Chapter 7
The R2-Fractional Trigonometry
The basis for the R1-trigonomtry (developed in Chapter 6) was to replace the parameter a by ia in the R-function. For the R2-trigonometry, we consider a as a real parameter and let t be imaginary, that is, 
in the R-function. Specifically, the basis for the R2-trigonometry is 
[73, 74, 75].
In the traditional trigonometry (Section 1.3), the common exponential function is expanded in integer powers of t, and we found that the real part (equation (1.18)) was the summation of even powers of t and the imaginary part (equation (1.19)) was the summation of odd powers of t.
In this chapter, we find that this is no longer the case when it is raised to fractional powers. This introduces families of new functions not found in the traditional trigonometry. We find that these new functions occur in a pairwise manner similar to the classical sine and cosine functions.
The development of the R2-trigonometry follows from Lorenzo and Hartley [74] with permission of Springer:
7.1 R2-Trigonometric Functions: Based on Real and Imaginary Parts
We start by considering the real and imaginary parts of
Now, for a complex number 
, we have from equation (3.122)
Therefore, let us write for the exponent of (it) in equation (7.1)
where 
and 
are assumed to be rational and irreducible and M/D is in minimal form. Then, equation (7.1) becomes
and by (7.2)
where k has been introduced into the R argument since it is now explicit on the right-hand side. We propose the following definitions:
and
Thus, we see that the fractional character of 
and 
gives rise to a family of 
-many fractional trigonometric functions for rational values of 
and 
and 
in minimal form. This situation is unparalleled in the ordinary trigonometry. The principal functions, 
, are defined as
and
The R2Tan-function is defined as the ratio
By substituting 
in equations (7.6) and (7.7), it can be seen that R2-trigonometric functions generalize the classical trigonometric functions, that is,
Figures 7.1 and 7.2 show the principal 
and 
and 
-functions for 
. Note that there is a reversal in the amplitude of the 
-function for q > 1.9 (Figure 7.3). Furthermore, because 
the tangent plot is limited to q = 1.9. The effect of the a parameter on the 
-function is shown in Figures 7.4 and 7.5 for order variable q = 0.25 and 0.75, respectively. It can be seen that as a increases, the functions become increasingly oscillatory, also the amplitudes grow, and the frequency of oscillation increases. Similar effects are observed for the 
-functions in Figures 7.6 and 7.7. The effect of the change in frequency is clearly seen for the tangent functions in Figures 7.8 and 7.9. Note, when q = 1 and v = 0, from equations (7.11) the 
-functions revert to the classical circular functions and a becomes exactly the frequency parameter.
Figure 7.1 
versus t-Time for q = 0.2–2.0 in steps of 0.1, with a = 1.0, v = 0, k = 0.
Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.
Figure 7.2 
versus t-Time for q = 0.2–2.0 in steps of 0.1, with a = 1.0, v = 0, k = 0.
Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.
Figure 7.3 
versus t-Time for q = 0.2–1.9 in steps of 0.1, with a = 1.0, v = 0, k = 0.
Figure 7.4 Effect of a for 
versus t-Time for a = 0.2–1.1 in steps of 0.1, with q = 0.25, k = 0, v = 0.0.
Figure 7.5 Effect of a for 
versus t-Time for a = 0.2–1.1 in steps of 0.1, with q = 0.75, k = 0, v = 0.0.
Figure 7.6 Effect of a for 
versus t-Time for a = 0.2–1.1 in steps of 0.1, with q = 0.25, a = 1.0, k = 0, v = 0.0.
Figure 7.7 Effect of a for 
versus t-Time for a = 0.2–1.1 in steps of 0.1, with q = 0.75, a = 1.0, k = 0, v = 0.0.
Figure 7.8 Effect of a for 
versus t-Time for a = 0.7–1.0 in steps of 0.1, with q = 0.25, a = 1.0, k = 0, v = 0.0.
Figure 7.9 Effect of a for 
versus t-Time for a = 0.7–1.0 in steps of 0.1, with q = 0.75, a = 1.0, k = 0, v = 0.0.
Figures 7.10–7.15 examine the effects of the differintegration of order v on the principal 
, 
, and 
-functions for v = −0.6–0.6 in steps of 0.6, with q = 0.25 and q = 0.75. It is interesting to observe that in all cases, after a short transient period of less than one cycle, the v = 0 case and the v = 0.6 case are nearly indistinguishable.
Figure 7.10 Effect of v for 
versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.25, a = 1.0, k = 0.
Figure 7.15 Effect of v for 
versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.75, a = 1.0, k = 0.
Figure 7.11 Effect of v for 
versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.75, a = 1.0, k = 0.
Figure 7.12 Effect of v for 
versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.25, a = 1.0, k = 0.
Figure 7.13 Effect of v for 
versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.75, a = 1.0, k = 0.
