# p-Trigonometric and p-Hyperbolic Functions in Complex Domain

Authors: Petr Girg; Lukáš Kotrla
Journal: Abstract and Applied Analysis
 (2016)

---

## Abstract

We study extension ofp-trigonometric functions s
i
n
p and c
o
s
p and of p-hyperbolic functions s
i
n
h
p and c
o
s
h
p to complex domain. Our aim is to answer the question under what conditions on p these functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example, s
i
n
(
z
)
=
-
i
·
sinh
⁡
i
·
z. In particular, we prove in the paper that for p
=
6,10,14
,
… the p-trigonometric and p-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series for p-trigonometric and p-hyperbolic functions.


---

## Body

## 1. Introduction

Thep-trigonometric functions are generalizations of regular trigonometric functions sine and cosine and arise from the study of the eigenvalue problem for the one-dimensional p-Laplacian.In recent years, thep-trigonometric functions were intensively studied from various points of views by many researchers; see, for example, monograph [1] for systematic survey and further references. The purpose of this paper is twofold. We begin with a short survey of results from [2, 3]. Then, we extend the ideas from [3] to define corresponding generalization of hyperbolic functions and study relations of p-trigonometric and p-hyperbolic functions on a disc in the complex domain.More precisely, our goal is to generalize the hyperbolic functions such that the relations(1)
sin
⁡
z
=
-
i
·
sinh
⁡
i
·
z
,
cos
⁡
z
=
cosh
⁡
i
·
z
,
(2)
cos
⁡
z
=
sin
′
⁡
z
,
cosh
⁡
z
=
sinh
′
⁡
z
,
(3)
cos
2
⁡
z
+
sin
2
⁡
z
=
1
,
cosh
2
⁡
z
-
sinh
2
⁡
z
=
1
,where z
∈
C, have their counterparts for generalized p-trigonometric and p-hyperbolic functions. It turns out that this goal can be achieved only for even integer p
>
2.Thep-trigonometric functions in the real domain R originate naturally from the study of the nonlinear eigenvalue problem(4)
-
u
′
p
-
2
u
′
′
-
λ
u
p
-
2
u
=
0
in
0
,
π
p
,
u
0
=
u
π
p
=
0
,where p
>
1,λ
∈
R is a parameter, and(5)
π
p
=
2
∫
0
1
1
1
-
s
p
1
/
p
d
s
=
2
π
p
sin
⁡
π
/
p
.It was shown in Elbert [4] that all eigenfunctions of (4) can be expressed in terms of solutions of the initial-value problem(6)
-
u
′
p
-
2
u
′
′
-
p
-
1
u
p
-
2
u
=
0
,
x
∈
R
,
u
0
=
0
,
u
′
0
=
1
.Indeed, (6) has the unique solution in R; see, for example, [5, Lemma A.1], [6, Section 3], and [4]. Denoting the solution of (6) by s
i
n
p
x, the set of all eigenvalues λ
k
∈
R and eigenfunctions u
k
∈
W
0
1
,
p
(
0
,
π
p
) of (4) can be written as (7)
λ
k
=
p
-
1
k
p
,
u
k
x
=
sin
p
⁡
k
·
x
,
where
k
∈
N
.A piecewise construction of the solution of (6) was provided in [4]. At first, one sets(8)
arcsin
p
⁡
x
≝
∫
0
x
1
1
-
s
p
1
/
p
d
s
,
x
∈
0,1
.Then, the restriction of s
i
n
p
x on [
0
,
π
p
/
2
] is the inverse function to a
r
c
s
i
n
p
x. For x
∈
(
π
p
/
2
,
π
p
], s
i
n
p
x satisfies s
i
n
p
x
=
s
i
n
p
(
π
p
-
x
), where clearly π
p
-
x
∈
[
0
,
π
p
/
2
), and s
i
n
p
x
=
-
s
i
n
p
(
-
x
) for x
∈
[
-
π
p
,
0
). Finally, s
i
n
p
x is a 2
π
p-periodic function on R.We also extenda
r
c
s
i
n
p
x from (8) to [
-
1,1
] as an odd function. Then, it is the inverse function to the restriction of s
i
n
p
x to [
-
π
p
/
2
,
π
p
/
2
], and we have(9)
sin
p
⁡
arcsin
p
⁡
x
=
x
,
∀
x
∈
-
1,1
.Finally, let us definec
o
s
p
x
≝
s
i
n
p
′
x for all x
∈
R. Then, the functions s
i
n
p
x and c
o
s
p
x satisfy the so-called p-trigonometric identity(10)
cos
p
⁡
x
p
+
sin
p
⁡
x
p
=
1for all x
∈
R; see, for example, [4–6].Note that there is an alternative definition of “c
o
s
p
x” in [7] and/or [8] which leads to different “p-trigonometric” identity. Yet another alternative generalization of trigonometric and hyperbolic functions motivated by geometrical point of view was introduced in [9]. Studies of relations between their respective generalizations of p-trigonometric and p-hyperbolic functions were suggested in [7] and in [9], respectively.Remark 1.
In the paper, we use Gauss’ hypergeometric functionF
1
2
(
a
,
b
,
c
,
z
), where a
,
b
,
c
∈
C are parameters and z
∈
C is variable (for definition see [10, 15.1.1. p. 556]), to express integrals of the type (11)
∫
0
x
1
1
±
s
p
1
/
p
d
s
,
∫
0
z
1
1
±
s
p
1
/
p
d
s
, for p
>
1,x
∈
R, andz
∈
C (by z
1
/
p we understand the principal branch thereof). Indeed, (12)
z
F
1
2
1
p
,
1
p
,
1
+
1
p
,
∓
z
p
=
∫
0
z
1
1
±
s
p
1
/
p
d
sfor |
z
|
<
1 (by comparing respective series expansions). By the uniqueness of analytic extension, the equation is valid for z
∈
C
∖
{
x
+
i
y
:
x
>
1
a
n
d
y
=
0
} (for analytic continuation of 2
F
1 see, e.g., [10, 15.3.1, p. 558] and [11, Theorem 2.2
.
1, p. 65]).
In the definition ofπ
p (i.e., (5)) and in (8), we need to evaluate integral (13)
∫
0
1
1
1
-
s
p
1
/
p
d
s
.By [11, Theorem 2.2
.
2, p. 66], this is possible, since R
[
c
-
a
-
b
]
=
1
+
1
/
p
-
1
/
p
-
1
/
p
=
1
-
1
/
p
>
0 for p
>
1.Notation 1.
This paper combines real variable and complex variable approach to thep-trigonometric and p-hyperbolic functions. Each of these approaches has its own natural way of how to define the functions s
i
n
p and s
i
n
h
p. Thus, we need to distinguish between real and complex definitions. By s
i
n
p
x and s
i
n
h
p
x, we mean functions defined by the real variable approach and by s
i
n
p
z and s
i
n
h
p
z we mean functions defined by the complex variable approach, throughout the paper.


