ACM Communications in Computer Algebra, Vol 48, No. 4, Issue 190, December 2014
Maple Package itsol for Formal Solutions of Iterative Functional Equations Hideaki Izumi Department of Mathematics Chiba Institute of Technology Shibazono 2-1-1, Narashino, Chiba 275-0023, Japan E-mail:
[email protected] Equations including unknown functions are called functional equations([1]). Among them, functional equations containing only one independent variable are called iterative functional equations([9]). On the other hand, functional equations, like Cauchy’s equation([10]), containing two or more independent variables for unknown functions of a single variable are called functional equations of several variables([2]). Among iterative functional equations, the problem of iterative roots is important. If the n-th iterate of a function f is equal to g, i.e. f n (x) = g(x), f is said to be an n-th iterative root of g. It is well-known that if g is given and g has a fixed point a, then we can compute f as a Taylor series at x = a ([8, Chap. 15 §6]). However, in fixed point-free situations or when we want global solutions, it is difficult to obtain analytic iterative roots. In the case where g(x) = exp(x), this equation of iterative roots attracted many mathematicians([7], [3] etc.) The author developed a new algorithm, called the prompter method, to obtain formal series solutions of polynomial-like iterative functional equations of the form N X
n=1
an f n (x) = sx + b0 + b1 e−p1 x + b2 e−p2 x + · · · ,
(1)
where N ∈ N, a1 , . . . , aN , s, b0 , b1 , . . . ∈ R and 0 < p1 < p2 < · · · → ∞. The right-hand side is the sum of an affine term and a Dirichlet series, and hence it expresses an analytic function if it converges. We call this kind of series standard exponential series and the linear term sx its prompter. The real numbers −p1 , −p2 , . . . are called exponents of the standard exponential series and if b1 6= 0, then −p1 is called initial exponent. We can define the formal composition f ◦ g(x) of two standard exponential series f (x) and g(x), when the coefficient of the prompter of g is positive. The formal composition is defined as follows. Let f (x) = sx + a0 + a1 e−p1 x + a2 e−p2 x + · · · , g(x) = tx + b0 + b1 e−q1 x + b2 e−q2 x + · · · ,
where s, t, ai , bi , pi , qi ∈ R with t > 0, 0 < p1 < p2 < · · · → ∞ and 0 < q1 < q2 < · · · → ∞. We first X2 Xn define e−pg(x) with p > 0. By using the Taylor series expansion eX = 1 + X + + ··· + + · · · and 2! n! expanding formally the infinite product of infinite series, we have e−pg(x) = e−ptx−pb0 −pb1 e
−q1 x −pb
e−q1 x
2e
−q2 x −···−pb
e−q2 x
je
−qj x
−··· −qj x
= e−ptx e−pb0 e−pb1 e−pb2 · · · e−pbj e ··· ! ∞ ! ∞ ∞ ij i2 i X X X 1 (−pb ) (−pb ) (−pb ) j 2 1 e−i1 q1 x e−i2 q2 x · · · · e−ij qj x · · · · = e−ptx e−pb0 i1 ! i2 ! ij ! i1 =0
i2 =0
192
ij =0
ISSAC software presentations
= e
−pb0 −ptx
e
+
∞ X s=1
X
i1 , i2 , . . . , is−1 ≥ 0, is ≥ 1
(
e
−pb0 (−p)
i1 +i2 +···+is bi1 bi2 1 2
i1 !i2 ! · · · is !
· · · biss
)
e−(pt+i1 q1 +i2 q2 +···+is qs )x .
This shows that the initial exponent of e−pg(x) is −pt. Hence the sequence of moduli of the initial exponents of e−p1 g(x) , e−p2 g(x) , e−p3 g(x) , . . . is strictly increasing and we can define the composition f ◦ g(x) by f ◦ g(x) = sg(x) + a0 + a1 e−p1 g(x) + a2 e−p2 g(x) + · · · .
This is a well-defined sum of infinitely many standard exponential series, since for each exponent −p the number of non-zero coefficients of e−px appearing in these standard series is finite. Utilizing formal compositions, we define the N -th formal iterate f N of a standard exponential series f with a prompter of positive coefficient in a recursive way: f 1 = f, f N +1 = f ◦ f N , N = 1, 2, . . . . A formal solution of the iterative functional equation(1) is a standard series that satisfies the equation, where iterates of the unknown function are interpreted as formal ones. Theorem 1 ([5]) Suppose that the system of characteristic equations N X k=1
ak λk = s,
N X k=1
ak (1 + λ + · · · + λk−1 )µ = b0
associated with the iterative functional equation (1) has a solution (λ, µ) with λ > 0 (if λ = 1, assume further that λ = 1 is a simple root of the first equation). Then there exists a unique formal solution f (x) with prompter λx and constant term µ. Example 1 Let us consider an iterative functional equation of second order f 2 (x) = x − 3e−x .
