Chapter 8
The R3-Trigonometric Functions
In the integer-order trigonometry, replacing the frequency parameter a by the imaginary parameter 
toggles the functions back and forth between the trigonometric and the hyperbolic functions. For example,
and for the cosine function
Parallels to this were found for the 
-trigonometry with the 
-hyperboletry. However, this is not the case relative to the 
-trigonometry. This chapter develops the effect of an imaginary frequency parameter and an imaginary time parameter in the R-function.
8.1 The R3-Trigonometric Functions: Based on Complexity
The basis for definition of the R1-trignometric functions is to allow 
. The R2-trigonometric functions are based on 
. The obvious question is: What is the nature of the results when both 
and 
? This assumption is the basis of the R3-trignometric functions. We start by separating 
into real and imaginary parts. Thus, we consider
Now, for rational q and v, we may write
Thus,
where 
, 
, and 
. Therefore, equation (8.2) may be written as
where k is included in the R-function argument to reflect its presence in 
on the right-hand side. Then, by similarity to equations (7.6) and (7.7), we define
Thus, these definitions are quite similar to those for 
and 
, differing from their R2 counterparts only by the presence of an extra “n” in the arguments of the circular functions. As with the R2-trigonometric functions, we define the principal functions for t > 0, as
Combining equations (8.4)–(8.6) gives
Figure 8.1 shows the principal 
- and the 
-functions for values of q between q = 0.2 and q = 1.0, with a = 1.0 and v = 0. We observe from Figure 8.1 that for q = 1 and 
we have 
, and we can see that 
. These results are readily verified by appropriate substitution into equations (8.7) and (8.8). The character of these functions changes considerably for 
, which is illustrated in Figure 8.2. For 
and greater, the principal 
- and 
-functions also grow in oscillation amplitude.
Figure 8.1 
and 
versus t-Time for a = 1, k = 0, v = 0, q = 0.2–1.0 in steps of 0.2.
Figure 8.2 
and 
versus t-Time for a = 1, k = 0, v = 0, q = 1.0–1.5 in steps of 0.1.
In Figures 8.3 and 8.4, we study the effect of changes in the a parameter on the principal 
- and the 
-functions for values of q = 0.25 and 0.75, respectively. As seen in the R1 and R2 trigonometries, the effect of increasing a leads to faster transient responses of the trigonometric variables. The effect of the differintegration order variable, v, is presented in Figures 8.5 and 8.6 for values of q = 0.25 and 0.75, respectively. Here, we see that v not only changes the transient rate of response of the trigonometric functions, but also the nature of the response, in particular when the sign of v changes.
Figure 8.3 Effect of a, 
, and 
versus t-Time for a = 0.25–1.0 in steps of 0.25, with q = 0.25, k = 0, v = 0.
Figure 8.4 Effect of a, 
, and 
versus t-Time for a = 0.25–1.0 in steps of 0.25, with q = 0.75, k = 0, v = 0.
Figure 8.5 Effect of v, 
, and 
versus t-Time for v = −0.6–0.6 in steps of 0.3, with q = 0.25, k = 0, a = 1.
Figure 8.6 Effect of v, 
, and 
versus t-Time for v = −0.6–0.6 in steps of 0.3, with q = 0.75, k = 0, a = 1.
The effect of k, for 
, with k = 0 to 6 in steps of 1, and q = 5/7, a = 1.0, v = 0 is shown in Figure 8.7. In this case, strong symmetry is again seen. It has been observed that the 
-function maintains this symmetry for 
with n an odd integer.
Figure 8.7 Effect of k, for 
, with k = 0–6 in steps of 1, q = 5/7, a = 1.0, v = 0.
Figure 8.8 shows a phase plane for 
versus 
for q = 1.0 to 2.0. In this figure, all cases start from the origin. The cases with 
return to the origin, while those for 
diverge with increasing rates.
Figure 8.8 Phase plane 
versus 
for q = 1.0–2.0 in steps of 0.2, with a = 1.0, v = 0, and t = 0–7.0.
8.2 The R3-Trigonometric Functions: Based on Parity
We now consider 
based on parity of the exponent of a. Then, equation (8.1) is written as
The summation becomes
Forming two summations by separating terms with even and odd powers of 
, we may write
We note in passing the identity
which after application of equation (3.119) becomes
For 
, equation (8.13) is rewritten as
where the summations are separated into even and odd powers of 
in (8.12). In parallel with the development of the 
-trigonometric functions, and based on equations (8.13) and (8.14), we define the 
-Corotation and 
-Rotation functions as the summations with even and odd powered a terms, respectively:
We now use the real and imaginary parts of these functions to define the four new real fractional trigonometric functions that parallel those defined for the 
-trigonometry.