## 2. Real Analyticity Results forsin
p
⁡
x and cos
⁡
p
⁡
x

It is well known that thep-trigonometric functions are not real analytic functions in general; see, for example, [12, 13]. Very detailed study of the degree of smoothness of the restriction of s
i
n
p
x to (
-
π
p
/
2
,
π
p
/
2
) was performed in [2] including the following two results. The first one concerns “generic” p
>
1.Proposition 2 (see [2], Theorem 3.2 on p. 105).
Letp
∈
R
∖
{
2
m
}, m
∈
N, p
>
1. Then, (14)
sin
p
⁡
x
∈
C
p
-
π
p
2
,
π
p
2
,but (15)
sin
p
⁡
x
∉
C
p
+
1
-
π
p
2
,
π
p
2
.Here,⌈
p
⌉
≝
m
i
n
{
k
∈
N
:
k
≥
p
}.The second result treats only the even integersp
>
2 and differs significantly from the previous case in an unexpected way.Proposition 3 (see [2], Theorem 3.1 on p. 105).
Letp
=
2
(
m
+
1
), m
∈
N. Then, (16)
sin
p
⁡
x
∈
C
∞
-
π
p
2
,
π
p
2
.Thus, the Maclaurin series ofs
i
n
p
x
(17)
M
p
x
=
d
e
f
∑
n
=
1
∞
1
n
!
sin
p
n
⁡
0
·
x
nis well defined for p
=
2
(
m
+
1
)
,
m
∈
N. Moreover, the following result establishes an explicit expression for the radius of convergence of M
p.Proposition 4 (see [2], Theorem 3.3 on p. 106).
Letp
=
2
(
m
+
1
) for m
∈
N. Then, the Maclaurin series M
p
(
x
) of s
i
n
p
x converges on (
-
π
p
/
2
,
π
p
/
2
).Thus, forp
=
2
(
m
+
1
), m
∈
N, we can compute approximate values of s
i
n
p
x using Maclaurin series. It turns out that the most effective method of computing coefficients in (17) is to use formal inversion of the Maclaurin series of(18)
arcsin
p
⁡
x
=
∫
0
x
1
1
-
s
p
1
/
p
d
s
=
x
·
F
1
2
1
p
,
1
p
,
1
+
1
p
,
x
p
=
∑
k
=
0
∞
Γ
k
+
1
/
p
k
p
+
1
k
!
Γ
1
/
p
x
k
p
+
1
,where p
=
2
(
m
+
1
) for some m
∈
N. The procedure of inverting power series is well known; see, for example, [14]. This task can be easily performed using computer algebra systems. In Pseudocode 1, we provide an example of computing the partial sum of M
4 up to terms of order 32 inMathematica® v. 9.0. In this way, we can easily get partial sums of M
p and get approximations of s
i
n
p for any p
=
2
(
m
+
1
),m
∈
N, up to terms of orders of hundreds. Note that this formal inverse can be applied also for p
=
2
m
+
1
,
m
∈
N. The question is what is the mathematical sense of the resulting formal series. Let(19)
T
p
x
≝
∑
n
=
1
∞
a
n
·
x
ndenote the series that is the formal inverse of (18) for p
=
2
m
+
1
,
m
∈
N. For p
=
2
m
+
1
,
m
∈
N, let us also define(20)
M
p
x
≝
∑
n
=
1
∞
1
n
!
lim
x
→
0
+
⁡
sin
p
n
⁡
x
·
x
n
,which is a formal Maclaurin series of some unknown function. It turns out that this unknow function is not s
i
n
p
x as the following result holds.Pseudocode 1:
Mathematica v. 9.0 code.
I
n
[
1
]
:=
Series
[
s*Hypergeometric2F1
[
1/4, 1/4, 5/4, s
∧
4
]
,
s, 0, 32
]
(* computes the Maclaurin series of arcsin_4 *)
Out
[
1
]
=
s
+
s
5
20
+
5
s
9
288
+
15
s
13
1664
+
195
s
17
34816
+
221
s
21
57344
+
4641
s
25
1638400
+
16575
s
29
7602176
+
O
s
33
In
[
2
]
:=
InverseSeries
[
%
]
(* computes the inverse series *)
Out
[
2
]
=  s
-
s
5
20
-
7
s
9
1440
-
463
s
13
374400
-
211741
s
17
509184000
-
104361161
s
21
641571840000
-
8978996213
s
25
128314368000000
-
7995735867463
s
29
248783228928000000
+
O
s
33
(*which is M_4 up to terms of order 32*)Proposition 5 (see [2], Theorem 3.4 on p. 106).
Letp
=
2
m
+
1, m
∈
N. Then, the formal Maclaurin series M
p converges on (
-
π
p
/
2
,
π
p
/
2
). Moreover, the formal Maclaurin series M
p converges towards s
i
n
p
x on [
0
,
π
p
/
2
) but does not converge towards s
i
n
p
x on (
-
π
p
/
2,0
).In AppendixA, we prove that T
p and M
p are identical.Theorem 6.
Letp
=
2
m
+
1, m
∈
N. Then,(21)
a
n
=
1
n
!
lim
x
→
0
+
⁡
sin
p
n
⁡
x
,
∀
n
∈
N
,and T
p
(
x
)
=
M
p
(
x
) for all x
∈
(
-
π
p
/
2
,
π
p
/
2
).It turns out that the pattern of zero coefficients ofM
p is the same as in the Maclaurin series of a
r
c
s
i
n
p
x; compare (18).Theorem 7.
Letp
>
2 be an integer. Then, a
i
=
0 for all i
∈
N such that i
-
1 is not divisible by p.The proof is technical and thus postponed to AppendixB. It is based on the formal inversion of (18). Note that the structure of powers in (18) does not allow any substitution that will transform it into a power series of new variable without zero coefficients. This makes the proof technically complicated.Using Theorem7, we can omit zero entries and rewrite the series M
p:(22)
M
p
x
=
∑
l
=
0
∞
α
l
·
x
l
p
+
1
,where α
l can be obtained by formal inversion of the Maclaurin series of a
r
c
s
i
n
p
x in (18). In particular, (23)
α
0
=
1
,
α
1
=
-
1
p
p
+
1
,
α
2
=
-
p
2
-
2
p
-
1
2
p
2
p
+
1
2
p
+
1
,
…
.