Figure 7.14 Effect of v for 
versus t-Time for v = −0.6 to 0.6 in steps of 0.6, with q = 0.25, a = 1.0, k = 0.
The effect of the k index on the 
is shown in Figure 7.16. Note the strong symmetric behavior for this particular case. This is not always found for general values of q.
Figure 7.16 Effect of k for 
versus t-Time for k = 0–6 in steps of 1, with q = 6/7, a = 1.0, k = 0, v = 0.
Figure 7.17 shows a phase plane plot for 
versus 
with order, q, varying between 0.40 and 2.0. For 
, the amplitudes of both the 
and 
-functions are asymptotic to 
. In other words, the principal functions are attracted to circles of radius 
, for 
. We observe that for values of q < 1, the functions originate at infinity while for value of q > 1, they begin at the origin.
Figure 7.17 Phase plane plot 
versus 
for q = 0.4–2.0 in steps of 0.2, a = 1.0, v = 0, k = 0.
Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.
Continuing from Ref. [74] with permission of Springer:
paralleling the form of equation (6.13) for the R1-trigonometry. Furthermore,
is also validated in Section 7.4 (equation (7.70).
7.2 R2-Trigonometric Functions: Based on Parity
Because the real part of 
is no longer the same as the even part of 
and the imaginary part is no longer the same as the odd part of 
, we now consider the terms of the series containing only even or odd powers of a, that is, the parity of the 
series. Expanding 
Separating this into summations of even and odd powers of a gives
from which we observe
Equation (7.16) then separates 
into series with even and odd powers of a. These complex series are of theoretical importance and are named the Corotation and the Rotation functions and are defined as
where 
and 
, both with respect to powers of a.
Now, the real and imaginary parts of equations (7.18) and (7.19) will be used to define four new real fractional R2-trigonometric functions. Continuing from Ref. [74] with permission of Springer:
From the
-function, we have
We now write
where
and
are assumed rational, and M/D is in minimal form. Then, equation (7.20) may be written as
where 
with t > 0, 
.
The Coflutter function is defined as the real part of the
-function, that is,
and the Flutter function is defined as the imaginary part of the 
-function, namely
In a similar manner, from
, we have for t > 0
We now write
where 
and 
are assumed rational, and M/D is in minimal form. Then, equation (7.24) may be written as
where 
with t > 0, and 
. From equation (7.25), the Covibration function is defined as, 
, that is,
Similarly, the vibration function is defined as,
, namely
By substituting 
, into the parity series definitions, we can determine the backward compatibility of these functions. Therefore, we have continuing from Ref. [74] with permission of Springer:
Thus, we see that complex series – the Rotation and the Corotation – revert to hyperbolic functions of complex arguments. The real functions defined from the Rotation and Corotation are observed to either degenerate to zero or in the cases of
and
, to be backward compatible with the circular functions. Thus, clearly, we could have defined the
- and
-functions to be the fractional generalizations of the traditional sine and cosine functions; however, we have given precedence to complexity over parity in our naming conventions.
We again use the notations as follows: 
to mean the terms of 
having even powers of a, and 
to mean the terms of 
having odd powers of a. Then, the following observations are readily made:
The commutative property of equations (7.38)–(7.41) are readily proved by taking the even and odd parts of equations (7.6) and (7.7) and relating them to the real and imaginary parts of equations (7.18) and (7.19), which are the functions on the left-hand side of equations (7.38)–(7.41). It is of interest to also write the equivalent relationships for the R1-functions:
It is noted that the R1-functions, that is, equations (7.44)–(7.47), exactly parallel the traditional circular and hyperbolic functions with regard to complexity and parity.
The principal, k = 0, functions, 
, 
, 
, and 
, are presented in Figures 7.18 and 7.19 for selected q values. Observe that 
as indicated by equations (7.32) and (7.35). We also see that 
and that 
. Figures 7.20 and 7.21 show phase plane plots for 
versus 
and 
versus 
, for various values of q, compare these to Figure 7.17. For the parity functions we observe that for values of q < 0.5 the functions originate at infinity, while for values of q > 0.5 they begin at the origin.
Figure 7.18 
and 
versus t-Time for q = 0.2–1.0 in steps of 0.2 and q = 0.5, with a = 1.0, v = 0, k = 0.
Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.
Figure 7.19 
and 
versus t-Time for q = 0.2–1.0 in steps of 0.2 and q = 0.5, with a = 1.0, v = 0, k = 0.
Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.
Figure 7.20 Phase plane plot versus for q = 0.2–1.0 in steps of 0.2, and q = 0.5, a = 1.0, v = 0, k = 0. Arrows indicate increasing t.
Figure 7.21 Phase plane plot versus for q = 0.2–1.0 in steps of 0.2, and q = 0.5, a = 1.0, v = 0, k = 0. Arrows indicate increasing t.
