotation functions.

Thus, the process described produces eight functions, six of which are real, and the complex corotation and rotation functions. It is noted that at level three, the leading term of the Laplace transform denominators is 2q while at level four it doubles to 4q. It is noted that the mathematical process observed here is fractal and may be repeated indefinitely to finer and finer resolution of .

9.11 Backward Compatibility to Earlier Fractional Trigonometries

The generalized fractional or meta-trigonometry as defined here contains both the R₂ and R₃-fractional trigonometries discussed previously. For example, to obtain the R₂-functions, we need only set and in the meta-functions; that is, the basis for the trigonometry is . Similarly, for the R₃-trigonobolic functions, we set , giving the basis .

The generalized R₁-trigonometry is not, however, directly compatible with definitions forwarded for the R₁-trigonometry, which was conceived prior to the generalized theory. However, the principal R₁-trigonometric functions are compatible with the meta-trigonometry. It is noted that both versions have the basis .

The choice q = 1, v = 0, , and will yield the functions and identities associated with the classical trigonometry (albeit with new names).

The Covibration function is the real and even part with respect to the powers of a of and, therefore, is a proper successor to the cosine function of the classical trigonometry and the Flut function is imaginary and odd part with respect to the powers of a and, therefore, is an appropriate successor for the sine function. Thus, the -function in the meta-trigonometry is only the real part of the -function and is only the imaginary part of the -function; that is, the parity operations have not been applied. The taxonomy chart of Figure 9.53 illustrates the relationships.

9.12 Discussion

The meta-trigonometry expands the fractional trigonometries and fractional hyperboletry from the four bases of Chapters 5–8 to the complete infinite set. The defined functions are useable for the study of many dynamic processes and for the solution of many classes of fractional differential equations via the Laplace transforms of the meta-functions. Considerable study will be required to determine which of the bases are important and where they may be applied. While the fractional trignoboletries release us from the constraints associated with the unit circle, many challenges must be met for the area to develop to mathematical maturity, some follow.

The foregoing material has only dealt with the primary functions: , , and so on. The ratio and reciprocal functions (i.e., tan, cotan, etc.) associated with the six primary functions still need to be defined and to have their properties developed. The task is daunting as there are 36 such functions. The next chapter provides some thoughts on this topic.

Development of inverse function definitions (in series form) is problematic since in most cases no repetitive principal cycle exists, and the inverse functions are one-to-many mappings. Software methods, however, may allow us to temporarily bypass this step for practical application. A more detailed discussion of this topic is found in Chapter 20.

The existence of orthogonal function sets outside of the classical trigonometry will need to be explored. Furthermore, will the classical definitions of orthogonality be adequate or will new fractional-based definitions be needed?

¹ Only for the principal functions for the R₁-trigonometry and hyperboletry.