## 3. Extension ofsin
p
⁡
z and cos
⁡
p
⁡
z to the Complex Domain for Integer p
>
1

The conclusion of this theorem follows from the discussion in [2].Theorem 8.
Letp
=
2
(
m
+
1
), m
∈
N. Then, the Maclaurin series of s
i
n
p
x converges on the open disc(24)
B
p
≝
z
∈
C
:
z
<
π
p
2
.Proof.
In fact, the Maclaurin series∑
l
=
0
∞
‍
α
l
·
x
l
·
p
+
1 converges towards the values of s
i
n
p
x on (
-
π
p
/
2
,
π
p
/
2
) absolutely for p
=
2
(
m
+
1
), m
∈
N.Forp
=
2
(
m
+
1
), m
∈
N, the expressions with powers in the initial-value problem (6) can be written without the absolute values. Thus, the resulting initial-value problem(25)
u
′
p
-
2
u
′
′
+
u
p
-
1
=
0
,
u
0
=
0
,
u
′
0
=
1makes sense also in the complex domain (the derivatives are understood in the sense of the derivative with respect to complex variable). Let us observe that, using the substitution u
′
=
v, we get the following first-order system:(26)
u
′
=
v
,
v
′
=
-
u
p
-
1
v
p
-
2
,
u
0
=
0
,
v
0
=
1
.By [15, Theorem 9.1, p. 76], there exists δ
p
>
0 such that problems (26) and hence (25) have the unique solution on the open disc z
<
δ
p.Now we will consider initial-value problems (25) and (26) also for p
=
2
m
+
1, m
∈
N.Theorem 9.
Letp
=
2
m
+
1,m
∈
N. The unique solution u
(
z
) of (25) restricted to open disc B
p is the Maclaurin series M
p.Proof.
Letu
(
z
)
=
∑
k
=
1
∞
‍
b
k
z
k be the unique solution of (25) in any point of the open disc |
z
|
<
δ
p. Observe that the solution u
(
z
)
=
∑
k
=
1
∞
‍
b
k
z
k solves also the real-valued Cauchy problem (6) for x
>
0 (where there is no need for ·). On the other hand, s
i
n
p
x is the unique solution of the real-valued Cauchy problem (6). Since M
p given by (20) converges towards s
i
n
p
x in 0
,
π
p
/
2, we find that M
p satisfies (6) in 0
,
π
p
/
2. Thus, M
p
(
x
)
=
u
(
x
+
i
·
y
) for x
∈
[
0
,
δ
p
) and y
=
0. In particular, taking a sequence of points z
n
=
δ
p
/
(
n
+
1
)
+
0
·
i, n
∈
N, we have M
p
(
z
n
)
=
u
(
z
n
); thus, we infer from Proposition A.2 that (27)
b
k
=
1
k
!
lim
x
→
0
+
⁡
sin
p
k
⁡
x
.
M
p has radius of convergence π
p
/
2 by Proposition 5, and so does u
(
z
). Thus, δ
p
=
π
p
/
2.Theorems8 and 9 enable us to extend the range of definition of the function s
i
n
p
x to the complex open disc B
p by M
p for p
=
2
(
m
+
1
) and for p
=
2
m
+
1, m
∈
N, respectively. Thus, we can consider p
=
m
+
2 in the following definition. Note that all the powers of z are of positive-integer order l
·
p
+
1 and the function s
i
n
p
z is an analytic complex function on B
p and thus is single-valued.Definition 10.
Letp
=
m
+
2,m
∈
N, and z
∈
B
p. Then,(28)
sin
p
⁡
z
≝
∑
l
=
0
∞
α
l
·
z
l
·
p
+
1
,
cos
p
⁡
z
≝
sin
p
′
⁡
z
=
d
d
z
sin
p
⁡
z
,where the derivative d
/
d
z is considered in the sense of complex variables.The following fundamental results were proved in [3] (providing explicit value for δ
p).Proposition 11 (see [3], Theorem 2.1 on p. 226).
Letp
=
2
(
m
+
1
),m
∈
N; then, the unique solution of the initial-value problem (25) on B
p is the function s
i
n
p
z.Proposition 12 (see [3], Theorem 3.1 on p. 229).
Letp
=
2
m
+
1, m
∈
N. Then, the unique solution u
(
z
) of the complex initial-value problem (25) differs from the solution s
i
n
p
x of the Cauchy problem (6) for z
=
x
∈
(
-
π
p
/
2,0
).In [3], it was shown that there is no hope for solutions of (25) to be entire functions for p
=
m
+
2,m
∈
N. This result follows from the complex analogy of the p-trigonometric (10).Lemma 13.
Letp
=
m
+
1, m
∈
N, and r
>
0 be such that the solution u of (25) is holomorphic on a disc D
r
=
{
z
∈
C
:
z
<
r
}. Then, u satisfies the complex p-trigonometric identity(29)
u
′
z
p
+
u
z
p
=
1on the disc D
r.Proof.
Multiplying (25) by u
′ and integrating from 0 to z
∈
D
r, we obtain (30)
u
′
z
p
-
u
′
0
p
+
u
z
p
-
u
0
p
=
0
.Now using the initial conditions of (25), we get (29), which is the first integral of (25) and we can think of it as complex p-trigonometric identity for holomorphic solutions of (25) for p
=
m
+
1,m
∈
N.Now we state the very classical result from complex analysis.Proposition 14 (see [16], Theorem 12.20 on p. 433).
Letf and g be entire functions and for some positive integer n satisfy identity (31)
f
n
+
g
n
=
1
.
(i)
Ifn
=
2, then there is an entire function h such that f
=
c
o
s
∘
h and g
=
s
i
n
∘
h.
(ii)
Ifn
>
2, then f and g are each constant.The following interesting connection between complex analysis (including the classical reference [16, Theorem 12.20]) and p-trigonometric functions was studied in [3]. We should point out that it was an interesting internet discussion [17] that called our attention towards this connection. It seems to us that this connection was overlooked by the “p-trigonometric community.” Thus, we provide its more precise proof here.Theorem 15.
The solutionu of complex initial-value problem (25) cannot be entire function for any p
=
m
+
2,m
∈
N.Proof.
Assume by contradiction that the solutionu of (25) is entire function. Then, we can choose r
>
0 arbitrarily large in Lemma 13. Thus, u and u
′ must satisfy (29) at any point z
∈
C. Note that u
′ is an entire function too. Thus, by Proposition 14
u and u
′ are constant which contradicts u
′
(
0
)
=
1. This concludes the proof.In particular, the solution of (25) is u
(
z
)
=
s
i
n
p
z with u
′
(
z
)
=
c
o
s
p
z. Thus, (29) becomes (32)
cos
p
p
⁡
z
+
sin
p
p
⁡
z
=
1and we see that s
i
n
p and c
o
s
p cannot be entire functions for p
=
m
+
2, m
∈
N.


