复域差分与差分方程

　　读者对象为从事复分析研究的学者及从事数学,物理,化学,工程工作的工程师. 

　　《复域差分与差分方程》主要介绍近七年来复域差分,差分方程的研究成果,其中包括:亚纯函数Nevanlinna理论的差分模拟,如对数导数的差分模拟,Clunie引理的差分模拟,涉及差分的多项式值分布性质,以及亚纯函数涉及差分,移动的分担值问题.　　同时介绍复域差分方程的研究成果,包括线性差分方程与非线性差分方程的一般理论,几种典型的非线性差分方程,如差分Riccati方程,差分Painleve方程的解析理论.　　还将介绍q-差分的Nevanlinna理论的差分模拟,及q-差分方程的解析理论.　　以上内容主要来自近七年Halburd,Korhonen,Laine,C.C.Yang,Bergwerler,Ablowitz,Y.M.Chiang,S.J.Feng,作者本人及其团队关于复域差分,差分方程的研究文献.

 

 

 

Chapter 1
Basic Properties of Complex Differences
1.1 Preliminaries
The Nevanlinna characteristic T(r, f), which encodes information about the distribution of values of f on the disk |z| <= r, plays a central role in the theory of meromorphc functions. It is a sum of two parts:
The proximity function m(r, f) is given by
where log+ x = max{0, logx}. The proximity function is an averaged measure of how large f becomes on the circle |z| = r. Define the counting function n(r, f) to be the number of poles of (counted according to multiplicities) in the circle \z\ = r. The integrated counting function, N(r, /)，is then defined to be
The Nevanlinna characteristic function T(r, f) is therefore the sum of a measure of how large f becomes on \z\ = r and a measure of the number of poles of / in \z\ < r.
Now, we recall the following results which play important roles in investigation of complex differences and difference equations.
Theorem 1.1.1 (The first main theorem of Nevanlinna theory, see [89, 185]) Let f be a meromorphic function，a G C. Then
The first main theorem implies that if f(z) takes a value a G C|J{oo} fewer times than average, i.e., the counting function is relatively
small, then the proximity function m(r, a) = m (r, 1/f-a) must be large, and
vice versa. Loosely speaking, if a meromorphic function assumes a certain value a relatively few times, the values of f(z) are “near” the value a in a large part of the complex plane.
Theorem 1.1.2(The second main theorem of Nevanlinna theory, see [89, 185]) Let f be a meromorphic function, be each other not equal. Then
Theorem 1.1.3(The logarithmic derivative lemma, see [89, 185]) Let f be a meromorphic function. Then
holds outside a possible set of finite linear measure.
It shows that the proximity function of logarithmic derivative of f(z) grows more slowly than the characteristic function of f(z).
Remark 1.1.1 A meromorphic function f is transcendental if and only if
Let f(z), g(z) and h(z) be meromorphic functions, and n E N. Then
T(r, f + g)^T(r, f)+T(r, 5)+ 0(1),
T(r, fg) < T(r, f)+T(r, g),
T(r, D = nT(r, /)，
T(r, fg + gh + hf)^T(r, f)+T(r, g)+T(r, h) + 0(l),
T(r + 1, /) = T(r, f) + S(jr, f) if f is of finite order.
Theorem 1.1.4(The Milloux theorem, see [89, 185]) Let f be a meromorphic function. Then
Theorem 1.1.5(The Wiman-Valiron theory, see [96, 121]) Let f be a transcen-dental entire function，letOholds. Then there exists a set F c M+ of finite logarithmic measure, i.e., I — <
Jf t
+oo, such that
holds for all m >=0 and all r ^ F.
Remark 1.1.2 Let f be an entire function whose Taylor expansion is
The maximum term is defined as
the central index is defined as
Theorem 1.1.6(Lemma of Clunie, see [89, 121]) Let f(z) be a transcendental meromorphic solution of equation
where P(z, f) and Q(z, f) are polynomials in f and its derivatives with meromorphic coefficients，say {a_\ : A G I}, such that m(r, a_\) = 5(r, f) for all X ? I. If the total degree of Q(z, f) as a polynomial in f and its derivatives is ，then
Theorem 1.1.7(Mohon,ko, see [136，121]) Let f(z) be a meromorphic funtion. Then for all irreducible rational funtions in f，
such that the meromorphic coefficients ai(z), bj(z) satisfies
then we have
T{r, R(z, /)) = max{p, q} T(r, f) + S(r, /)?
Theorem 1.1.8(The Weierstrass factorization theorem, see [121]) Let f(z) be an entire function，with a zero of multiplicity m 彡 0 at z = 0. Let the other zeros of f be ai, a2, ，each zero being repeated as many times as its multiplicity implies. Then f has the representation
for some entire function g and some integers mn. If (an)nen has a finite exponent of convergence A, then mn may be taken as k = [A] > A — 1.
Remark 1.1.3 The Weierstrass primary factors is defined as
Theorem 1.1.9(See [121]) Let g : (0, + oo) —> R, h : (0, + oo) —>R be monotone increasing functions such that g(r)<= h(r) outside of an exceptional set E of finite linear measure (or finite logarithmic measure). Then, for any a > 1，there exists r。> 0 such that g(r)<= h((ar) (or g(r)<= h((ra)) holds for all r > r〇.
Theorem 1.1.10(cosira theorem, see [7-9]) Let h(z) be an entire function with
order cr(h) = cr < -, set
where the lower logarithmic density log densH of subset H C (1, +oo) is defined by
and the upper logarithmic density log densH of subset H c (1，+oo) is defined by
where C{a, 8, ck) is a positive constant only dependent on a, 8, a.
By definitions of the logarithmic measure and the logarithmic density, we see that if the upper logarithmic density log densH > 0, then the logarithmic measure ImH = +oo.
1.2 Difference Analogue of the Lemma on the Logarithmic Derivative
The lemma on the logarithmic derivative states that outside of a possible small exceptional set
This is undoubtedly one of the most useful results of Nevanlinna theory, having a vast number of applications in the theory of meromorphic functions and in the theory of ordinar