7.3 Laplace Transforms of the R2-Trigonometric Functions
7.3.1 R2Cosq,v(a, k, t)
Using definition (7.6), we first consider
Because the series converges uniformly (see Sections 3.16 and 14.3), we may integrate term-by-term; thus,
where 
and 
. Then,
Applying the cosine definition (equation (1.18)), this yields
This may be written as
Based on equation (2.5), we write
Applying this relation, equation (7.51) may be written as
or inverse transforming
After some algebraic manipulation of equation (7.53), we obtain
The bracketed terms are recognized as cosine terms, giving
where 
, 
, 
, and 
. The following is an alternate statement of equations (7.55) and (7.53):
The development of the Laplace transforms of the remaining functions follows the same pattern used here; the key results are therefore presented without detailed derivation. For the transforms that follow, the general notations 
and 
apply. Furthermore, 
, 
,
, 
, and 
unless otherwise noted. The Laplace transforms are provided in the normal format and in a factored format.
7.3.2 R2Sinq,v(a, k, t)
7.3.3 L{R2Coflq,v(a, k, t)}
where 
and 
.
7.3.4 L{R2Flutq,v(a, k, t)}
where 
and 
.
7.3.5 L{R2Covibq,v(a, k, t)}
where 
and 
.
7.3.6 L{R2Vibq,v(a, k, t)}
where 
and 
.
7.4 R2-Trigonometric Function Relationships
Various properties of the 
-trigonometric functions in terms of the fractional exponential functions are derived here. Among these results are identities that parallel the basic equations (1.18) and (1.19). Some special R-function relationships also result. The results derived in the following are valid only for t > 0.
7.4.1 R2Cosq,v(a, k, t) and R2Sinq,v(a, k, t) Relationships and Fractional Euler Equation
We observe from equations (7.5)–(7.7) that
This is similar to the form of equation (6.13) for the R1-trigonometry.
Consider
Now,
Thus, equation (7.68) becomes
where 
. Although k and 
have the same range, they may not be enumerated in the same direction; however, it is clear that when k = 0 and 
, the principal values are equal. Therefore, it may be seen that
Consider now the 
-function as defined in equation (7.6), that is,
Continuing from Ref. [74] with permission of Springer:
Applying definition (1.18), we may write
Now, by equation (3.124),
, and (3.127), we have
This may be written as
Then, combining terms
or
Recognizing the summations as R-functions, we write
This equation exactly parallels equation (1.18) in the ordinary trigonometry. It could also serve as a definition of 
. The development for the 
-function follows in a similar manner, and thus, will not be detailed. The result for the 
-function is
This equation exactly parallels equation (1.19) in the classical trigonometry. It could also serve as a definition of 
. Continuing from Ref. [74] with permission of Springer:
and
These equations parallel equations (6.23) and (6.24) and generalize the identities
and
. We rewrite equation (7.73) and (7.74) as
and
Adding equations (7.79) and (7.80) returns the fractional Euler equations for the R2-trigonometry
This equation is the fractional generalization of equation (1.29) and contains it as a special case, that is, when
. Subtracting equation (7.80) from (7.79) gives the complementary equation
7.4.2 R2Rotq,v(a, t) and R2Corq,v(a, t) Relationships
As previously mentioned, 
and 
are complex functions. From the definitions (equations (7.17)–(7.19), we have
The summations of equations (7.18) and (7.19) are R-functions of complex arguments, that is,
Combining the results of equations (7.84) and (7.85) with equation (7.83) gives the following identity:
7.4.3 R2Coflq,v(a, t) and R2Flutq,v(a, t) Relationships
The analysis starts from the Laplace transform of 
, that is, equation (7.63)
where 
and 
. Continuing from Ref. [74] with permission of Springer:
Performing the inverse transform using equation (3.22), we have
Applying equation (3.118), we write
and
Now
, also
and
. With these results, equation (7.88) may be written 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 (7.89) and (7.90) as
Adding equations (7.91) and (7.92) gives
This equation might be considered as a “fractional semi-Euler” equation as discussed later. Subtracting equation (7.92) from equation (7.91) yields the complimentary equation
Letting 
in equation (7.93) gives
Alternatively, we may write
Equating the real parts to real parts and imaginary parts to imaginary parts with the fractional Euler equation (7.67), namely
we have the relationships
and
Forms paralleling equations (7.77) and (7.78) have also been derived for the 
and 
-functions [74]. These are
and
7.4.4 R2Covibq,v(a, t) and R2Vibq,v(a, t) Relationships
This analysis also starts with the Laplace transform of the 
-function. From equation (7.63)
where 
and 
. Performing the inverse transform using equation (3.22), we have
Applying equation (3.118), we write
and
Now 
, also 
and 
. With these results, equation (7.103) may be written as
The derivation for the 
follows in a similar manner, giving
We now derive some additional properties of these functions. Rewrite equations (7.104) and (7.105) as
Adding equations (7.106) and (7.107) gives an additional Euler-like equation:
This equation may also be considered as a “fractional semi-Euler” equation as discussed later. Subtracting equation (7.107) from equation (7.106) yields the complimentary equation
Let 
in equation (7.108), then
This may also be written as
another fractional semi-Euler equation. Comparing the real and imaginary parts with the fractional Euler equation (7.67)
we have the relationships
and
Then, using equations (7.67) (7.96) together with equation (7.111), we obtain
Forms paralleling equations (7.115) and (7.116) have also been derived for the 
and 
-functions [74]. These are
and
Many more such relationships are possible for the 
-trigonometric functions. Relationships such as the multiple-angle and fractional-angle formulas from the integer trigonometry and more are yet to be developed. The following section derives some fractional calculus relationships.