## 4. Generalized Hyperbolometric Functionargsinh
p
⁡
x and Generalized Hyperbolic Function sinh
⁡
p
⁡
x in the Real Domain for Real p
>
1

In analogy top
=
2, we define s
i
n
h
p
x for p
>
1 as the solution to the initial-value problem: (33)
-
u
′
p
-
2
u
′
′
+
p
-
1
u
p
-
2
u
=
0
,
x
∈
R
,
u
0
=
0
,
u
′
0
=
1
.The uniqueness of the solution of this problem can be proved in the same way as in the case of (6) using the first integral (see [4]). Note that the first integral of the real-valued initial-value problem (33) is the real p-hyperbolic identity (34)
1
+
u
p
=
u
′
p
,for p
>
1; compare [4]. Thus, u
′
≥
1 on the domain of definition of solution to (33). Since u
′
(
0
)
=
1 and u
′ must be absolutely continuous, we find that u
′
>
0 on the domain of definition of solution to (33) and the real p-hyperbolic identity can be rewritten in equivalent form (35)
u
′
=
1
+
u
p
1
/
p
, which is a separable first-order ODE in R. By the standard integration procedure, we obtaininverse function of the solution u (cf. [4]).Therefore, it is natural to define(36)
argsinh
p
⁡
x
≝
∫
0
x
1
1
+
s
p
1
/
p
d
s
,
x
∈
R
,for any p
>
1, in the real domain (cf., e.g., [18–21]). Note that the integral on the right-hand side can be evaluated in terms of the analytic extension of Gauss’s F
1
2 hypergeometric function to C
∖
{
s
+
i
t
:
s
>
1
,
t
=
0
} (see, e.g., [10, § 15.3.1, p. 558] and [11, Theorem 2.2
.
1, p. 65]); thus, (taking into account that integrand in (36) is even)(37)
argsinh
p
⁡
x
=
x
·
F
1
2
1
p
,
1
p
,
1
+
1
p
,
-
x
p
,
x
∈
0
,
+
∞
-
argsinh
p
⁡
-
x
,
x
∈
-
∞
,
0
.Sincea
r
g
s
i
n
h
p
:
R
→
R is strictly increasing function on R, its inverse exists and it is, in fact, s
i
n
h
p
x by the same reasoning as in [4] (cf., e.g., [20]).


## 5. Generalized Hyperbolic Functionssinh
⁡
p
⁡
z and cosh
⁡
p
⁡
z in Complex Domain for Integer p
>
1

In the previous section, we introduced real-valued generalization ofsinh
⁡
x called s
i
n
h
p
x. Our aim is to extend this function to complex domain. It is important to observe that, for p
=
2, the following relations between complex functions sin
⁡
z and sinh
⁡
z are known:(38)
sin
⁡
z
sinh
⁡
z
u
′
′
+
u
=
0
u
′
′
-
u
=
0
u
0
=
0
u
0
=
0
u
′
0
=
1
u
′
0
=
1
sin
⁡
z
=
-
i
·
sinh
⁡
i
·
z
,where z
∈
C and the equations are understood in the sense of ordinary differential equations in the complex domain.Since the function·
:
C
→
[
0
,
+
∞
) (complex modulus) is not analytic at 0
∈
C, we cannot work with (6) and (33), but we need to consider (25) and(39)
u
′
p
-
2
u
′
′
-
u
p
-
1
=
0
,
u
0
=
0
,
u
′
0
=
1in our discussion in the complex domain. Thus, the direct analogy of the classical relations summarized in the table above for p
≠
2 is stated in the following table: (40)
sin
p
⁡
z
sinh
p
⁡
z
u
′
p
-
2
u
′
′
+
u
p
-
1
=
0
u
′
p
-
2
u
′
′
-
u
p
-
1
=
0
u
0
=
0
u
0
=
0
u
′
0
=
1
u
′
0
=
1
sin
p
⁡
z
=
?
-
i
·
sinh
p
⁡
i
·
z
,where z belongs to some complex disc centred at 0
∈
C with radius small enough such that both complex initial-value problems are solvable. However, it turns out (see below) that if we define s
i
n
h
p
z as the solution (39), then the “p-analogies” of (2)-(3) are satisfied, but the “p-analogy” of the identity (1), that is,(41)
sin
p
⁡
z
=
-
i
·
sinh
p
⁡
i
·
z
,is not satisfied in general. Our aim is to provide conditions when (41) holds as well.Let us formalize the above-stated ideas. Denote(42)
D
p
≝
z
∈
C
:
z
<
γ
pan open disc in C, where γ
p
>
0 is given radius. At first we prove unique solvability of (39) in D
p.Lemma 16.
Letp
=
m
+
2,m
∈
N. Then, there exists a complex disc D
p such that the initial-value problem in complex domain (39) has a unique solution on D
p.Proof.
Using the substitutionu
′
=
v, we get the following first-order system:(43)
u
′
=
v
,
v
′
=
u
p
-
1
v
p
-
2
,
u
0
=
0
,
v
0
=
1
.By [15, Theorem 9.1] on page 76, the statement of the lemma follows.Now we can defines
i
n
h
p
:
D
p
→
C for any integer p
>
2.Definition 17.
Letp
=
m
+
2,m
∈
N. The complex function s
i
n
h
p
z is defined on D
p as the unique solution of the initial-value problem (39) and c
o
s
h
p
z
≝
sinh
p
′
⁡
z for all z
∈
D
p.Lemma 18.
Letp
=
m
+
1, m
∈
N, and r
>
0 be such that the solution u of (25) is holomorphic on a disc D
r
=
{
z
∈
C
:
z
<
r
}. Then, u satisfies the complex “p-hyperbolic” identity(44)
u
′
z
p
-
u
z
p
=
1on the disc D
r.The proof of Lemma18 is analogous to the proof of Lemma 13 and thus it is omitted.Remark 19.
Les us note that the real-valued identity(45)
u
x
′
p
-
u
x
p
=
1for general p
>
0 already appeared in [22], where the formal Maclaurin power series expansion of the solution to this identity was treated. Interesting recurrence formula for the coefficients of the Maclaurin power series can be found there. It will be very interesting to use the following relations between s
i
n
p
z and s
i
n
h
p
z to find the analogous recurrence formulas for s
i
n
p
z.Now we are ready to state main results of Section5.Theorem 20.
Letp
=
4
l
+
2, l
∈
N. Then,(46)
s
i
n
p
z
=
-
i
·
s
i
n
h
p
i
·
z
,
(47)
cos
p
⁡
z
=
cosh
p
⁡
i
·
zfor all z
∈
B
p. Moreover,(48)
s
i
n
h
p
z
=
∑
k
=
0
∞
-
1
k
·
α
k
·
z
k
p
+
1
.On the other hand, we have also the following surprising result.Theorem 21.
Letp
=
4
l,l
∈
N. Then,(49)
sin
p
⁡
z
=
-
i
·
sin
p
⁡
i
·
z
,
cos
p
⁡
z
=
cos
p
⁡
i
·
zfor all z
∈
B
p.The statement of the previous theorem is closely related to similar result forp-hyperbolic functions.Theorem 22.
Letp
=
4
l, l
∈
N. Then,(50)
sinh
p
⁡
z
=
-
i
·
sinh
p
⁡
i
·
z
,
(51)
cosh
p
⁡
z
=
cosh
p
⁡
i
·
zfor all z
∈
D
p.Proof of Theorem20.
Letp
=
4
l
+
2, l
∈
N, and u
(
z
)
=
s
i
n
h
p
z be the unique solution of the initial-value problem (39) on D
p.
We show thatw
(
z
)
=
-
i
·
u
(
i
·
z
) satisfies (25) on D
p
∩
B
p. Due to uniqueness of solution of (25), the identity (46) must hold on D
p
∩
B
p.
Indeed, plugging into the left-hand side of (25), we get (52)
w
′
z
p
-
2
w
′
′
z
+
w
z
p
-
1
=
d
d
z
w
z
p
-
2
d
2
d
z
2
w
z
+
w
z
p
-
1
=
d
d
z
-
i
·
u
i
·
z
p
-
2
d
2
d
z
2
-
i
·
u
i
·
z
+
-
i
·
u
i
·
z
p
-
1
=
i
·
d
d
i
·
z
u
i
·
z
p
-
2
d
2
d
i
·
z
2
u
i
·
z
+
-
i
·
u
i
·
z
p
-
1
=
i
·
d
d
s
u
s
p
-
2
d
2
d
s
2
u
s
-
i
·
-
i
p
-
2
·
u
s
p
-
1
=
i
·
d
d
s
u
s
p
-
2
d
2
d
s
2
u
s
-
-
i
p
-
2
·
u
s
p
-
1
=
i
·
d
d
s
u
s
p
-
2
d
2
d
s
2
u
s
-
u
s
p
-
1
=
0
.Note that for p
=
4
l
+
2,l
∈
N, -
i
p
-
2
=
1. The last equality then follows from (39). The right-hand side of (25) is zero. So we have verified that w
(
z
)
=
-
i
·
s
i
n
h
p
(
i
·
z
) satisfies the differential equation in (39). The initial conditions of (25) are also satisfied by w
(
z
)
=
-
i
·
s
i
n
h
p
(
i
·
z
), since u
(
z
)
=
s
i
n
h
p
z satisfies the initial conditions of (39).
Now it remains to show thatD
p
=
B
p and hence s
i
n
p
z
=
-
i
·
s
i
n
h
p
(
i
·
z
) on B
p. To this end, let us write s
i
n
h
p
z
=
∑
n
=
1
∞
‍
c
n
·
z
n for yet unknown c
n
∈
C. Then, (53)
∑
n
=
1
∞
a
n
·
z
n
=
sin
p
⁡
z
=
-
i
·
sinh
p
⁡
i
·
z
=
∑
n
=
1
∞
c
n
·
i
n
-
1
·
z
non D
p
∩
B
p. From here, (54)
c
n
=
a
n
i
n
-
1
=
i
3
n
+
1
·
a
n
.Since |
i
3
n
+
1
|
=
1, D
p
=
B
p.
Now taking into account thata
n
=
0 for n
-
1 not divisible by p, we immediately get c
n
=
0 for n
-
1 not divisible by p. Now using our notation α
k
=
a
k
p
+
1, k
∈
N, we find that (55)
c
k
p
+
1
=
i
3
k
p
+
1
+
1
α
k
=
-
1
k
α
kwhich establishes (48).
Equation (47) follows directly from c
o
s
h
p
z
=
sinh
p
′
⁡
z and (46).Proof of Theorem21.
Letp
=
4
l,l
∈
N, and u
(
z
)
=
s
i
n
p
z. Now, plugging w
(
z
)
=
-
i
·
u
(
i
·
z
) into the left-hand side of (25), we proceed in the same way as in the proof of Theorem 20. The most important difference is that for p
=
4
l,l
∈
N, -
i
p
-
2
=
-
1. Then, all following steps are analogous to those in proof of Theorem 20 with several obvious changes.Proof of Theorem22.
The proof is almost identical to the proof of Theorem20 with obvious changes (cf. the proof of Theorem 21).


