Mittag-Leffler's Theorem: Let ao, a., . . . be any set of points in the complex plane such that lirn a.=--so when n --= . In each of these points let an arbitrary polynomial in 1/ : A i(w) 212(*) A (n) fn(z)— z — (z—an)2 (Z an)te be chosen. Then there exists a single-valued function F (z) analytic throughout the whole plane except at the points an and behaving in each of these points like fn(z); i.e., the differ ence F (z) — fn(z) has no singularity in the point a.. The most general such function is given by the formula F (z) (z) G (z), where a denotes a particular function of the class in question and G is an integral function, rational or transcendental.
17. Elliptic and Abelian Integrals.— The motion of a simple pendulum and the length of the arc of an ellipse lead respectively to the elliptic integrals rxdx, ydx joY jo where 3,1— (1—x') (1 — k'x'), 0
If we set the first of these integrals equal to w and let x take on complex values s, then the inverse function z defined by dz _ rs 0 ( 1—z2) ( 1—kiss) turns out to be a single-valued function of its argument w having two linearly independent periods (§ 18) and being analytic throughout the whole w-plane except for poles.
A generalization which now presents itself is the following: The integrand in each of tile foregoing cases is a rational function of x and y, and these variables are connected by an alge braic equation. Consider, therefore, generally an algebraic equation G (w, z)=0, where G denotes an irreducible pol3nmmial in w and z. and construct the Riemann's surface corresponding to it. Let R (w, z) be any rational function of w and z. Then, setting w equal to the algebraic function of z defined by the fore going equation, we have in R(w, z) a function which is singled-valued and in general con tinuous on the above surface. If now we ex tend the definite integral roz. z) R(w,z)clz J (wo. so) along an arbitrary path on the surface, we have before us an Abelian Integral. lt is obvious that the logarithm and the inverse trigonometric functions, e.g., sdz sin—is=4 s2 + w2--= 1, o w are special cases of Abelian integrals. When the equation GO is of the form le= f (z), where f (z) denotes a polynomial in z of degree >4 having all distinct linear factors (or can be thrown into this form by means of birational transformations), the integral is called a hyper elliptic integral.
The only Abelian integrals whose inverse function is single-valued are those which lead to the exponential (including trigonometric) and the elliptic functions. The Abelian Functions
are single-valued periodic functions, not of a single argument, but of p independent variables, where p denotes what is known as the de ficiency of the corresponding Riemann's surface G(w, z)=. O. They have 2p periods. These functions arise as the inverse functions Pc (tm,... Wp), ( r= 1, ... , p) of a system of p equations: xi xi, wg -4 4),cdst +.. • +f Ozdzp, al 09 i where the integrals sb,ds are p linearly inde pendent Abelian integrals on the Riemann's sur face for G(w, z)= O.
18. Automorphic Functions.— A function f(z) is said to admit a linear transformation Into itself if , taz + g\ _f(o.
7 \Ys + di where a , . . . (5 are constants and a6—fly t O. A familiar class of such functions are the periodic functions. Here the transformation reduces to a translation: f(2+ w)=I(2).
The most general single-valued periodic func tions of a single variable a, which are in general analytic, are the and doubly periodic functions, i.e., those functions whose periods can all be expressed as integral multiples of a single primitive period or else in the form rur + n'w, where w and form a primitive pair of periods. In the latter case w/w cannot be a real quantity.
If the coefficients are integers and we have the elliptic modular functions. More generally the coefficients may have non-integral values, being restricted only in such a way that the transformations form a "discrete group.° These most general functions, including the special cases above mentioned, are known as the automorphic functions.
19. Definite Integrals.—The (-function may be defined by the definite integral f(0.---f PA-Je-idt, the variable of integration t being real and posi tive. This integral converges only for values of z which lie to the right of the axis of pure imaginaries: z=x--yi,x>0. A new definite in tegral can be formed as follows which converges for all values of z÷—n, where n denotes a positive integer, namely, C extended over the loop indicated in the figure.
It is readily shown that the second integral is equal to the first multiplied by for all values of a for which the first integral con verges. Hence we have generally 1 — c Tts)— By means of this integral the (-function has been systematically treated.