7.5 Fractional Calculus Operations on the R2-Trigonometric Functions
7.5.1 R2Cosq,v(a, k, t)
The 
-order differintegral of 
is determined as
Based on Section 3.16, we may term-by-term differintegrate
Applying equation (5.37), that is,
Continuing from Ref. [74], with permission of Springer:
Thus, equation (7.120) becomes
From the sum and difference formulas for the integer-order trigonometry, the following identities may be derived:
and
Now, in equation (7.122), let
Then, applying equation (7.123),
The summations are recognized as
and
, respectively, yielding the final result
where 
. Taking 
, we have 
and 
, giving
We may check backward compatibility by taking
which evaluates to
as expected.
An interesting alternative development of the result of equation (7.127) is obtained as follows:
By the differentiation equation (3.114),
When 
, we have
which is recognized as
7.5.2 R2Sinq,v(a, k, t)
Determination of the derivative for the 
-function proceeds in a manner similar to that of 
. The key results for the 
-order differintegral of 
follow. Using equation (7.124) with 
, 
, 
, and 
, we have
The summations are recognized as 
and 
, respectively, yielding the final result
where 
. Taking 
, we have 
and 
, giving
Backward compatibility is obtained by taking 
; thus,
which evaluates to
again the expected result.
An alternative R-function-based development using equation (7.74) yields
when 
, we have
which is recognized as
7.5.3 R2Corq,v(a, t)
The 
-order differintegral of 
is determined as
Application of equation (7.121) to this equation gives
with 
. The summation is recognized as both an R-function and as an 
-function, yielding the final result
7.5.4 R2Rotq,v(a, t)
Similarly, the 
-order differintegral of 
is
Only key results are given for the remaining functions.
7.5.5 R2Coflutq,v(a, t)
The 
-order differintegral of 
is determined as
where 
. Taking 
, 2q > v, we have 
and 
, giving
Taking 
gives
and with 
, we have
However, 
; thus,
The alternative R-function-based development gives
7.5.6 R2Flutq,v(a, k, t)
The 
-order differintegral of 
is given as
where 
. Taking 
, we have 
and 
, with 2q > v, gives
Taking 
, in equation (7.150), yields
When we also have 
, this becomes
However, from equation (7.33),
, thus indicating that
The alternative R-function-based development yields
7.5.7 R2Covibq,v(a, k, t)
The 
-order differintegral of 
is
where 
. With 
, we have 
and 
, giving
In addition, with 
, we obtain
From equation (7.34), 
; thus, we have
Alternatively,
When 
, we have
7.5.8 R2Vibq,v(a, k, t)
The 
-order differintegral of 
is determined as
where 
. Taking 
, we have cos(y) = 0 and sin(y) = 1, giving
Now, taking 
, in equation (7.161), gives
With both 
and with 
, we have
However, 
; therefore, we have
The alternative R-function-based development of the results of equation (7.105) is obtained as
When 
, we have
which is recognized as
Tables 7.1 and 7.2 summarize the various properties of the R2-trigonometric functions. Continuing from Ref. [74], with permission of Springer:
Table 7.1 Summary of R2-functions
Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.
For this table 
, 
, 
, and 
.
Table 7.2 Summary of R2-functions
Source: Lorenzo and Hartley 2004b [74]. Adapted with permission of Springer.
For this table 
, 
.
7.5.9 Summary of Fractional Calculus Operations on the R2-Trigonometric Functions
For ease of reference, the fractional calculus operations are summarized here. The derivations are for t > 0, 
, and 
. For all cases, except the Flut and Coflut, q > v, for the Flut and Coflut, we have 2q > v.
7.6 Inferred Fractional Differential Equations
Because the approach to obtaining special classes of FDEs for the 
-trigonometry is so similar to that for the R1-trigonometry, one only needs to change the functions in that analysis. We only point out that combinations of 
- and 
-functions may also be used.