## 6. Real Restrictions of the Complex Valued Solutions of (25) and (39) and Their Maximal Domains of Extension as Real Initial-Value Problems

Let us denote the restriction of the complex valued solution of (25) and (39) to the real axis by s
^
p
(
x
) and by s
h
^
p
(
x
), respectively. Since the equation in (25) and (39) contains only integer powers of the solution and its derivatives, all coefficients in the equation are real, and the initial conditions in (25) and (39) are real, the value of s
i
n
p
z and s
i
n
h
p
z must be a real number for z
=
x
+
i
y with -
π
p
/
2
<
x
<
π
p
/
2 and -
γ
p
<
x
<
γ
p and y
=
0, respectively. Hence, s
^
p
(
x
) and s
h
^
p
(
x
) attain only real values.We start with the slightly more complicated case, which iss
h
^
p
(
x
). Moreover, since the solution of (39) is an analytic function, it has the series representation(56)
sinh
p
⁡
z
=
∑
k
=
1
∞
c
k
z
k
,
z
∈
D
p
,where c
k
∈
R,k
∈
N (note that s
i
n
h
p
z must be a real number for any z
=
x
+
i
y with -
γ
p
<
x
<
γ
p and y
=
0).Now we show thats
h
^
p
(
x
) solves (33) (in the sense of differential equations in real domain) for p
=
2
(
m
+
1
),m
∈
N, and does not solve (33) (in the sense of differential equations in real domain) for p
=
2
m
+
1, m
∈
N, and x
<
0. For this purpose, we use an interesting consequence of omitting of the modulus function.Theorem 23.
Letp
=
2
m
+
1,m
∈
N, and x
∈
(
-
π
p
/
2
,
π
p
/
2
). Then,(57)
s
h
^
p
x
=
sinh
p
⁡
x
,
x
∈
0
,
π
p
2
,
sin
p
⁡
x
x
∈
-
π
p
2
,
0
.Proof.
Forx
∈
[
0
,
γ
p
), the statement of Theorem follows directly from the definition of real function s
i
n
h
p
x and the facts that s
i
n
h
p
x
≥
0 and sinh
p
′
⁡
x
≥
0 on [
0
,
γ
p
).
By the definition, the functions
i
n
p
x is the unique solution of (6); that is, (58)
-
u
′
p
-
2
u
′
′
-
p
-
1
u
p
-
2
u
=
0
,
u
0
=
0
,
u
′
0
=
1
.Assume that x
∈
(
-
π
p
/
2,0
). Then, s
i
n
p
x
<
0 and sin
p
′
⁡
x
>
0. Hence, we can rewrite (6) as(59)
u
′
p
-
2
u
′
′
-
u
p
-
1
=
0
,
u
0
=
0
,
u
′
0
=
1
,which is formally (39) but here considered in real domain.
By Lemma16, (39) has the unique solution on D
p. Its restriction to (
-
γ
p
,
0
)
∩
(
-
π
p
/
2,0
) clearly satisfies (59). Hence,(60)
sin
p
⁡
x
=
∑
k
=
1
∞
c
k
x
k
=
s
h
^
p
x
,
x
∈
-
γ
p
,
0
∩
-
π
p
2
,
0
.Moreover, s
i
n
p
x
=
∑
k
=
0
∞
‍
α
k
·
x
·
x
k
p, which is generalized Maclaurin series of s
i
n
p
x (see [2, Remark 6.6, p. 125]) convergent on (
-
π
p
/
2
,
π
p
/
2
). For (
-
π
p
/
2,0
), we obtain (61)
∑
k
=
0
∞
α
k
·
x
·
x
k
p
=
∑
k
=
0
∞
α
k
·
-
1
k
p
·
x
k
p
+
1
≕
⁡
G
x
. Hence, the Maclaurin series G
(
x
) converges on (
-
π
p
/
2
,
π
p
/
2
) (but not towards s
i
n
p
x for x
>
0). From (60) we get (62)
∑
k
=
1
∞
c
k
x
k
=
G
x
on
⁡
-
γ
p
,
0and using Proposition A.2 we obtain γ
p
=
π
p
/
2.Corollary 24.
Letp
=
2
m
+
1, m
∈
N. Then, s
h
^
p
(
x
) does not solve (33) for x
∈
(
-
π
p
/
2,0
).Proof.
Sinces
i
n
p
x
≠
s
i
n
h
p
x for x
≠
0, the statement of Corollary follows directly from Theorem 23 and the uniqueness of solution of (33).Theorem 25.
Letp
=
2
(
m
+
1
), m
∈
N. Then, s
h
^
p
(
x
) solves (33) for x
∈
(
-
γ
p
,
γ
p
). In particular, γ
p
=
π
p
/
2 for p
=
4
m
+
2, m
∈
N.Proof.
Sincep is even, we can drop the absolute values in (33) obtaining (59), which is formally (39) but here considered in real domain. Since s
i
n
h
p
(
z
) solves (39) on D
p, its restriction s
h
^
p
(
x
) to (
-
γ
p
,
γ
p
) must solve (59) on (
-
γ
p
,
γ
p
).
Forp
=
4
m
+
2, m
∈
N, we get γ
p
=
π
p
/
2 by (46) in Theorem 20.Theorem 26.
Letp
=
2
m
+
1,m
∈
N, and x
∈
(
-
π
p
/
2
,
π
p
/
2
). Then,(63)
s
^
p
x
=
sin
p
⁡
x
,
x
∈
0
,
π
p
2
,
sinh
p
⁡
x
,
x
∈
-
π
p
2
,
0
.Proof.
The proof follows the same steps as the proof of Theorem23 with obvious modifications.Now we will consider (25) and (39) as real-valued problems and find their maximal domains of extension. Let s
↔
p
(
x
) and s
h
↔
p
(
x
) denote solutions with maximal domains of (25) and (39), respectively. We also define c
↔
p
(
x
)
≝
d
/
d
x
s
↔
p
(
x
) and c
h
↔
p
(
x
)
=
d
e
f
d
/
d
x
s
h
↔
p
(
x
).Theorem 27.
Letp
=
2
m
+
1, m
∈
N. Then,(64)
s
↔
p
x
=
sinh
p
⁡
x
,
x
∈
-
∞
,
0
,
sin
p
⁡
x
,
x
∈
0
,
π
p
2
,
s
h
↔
p
x
=
sin
p
⁡
x
,
x
∈
-
π
p
2
,
0
,
sinh
p
⁡
x
,
x
∈
0
,
+
∞
.Theorem 28.
Letp
=
2
(
m
+
1
), m
∈
N. Then, (65)
s
↔
p
x
=
sin
p
⁡
x
,
x
∈
R
,
s
h
↔
p
x
=
sinh
p
⁡
x
,
x
∈
R
.Proof of Theorems27 and 28.
The solutions with maximal domain of extension are known for (6) and (33). The proof uses uniqueness of the solutions of real initial-value problems (6) and (33) and initial-value problems (25) and (39) considered in real domain and the fact that (6) and (33) can be rewritten as (25) and (39) depending on u
(
x
)
≶
0,u
′
(
x
)
≶
0, and the parity of the positive integer p. The main ideas of how to combine these ingredients are contained in the proof of Theorem 23.It easily follows from Theorems27 and 28 that c
↔
p
(
x
) is defined on (
-
∞
,
π
p
/
2
) for p
=
2
m
+
1 and on R for p
=
2
(
m
+
1
),m
∈
N. Similarly, c
h
↔
p
(
x
) is defined on (
-
π
p
/
2
,
+
∞
) for p
=
2
m
+
1 and on R for p
=
2
(
m
+
1
),m
∈
N. Moreover, functions s
↔
p
(
x
) and c
↔
p
(
x
) satisfy the complex p-trigonometric identity (29); that is,(66)
s
↔
p
x
p
+
c
↔
p
x
p
=
1
.Analogously, functions s
h
↔
p
(
x
) and c
h
↔
p
(
x
) satisfy the complex p-hyperbolic identity (44); that is,(67)
c
h
↔
p
x
p
-
s
h
↔
p
x
p
=
1
.