The 
-function is rewritten as
Applying equation (3.123) to 
with q and v rational, we have
where 
and 
are assumed to be rational and irreducible and M/D is in minimal form. Thus,
where 
. Then 
may be written as
The real part of the 
is defined as the 
-Covibration function
The imaginary part of equation (8.19) defines the 
-Vibration function
We now examine the nature of these functions when 
and 
:
and
The 
-function (equation (8.17) is now written as
Applying equation (3.123) to 
with q and v rational, we have
where 
and 
are assumed to be rational and irreducible and M/D is in minimal form. Thus
where 
. Then, 
may be written as
Thus, we define the real part as the 
-Coflutter function
and the imaginary part as the 
-Flutter function
Again, we examine the nature of these functions when 
, and 
, then
and
The backward compatibility, that is, the 
, and 
evaluation, of the 
-functions for t > 0, is summarized as
Thus, we see that these functions are backward compatible with the common exponential and the hyperbolic functions and indeed might be considered as the 
-hyperbolic functions.
Here, the complexity and parity properties of 
in terms of the 
-trigonometric functions are summarized; for t > 0
where 
and 
refer to the forms with terms containing the even and odd powers of a. Also we see that, by design, these forms exactly parallel those of the related 
-functions. Similar to the 
case, the commutative property of equations (8.39)–(8.42) are readily proved by taking the real and imaginary parts of equations (8.16) and (8.17) and relating them to the even and odd parts of equations (8.7) and (8.8).
Figure 8.10 
versus t-Time for a = 1, v = 0, k = 0, and q = 0.1–0.5 in steps of 0.1.
Figure 8.11 
versus t-Time for a = 1, v = 0, k = 0, and q = 0.1–0.5 in steps of 0.1.
The 
-, 
-, 
-, and 
-functions are presented in Figures 8.9–8.12 for 
, with a = 1, and v = 0. In the figures, and validated by substitution in the appropriate series, we observe 
and 
. Figure 8.13 presents a phase plane 
versus 
for q = 0.5–1.5. For q > 0.5, the functions spiral out from the origin with increasing rate as q increases.
Figure 8.9 
versus t-Time for a = 1, v = 0, k = 0, and q = 0.1–0.5 in steps of 0.1.
Figure 8.12 
versus t-Time for a = 1, v = 0, k = 0, and q = 0.1–0.5 in steps of 0.1.
Figure 8.13 Phase plane 
versus 
for q = 0.5–1.5 in steps of 0.1, a = 1.0, k = 0, v = 0, t = 0–10.
The phase plane 
versus 
for q = 0.5–1.5 is presented in Figure 8.14. For 
the curves approach from infinity and are repelled before reaching the unit circle at q = 0.5 outward back toward infinity. For q > 1, the curves leave the origin and spiral out to infinity with increasing rate. The q = 1 case starts at the unit circle and proceeds to infinity. Also, it may be shown that 
.
Figure 8.14 Phase plane 
versus 
for q = 0.5–1.5 in steps of 0.1, a = 1.0, k = 0, v = 0, t = 0–12.
8.3 Laplace Transforms of the R3-Trigonometric Functions
The development of the Laplace transform for the R3-trigonometric functions parallels that of the 
-functions. Thus, we state the results in normal and factored form without development.
where 
and 
, with 
. Comparison of these transforms with the parallel results for the R2-trigonometric functions shows that the transforms are structurally the same and vary from each other primarily in the coefficient of the 
term in the denominator. We will later see that this influences the damping behavior of the transform.
Note that most of the figures presented in this and previous chapters are of results in the stable domain that is of trigonometric order, 
for the complexity functions. The reader is encouraged to numerically explore values of q > 1 as many important applications are found there; see, for example, Chapters 17–20. For the parity functions, explore q > 1/2.
8.4 R3-Trigonometric Function Relationships
This section determines expressions for the 
-trigonometric functions in terms of the fractional exponential functions. These expressions parallel the basic equations (1.18) and (1.19). Some special R-function relationships also result. The organization of the subsections of this section has been set by the ease of development of relationships.