(2)
The characteristic equation is λ2 = 1, so the positive characteristic is λ = 1. We put f (x) = x + a1 e−x + a2 e−2x + a3 e−3x + · · · . Then we can compute the second formal iterate as a1 3 f 2 (x) = x + 2a1 e−x + (2a2 − a1 2 )e−2x + 2a3 + − 3a1 a2 e−3x + · · · . 2 Comparing the coefficients with (2), we have a1 3 − 3a1 a2 = 0, . . . . 2 Hence we can determine an recursively and up to the exponent 10 we have a formal solution 2a1 = −3, 2a2 − a1 2 = 0, 2a3 +
3 9 27 189 −4x 567 −5x 3159 −6x f (x) = x − e−x + e−2x − e−3x + e − e + e 2 8 16 64 128 2560 143613 −7x 4877739 −8x 8636463 −9x 79218243 −10x e − e + e + e . + 5120 35840 28672 229376 The graphs of f (x), f (f (x)) and x − 3e−x are in Fig. 1 and these graphs suggest that the solution f (x) would converge for x > 1. 193
Hideaki Izumi
Figure 1 The prompter method also applies to the trigonometric functions sin ax and cos ax, a ∈ R. A standard trigonometric series is a formal double series X f (x) = µx + am, n (sin ax)m (cos ax)n , m, n≥0
where µ is either 1 or −1. The composition of two standard trigonometric series is defined by using Taylor expansions and the addition theorems of sine and cosine functions. Theorem 2 ([5]) For an iterative functional equation X f N (x) = µx + am, n (sin ax)m (cos ax)n ,
(3)
m, n≥0
where µ is either 1 or −1, we have the following. (i) If N is even and µ = 1, then there exists two formal solutions of (3). Both of them are standard trigonometric series; one has the prompter x, and the other has −x. (ii) If N is odd, then there exists a formal solution of (3), and its prompter is µx. This theorem also holds for standard hyperbolic series, where we replace sin and cos in (3) by sinh and cosh. Example 2 For an iterative functional equation of second order f 2 (x) = x + sin x, we have a formal solution up to degree 5 (Fig. 2): 1 1 1 1 5 sin x − (sin x)(cos x) + (sin x)(cos x)2 − (sin x)3 (cos x) − (sin x)(cos x)3 2 8 16 384 128 1 1 7 − (sin x)5 + (sin x)3 (cos x)2 + (sin x)(cos x)4 . 256 192 256 194
f (x) = x +
ISSAC software presentations
Figure 2 and the other solution is −f (x).
Even if standard trigonometric series or standard hyperbolic series lack their prompters, we can manage to obtain iterative roots with phantom prompter techniques as the following example shows. Example 3 To solve f 2 (x) = sin x
(4)
within our framework, we express x by a polynomial of sin x. To do this, let us recall the Taylor expansion of arcsin x: 1 3 5 7 35 9 arcsin x = x + x3 + x5 + x + x + · · · . (−1 ≤ x ≤ 1) (5) 6 40 112 1152 If we substitute sin x into x in (5), we have π π 3 5 35 1 sin5 x + sin7 x + sin9 x + · · · . − ≤x≤ (6) x = sin x + sin3 x + 6 40 112 1152 2 2 By using (6), we can transform the prompter-free iterative functional equation (4) into a functional equation with a prompter: 1 3 5 35 f 2 (x) = x − sin3 x − sin5 x − sin7 x − sin9 x − · · · . (7) 6 40 112 1152 From Theorem 2, we have a solution of (7) 3 1 5 35 1 1 sin3 x − sin5 x − sin5 x cos x − sin7 x − sin9 x + sin7 x cos2 x. 12 80 96 224 2304 1152 We apply again (6) to (8) to make the solution periodic. We get (Fig. 3) f (x) = x −
f (x) = sin x +
1 3 1 5 35 1 sin3 x + sin5 x − sin5 x cos x + sin7 x + sin9 x + sin7 x cos2 x. 12 80 96 224 2304 1152 195
(8)
Hideaki Izumi
Figure 3 The Maple package itsol based on the prompter method and the dimensioned number method([6]), which is suitable for expressing iterated exponentials, will be demonstrated.
References [1] Acz´el, J´anos. Lectures on functional equations and their applications. Academic Press, 1966. [2] Acz´el, J´anos.; Dhombres, Jean. Functional equations in several variables. Encyclopedia of Mathematics and its Applications 31, Cambridge University Press, 1989. [3] Belitskii, Genrikh Ruvimovich.; Tkachenko, Vadim. One-Dimensional Functional Equations. Operator theory: Advances and Applications 144, Birkh¨ auser Verlag, 2003. [4] Izumi, Hideaki. Analytic Solutions of Iterative Functional Equations. Talk at Formal and Analytic Solutions of Differential and Difference Equations, 2011, Poland. http://www.impan.pl/~fasde/ presentations/Izumi.pdf [5] Izumi, Hideaki. Formal solutions of iterative functional equations. (preprint) [6] Izumi, Hideaki. Applications of dimensioned numbers to functional equations. Accepted for publication in ESAIM: Proceedings and Surveys. [7] Kneser, Hellmuth. Reelle analytische L¨ osungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen. Journal f¨ ur die reine und angewandte Mathematik 187: 56–67, 1950. [8] Kuczma, Marek. Functional equations in a single variable. PWN-Polish Scientific Publishers, 1968. [9] Kuczma, Marek.; Choczewski, Bogdan.; Ger, Roman. Iterative Functional Equations. Encyclopedia of Mathematics and its Applications 32, Cambridge University Press, 1990. [10] Kuczma, Marek. An introduction to the theory of functional equations and inequalities: Cauchy’s equation and Jensen’s inequality, 2nd ed. Birkhauser, 2009. 196