## 7. Visualizations

In this section, we provide visualizations of theoretical results from previous sections. To generate graphical output, we need to approximate special functions from previous sections numerically. Note that the standard numerical methods (available inMathematica orMatlab®) can handle only initial-value problems on real intervals. Thus, in our numerical calculations we need to consider initial-value problems in real domain. This is not a problem for (6) and (33). For (25) and (39), we calculate either the partial sum of the Maclaurin series of solutions or we calculate functions s
↔
p
(
x
) and s
h
↔
p
(
x
) which come from real initial-value problems. In our graphical outputs, the solutions of real initial-value problems are numerically approximated by the NDSolve command ofMathematica, version 9.0. For the convenience of the reader, we provide some source code. In Pseudocode 2, we list source code for approximation of functions s
↔
3
(
x
), s
h
↔
3
(
x
), and s
i
n
3
x.Pseudocode 2:
Mathematica, v. 9.0 code. Code for p
ip
[
p_
] computes π
p
/
2 for given argument p. Code for s3
[
x
_
] computes s
↔
3
(
x
) for x
∈
(
-
5
,
π
3
/
2
)
⊊
(
-
∞
,
π
3
/
2
), code for sh3
[
x_
] computes s
h
↔
3
(
x
) for x
∈
(
π
3
/
2,5
)
⊊
(
π
3
/
2
,
+
∞
), and code for sin3
[
x_
] computes real-valued function s
i
n
3
(
x
) for x
∈
(
-
2
π
3
,
2
π
3
)
⊊
R.
I
n
[
1
]
:=
pip
[
p_
]
=
Integrate
[
1/(1 - s
∧
p)
∧
(1/p),
s, 0, 1
,
Assumptions -> p > 1
]
(* assigns definition to function pip which returns pi_p/2 *)
Out
[
1
]
=
π
c
s
c
π
/
p
p
In
[
2
]
:=
s3
[
x_
]
=
(u
[
x
]
/.
NDSolve
[
u
＇
[
x
]
==
(v
[
x
]
)
∧
(1/2), v
＇
[
x
]
==
- 2 u
[
x
]
∧
2,
u
[
0
]
==
0, v
[
0
]
==
1
,
u, v
,
x, -5, pip
[
3
]
]
[
[
1
]
]
)
(* assigns definition to auxiliar function s3 *)
In
[
3
]
:=
sh3
[
x_
]
= (u
[
x
]
/.
NDSolve
[
u
＇
[
x
]
==
(v
[
x
]
)
∧
(1/2), v
＇
[
x
]
==
2 u
[
x
]
∧
2,
u
[
0
]
==
0, v
[
0
]
==
1
,
u, v
,
x, -pip
[
3
]
, 5
]
[
[
1
]
]
)
(* assigns definition to auxiliar function sh3 *)
In
[
4
]
:=
sin3
[
x_
]
=
(u
[
x
]
/.
NDSolve
[
u
＇
[
x
]
==
(Abs
[
v
[
x
]
]
)
∧
(1/2) Sign
[
v
[
x
]
]
,
v
＇
[
x
]
==
-2 Abs
[
u
[
x
]
]
*u
[
x
]
,
u
[
0
]
==
0, v
[
0
]
==
1
,
u,
v
,
x, -4 pip
[
3
]
, 4 pip
[
3
]
]
[
[
1
]
]
)
(* assigns definition to auxiliar function sin3 *)Figure1 compares graphs of s
i
n
3
x and s
↔
3
(
x
) for x
∈
(
-
π
3
/
2
,
π
3
/
2
). Figure 2 compares graphs of s
↔
3
(
x
) and M
^
3
;
28
(
x
) for x
∈
(
-
π
3
/
2
,
π
3
/
2
). Here, M
^
3
;
28
(
x
) is partial sum of M
3
(
x
) up to the order 28, which is(68)
M
^
3
;
28
x
=
x
-
x
4
12
-
x
7
252
-
83
x
10
90
720
-
1
817
x
13
7
076
160
-
199
691
x
16
2
377
589
760
-
12
324
719
x
19
406
567
848
960
-
22
008
573
061
x
22
1
878
343
462
195
200
-
107
355
387
043
x
25
22
540
121
546
342
400
-
89
152
153
354
993
x
28
44
304
862
911
490
621
440
.Since the difference s
↔
3
x
-
M
^
3
;
28
x varies in order of several magnitudes throughout the radius of convergence π
3
/
2, we use logarithmic scale on the vertical axis.Figure 1
(a) Real functions
i
n
3
x (short-dashed line) versus s
↔
3
(
x
) (dashed line). (b) Plot of s
i
n
3
x
-
s
↔
3
(
x
).
(a)
(b)Figure 2
(a)s
↔
3
(
x
) versus M
^
3
;
28 (the partial sum of M
3
(
x
) up to the order 28, which is (68)). (b) The logarithmic plot of s
↔
3
x
-
M
3
x.
(a)
(b)We can also compute the functionss
↔
p
(
x
) and s
h
↔
p
(
x
) by inverting formulas (8) and (36) for a
r
c
s
i
n
p
x and a
r
g
s
i
n
h
p
x, respectively. It turns out that this approach provides more precision than solving differential equation and enables computing values of c
↔
p
(
x
) and c
h
↔
p
(
x
) using identity (66) and identity (67), respectively. We provide sample code for computing s
h
↔
3
(
x
) for x
∈
(
π
3
/
2,5
)
⊊
(
π
3
/
2
,
+
∞
) in Pseudocode 3. Analogously, we wrote a code for computing s
h
↔
4
(
x
), s
h
↔
30
(
x
), s
h
↔
31
(
x
), s
↔
3
(
x
), s
↔
4
(
x
), s
↔
30
(
x
), and s
↔
31
(
x
).Pseudocode 3:
Mathematica v. 9.0 code. Function s
h3inv returns values of s
h
↔
3
(
x
) for x
∈