8.4.1 R3Cosq,v(a, t) and R3Sinq,v(a, t) Relationships and Fractional Euler Equation
This section develops some important properties of the R3-trigonometric functions. We observe from equations (8.4)–(8.6) that
This equation parallels the form of equations (6.15) and (7.12) for the 
and R2-trigonometries, respectively. Consider now the 
-function as defined in equation (8.5), that is,
Applying definition (1.18), we may write
Now, by equation (3.124), 
, and (3.127), we have
This may be written as
Recognizing the summations as 
-functions, we have
or applying equation (3.119)
This equation generalizes equation (1.18) in the ordinary trigonometry. It could also serve as a definition of 
.
The development for the 
-function follows in similar manner, yielding
This equation exactly parallels equation (1.19) in the ordinary trigonometry. It could also serve as a definition of 
. Solving for 
using both equations (8.58) and (8.59) yields equation (8.51) and subtracting (8.59) from (8.58) gives, for t > 0
the companion to equation (8.51). Now, squaring equations (8.58) and (8.59) yields
Adding and subtracting equations (8.61) and (8.62) gives
and
These equations parallel equations (6.23) and (6.24) and provide additional generalizations of the equations 
and 
.
8.4.2 R3Rotq,v(a, t) and R3Corq,v(a, t) Relationships
As previously mentioned, 
and 
are complex functions. From the definitions (equations (8.15)–(8.17), we have for t > 0
We observe that the summations of equations (8.16) and (8.17) are R-functions of complex arguments, and using equation (3.118), we may write
Combining the results of equations (8.66) and (8.67) with equation (8.65) gives the following identity:
8.4.3 R3Coflq,v(a, t) and R3Flutq,v(a, t) Relationships
The analysis starts by replacing the cos function in the definition of 
in equation (8.26). Then, using logic similar to the process used for the 
, that is, equations (8.53) to (8.57), we have for t > 0
where 
, 
with t > 0, and 
. Applying equation (3.124) to the exponential functions, we have 
and 
. Thus, we have
Applying equation (3.119) to these R-functions, we write the final result as
The 
-function is derived in a similar manner. We give the final result
We now derive some additional properties of these functions. Rewrite equations (8.70) and (8.71) as
Adding equations (8.72) and (8.73) gives
This equation may be considered as a fractional “semi-Euler” equation. Subtracting equation (8.72) from equation (8.73) and using (8.17) yields the complimentary equation
Now, in equations (8.74) and (8.75), let 
; then,
and
which may be considered as fractional “semi-Euler” equations. Replacing 
in equation (8.9) gives
Comparing the real and imaginary parts of this equation with equation (8.76), we observe that
and
8.4.4 R3Covibq,v(a, t) and R3Vibq,v(a, t) Relationships
This analysis starts by rewriting equation (8.20), the 
-function, as
where 
and 
. This becomes
Since 
and 
, we have
Applying equation (3.119) to both terms gives
The derivation for the 
follows in a similar manner; thus,
Additional properties of these functions are now determined. Rewrite equations (8.81) and (8.82) as
Adding equations (8.83) and (8.84) gives
This equation may also be considered as a fractional “semi-Euler” equation. Subtracting equation (8.84) from equation (8.83) yields the complimentary equation
Now, in equation (8.85), let 
; then,
Alternatively, we may write
Replacing a in equation (8.51) by 
gives
Comparing this result with equation (8.88), we observe that
and
Rewrite equation (8.85) as
Relating the real and imaginary parts of equations (8.92) and (8.75), we obtain
exposing relationships between the parity functions. It should be remembered, because of the role of the v as an order variable, that these relationships also represent fractional differintegration relationships.
These are but a few of the many relationships possible for the 
-trigonometric functions. Many other relations paralleling the multiple-angle and fractional-angle formulas from the integer-order trigonometry and more are yet to be derived.