(
π
3
/
2,5
)
⊊
(
π
3
/
2
,
+
∞
) using inversion of the formula (36) for a
r
g
s
i
n
h
3
x. This approach provides more precision than solving differential equation.
I
n
[
2
]
:=
auxAgSh3
[
s_?NumberQ
]
:=
s Hypergeometric2F1
[
1/3,
1/3,
1 + 1/3,
-s
∧
3
]
(* assigns definition to auxiliar function auxAgSh3 *)
In
[
3
]
:=
sh3inv
[
x_?NumericQ
]
:=
(s/. FindRoot
[
auxAgSh3
[
s
]
-
x,
s, 0, -1, 20
]
)
(* assigns definition to function sh3inv *)
In
[
4
]
:=
Plot
[
sh3inv
[
x
]
,
x, -pip
[
3
]
, 2
,
PlotStyle -> Thick,
PlotRange
->
-2, 7
]
(* plots the graph *)In the same way, as we defined real functiona
r
g
s
i
n
h
p
x by (36), we can define complex valued function a
r
g
s
i
n
h
p
z by (69)
argsinh
p
⁡
z
≝
∫
0
z
1
1
+
s
p
1
/
p
d
s
=
z
F
1
2
1
p
,
1
p
,
1
+
1
p
,
-
z
p
,
z
∈
D
⊊
Cfor any p
=
m
+
2. Note that the integrand has poles at z satisfying 1
+
z
p
=
0. Thus, the function a
r
g
s
i
n
h
p
z is not an entire function. In particular, for p
=
2
m
+
1, there is a pole at z
=
-
1. In Figure 3, we compare graphs of a
r
g
s
i
n
h
p
x for p
=
2,3
,
31 and x
∈
(
-
3,3
)
⊊
R, with the restriction of the complex valued function a
r
g
s
i
n
h
p
z for p
=
2,3
,
31, where z
=
x
+
i
y, where y
=
0, x
∈
(
-
3,3
) for p
=
2 and x
∈
(
-
1,3
) for p
=
3,31, and with a
r
g
s
i
n
h
p
x
=
a
r
g
s
i
n
h
p
z for p
=
2,4
,
30, x
∈
(
-
3,3
), y
=
0, and z
=
x
+
i
y.Figure 3
(a)a
r
g
s
i
n
h
p
x
=
x
2
F
1
(
1
/
p
,
1
/
p
,
1
+
1
/
p
,
-
x
p
) for p
=
2,3
,
31, and x
∈
(
-
3,3
). (b) a
r
g
s
i
n
h
p
z
=
z
2
F
1
(
1
/
p
,
1
/
p
,
1
+
1
/
p
,
-
z
p
) for p
=
2,3
,
31. Here, z
=
x
+
i
y, where y
=
0, x
∈
(
-
3,3
) for p
=
2, and x
∈
(
-
1,3
) for p
=
3,31. (c) a
r
g
s
i
n
h
p
x
=
a
r
g
s
i
n
h
p
z for p
=
2,4
,
30, x
∈
(
-
3,3
), y
=
0, and z
=
x
+
i
y.
(a)
(b)
(c)In Figures4, 5, and 6, we compare graphs of real-valued functions: sinh
⁡
x, s
i
n
h
4
x, s
i
n
h
30
x, cosh
⁡
x, c
o
s
h
4
x, c
o
s
h
30
x (see Figure 4), sinh
⁡
x, s
i
n
h
3
x, s
i
n
h
31
x, cosh
⁡
x, c
o
s
h
3
x, c
o
s
h
31
x (see Figure 5), s
h
↔
3
(
x
), s
h
↔
31
(
x
), c
h
↔
3
(
x
), and c
h
↔
31
(
x
) (see Figure 6).Figure 4
(a)sinh
⁡
x (short-dashed line), s
i
n
h
4
x (dashed line), and s
i
n
h
30
x (solid line). (b) cosh
⁡
x (short-dashed line), c
o
s
h
4
x (dashed line), and c
o
s
h
30
x (solid line).
(a)
(b)Figure 5
(a)sinh
⁡
x (short-dashed line), s
i
n
h
3
x (dashed line), and s
i
n
h
31
x (solid line). (b) cosh
⁡
x (short-dashed line), c
o
s
h
3
x (dashed line), and c
o
s
h
31
x (solid line).
(a)
(b)Figure 6
(a)sinh
⁡
x (short-dashed line), s
h
↔
3
(
x
) (dashed line), and s
h
↔
31
(
x
) (solid line). (b) cosh
⁡
x (short-dashed line), c
h
↔
3
(
x
) (dashed line), and c
h
↔
31
(
x
) (solid line). Note that s
h
↔
p
(
x
) and c
h
↔
p
(
x
) are defined on (
-
π
p
/
2
,
+
∞
) for integer p
>
1 odd.
(a)
(b)Figure7 illustrates the relation between s
i
n
3
x, s
i
n
h
3
x, s
↔
3
(
x
), and s
h
↔
3
(
x
), which are due to Theorem 27.Figure 7
(a)s
i
n
3
x, s
i
n
h
3
x, and s
↔
3
(
x
). (b) s
i
n
3
x, s
i
n
h
3
x, and s
h
↔
3
(
x
).
(a)
(b)It follows from identity (66) and identity (67) that the pairs of functions s
↔
p
(
x
)
,
c
↔
p
(
x
) and s
h
↔
p
(
x
)
,
c
h
↔
p
(
x
), respectively, are parametrizations of Lamé curves restricted to the first and fourth quadrant, see [23, Book V, Chapter V, pp. 384–407] and [24]. Since the Lamé curves are frequently used in geometrical modeling, we provide graphical comparison of the Lamé curves and phase portraits of initial-value problems (25) and (39) in real domain on Figures 8 and 9, respectively.Figure 8
Bold lines: dependence ofu
=
s
↔
p
(
x
) on u
′
=
c
↔
p
(
x
). Both bold and thin lines: dependence of restriction to real axes of derivative of solution of (29) on the restriction to real axes of the solution itself (Lamé curves).Figure 9
Bold lines: dependence ofu
=
s
h
↔
p
(
x
) on u
′
=
c
h
↔
p
(
x
). Both bold and thin lines: dependence of restriction to real axes of derivative of solution of (44) on the restriction to real axes of the solution itself (Lamé curves).


## 8. Conclusion

We have discussed real and complex approaches of how to define generalized trigonometric and hyperbolic functions. The real approach is motivated by minimization of Rayleigh quotient (see, e.g., [1, Equation (
3.4
), p. 51] and references therein): (70)
∫
0
π
p
u
′
p
d
x
∫
0
π
p
u
p
d
xin W
0
1
,
p
(
0
,
π
p
). This leads to (4) with λ
=
p
-
1 and to initial-value problem (6) in turn. Thus, from this point of view, the solution of (6) and its derivative can be seen as natural generalizations of the functions sine and cosine. Unfortunately, presence of absolute value in (6) does not allow for extension to complex domain for general p
>
1. It was shown in [2] that functions s
i
n
p
x are real analytic functions forany even integer
p
>
2. Moreover, there is no need to write absolute value in (6) for x
∈
[
-
π
p
/
2
,
π
p
/
2
] provided p
>
2 is aneven integer.It turns out that the relation between the real and complex approach is not as smooth as in the classical casep
=
2. Thus, we summarize our results in Tables 1 and 2.Table 1
Summary of results according to discussed functions and their domain.
p
∈
Function
Domain
IVP
Results
(1, +∞)
sin
⁡
p
⁡
x
R
(6)
Propositions2–5; Theorems 6, 7, 8, 23, and 26–28
(1, +∞)
sinh
⁡
p
⁡
x
R
(33)
Theorems23 and 26–28
N ∖1
sin
⁡
p
⁡
z
B
p
(25)
Propositions11 and 12; Lemma 13; Theorems 6, 7, 8, 9, 15, 20, and 21
N ∖1
sinh
⁡
p
⁡
z
D
p
(39)
Lemmas16 and 18; Theorems 20 and 22Table 2
Interrelations betweenp-trigonometric and p-hyperbolic functions. Note that the relations must be symmetric.
sin
p
⁡
x
sinh
⁡
p
⁡
x
sin
p
⁡
z
sinh
p
⁡
z
sin
p
⁡
x
—
(57), (63)–(64)
—
—
sinh
p
⁡
x
sym.
—
—
—
sin
p
⁡
z
sym.
sym.
(49)
(46)
sinh
p
⁡
z
sym.
sym.
sym.
(50)We also discussed the Lamé curves, which are important curves in geometrical modeling. We hope this will stimulate interest inp-trigonometric and p-hyperbolic functions among the geometric-modeling community.



---

*Source: 3249439-2016-04-27.xml*