8.5 Fractional Calculus Operations on the R3-Trigonometric Functions
8.5.1 R3Cosq,v(a, k, t)
The 
-order differintegral of 
is determined for t > 0 as
Based on Section 3.16, we may differintegrate term-by-term:
Applying equation (5.37), that is,
valid for all 
. Differintegrating equation (8.96) term-by-term yields
From the sum and difference formulas for the integer-order trigonometry, the following identities are derived:
and
Now, in equation (8.98), let 
, 
, also, let 
, then applying equation (8.99)
The summations are recognized as 
and 
, respectively, yielding the final result
where 
. Taking 
we have 
and 
giving
Now, taking 
gives
which evaluates to
An alternative development of the result of equation (8.103) is obtained as follows:
by the differintegration equation (3.114)
When 
, we have
which is recognized as
8.5.2 R3Sinq,v(a, k, t)
Determination of the differintegral for the 
-function proceeds in a similar manner to the 
. Then, the 
-order differintegral of 
is determined as
Application of equation (8.97) to this equation gives
Now, in equation (8.112), let 
, 
, also, let 
; then applying equation (8.100), we have
The summations are recognized as 
and 
, respectively, yielding the final result
where 
. Taking 
, we have 
and 
, giving
Taking 
yields
which evaluates to
again the expected result.
The alternative R-function-based development of the result of equation (8.114) is obtained based on equation (8.59), as
by the differintegration equation (3.114)
When 
, we have
which is recognized as
8.5.3 R3Corq,v(a, t)
Determination of the derivative for the 
-function proceeds in similar manner to 
. Then, the 
-order differintegral of 
is determined, using definition (8.16), as
Application of equation (8.97) to this equation gives
The summation is recognized as both an R-function and as an R3Rot-function, yielding the final result
8.5.4 R3Rotq,v(a, t)
Determination of the derivative for the 
-function proceeds in a manner that is similar to 
. Then, the 
-order differintegral of 
is determined as
Application of equation (8.97) to this equation gives
The summation is recognized as both an R-function and as an R3Rot-function, yielding
8.5.5 R3Coflutq,v(a, k, t)
In this section, we determine the derivative of the 
-function. The 
-order differintegral of 
is determined as
Differintegrating term-by-term,
Application of equation (8.97) to this equation gives
Now, in equation (8.126), let 
, 
, also, let 
; then applying equation (8.99), we have
The summations are recognized as 
and 
yielding the desired derivative form
where 
. Taking 
, we have 
and 
, giving
Taking 
, in equation (8.127), gives the derivative as
and with 
we have
However, 
, thus
The alternative R-function-based development uses equation (8.70) to obtain
by the differintegration equation (3.114)
When 
, we have for t > 0
as seen in equation (8.128).
8.5.6 R3Flutq,v(a, k, t)
The 
-order differintegral of 
is determined as
Application of equation (8.97) to this equation gives
Now, let 
, also let 
; then, applying equation (8.100), we have
The summations are recognized as 
and 
, respectively, yielding the final result
where 
. Taking 
, we have 
and 
, giving
Taking 
, in equation (8.135), we obtain
When we also have 
, this becomes
However, 
, thus indicating that
The alternative R-function-based development uses equation (8.71) to obtain
by the differintegration equation (3.114)
When 
, we have
which is recognized as
8.5.7 R3Covibq,v(a, k, t)
In this section, we determine the derivative of the 
-function. The 
-order differintegral of 
is determined as
Differintegrating term-by-term gives
Application of equation (8.97) to this equation gives
Now, in equation (8.140), let 
, 
, also let 
; then, applying equation (8.99), we have
The summations are recognized as 
and 
, respectively, yielding the final result:
where 
. Taking 
, we have 
and 
, giving
Setting 
, we obtain
Because 
, we have
The alternative R-function-based development of the results of equation (8.82) is obtained as
By the differintegration equation (3.114)
When 
, we have
and the equation is the same as equation (8.142).
8.5.8 R3Vibq,v(a, k, t)
The 
-order differintegral of 
is determined as
Application of equation (8.97) to this equation gives
Now, in equation (8.148), let 
, 
, also let 
; then, applying equation (8.100), we have
The summations are recognized as 
and 
, respectively, yielding the final result:
where 
. Taking 
, we have 
and 
, giving
Now, taking 
in equation (8.149),
With 
, 
, we have
However, 
; therefore, we have
The alternative R-function-based development uses equation (8.82) giving
by the differentiation equation (3.114)
When 
, we have
which is
Tables 8.1 and 8.2 summarize the various properties of the R3-trigonometric functions.
Table 8.1 Summary of R3-functions
For this table, 
, 
, 
, and 
.
Table 8.2 Summary of R3-functions
For this table, 
, 
.
8.5.9 Summary of Fractional Calculus Operations on the R3-Trigonometric Functions
For ease of reference, the fractional calculus operations are summarized here. The derivations are for t > 0, 
, and 
:
