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CONCEPT AND TOPOLOGY OF RIEMANN SURFACES
§ 1. Weierstrass' concept of an analytic function
Let z be a complex variable and a a fixed complex number. With Weierstrass we say that any power series
(1.1) B(z  a) = A0 + A1 (z  a) + A2 (z  a)2 + ...,
with positive radius of convergence, is a function element with center a. The coefficients A0, A1, A2 ... are arbitrary complex numbers. The region of convergence of such a power series consists either of the whole zplane or of a disc z  a < r(r > 0), the "convergence disc," and a subset of the periphery [z  a = r of that disc.
In its convergence disc (which may be the whole plane regarded as a disc of radius r = ∞), such a function element represents a regular analytic function in the sense of Cauchy. Conversely, it is known from elementary function theory that a uniform regular analytic function may be expanded in a convergent power series (1.1) in any neighborhood z  a < r which is contained in the domain of regularity of the function. A power series then serves to represent the function only in a circular part of its domain.
If one starts with a power series which defines the function only in the convergence disc of the series (1.1), then the goal must be to define the function in larger domains of the zplane without losing the analytic character of the function. The method for this is Weierstrass' principle of analytic continuation. It turns out that the plan to conquer a largest possible domain of the zplane, for the function to be defined, is possible in only one way. But the uniformity (singlevaluedness) of the function is usually lost in the process of analytic continuation. This is not to be regarded as a defect; rather it is a great merit that in this fashion also the manyvalued analytic functions become amenable to an exact treatment.
If b is a value of z in the convergence disc z  a < r, then, as one knows, a rearrangement of the series (1.1) in powers of z  b gives a new power series,
(1.2) Q (z  b) = B0 + B1 (z  b) + B2 (z  b)2 + ...,
which converges at least in the disc [z  b < r  b  a; its convergence disc may have a radius greater than r  b  a. Since (1.1) and (1.2) take the same values at each point common to their two convergence discs, (1.2) provides an extension of the definition of our analytic function beyond the original domain. We shall say that (1.2) is an immediate analytic continuation of (1.1). The general process of (mediate) analytic continuation consists in applying immediate analytic continuation not just once, but an arbitrary finite number of times in a sequence – which is approximately analogous to the fact of projective geometry that the general projective transformation may be obtained by a sequence of an arbitrary number of immediate projective, i.e., perspective, transformations.
Analytic continuation may be undertaken along a given curve c. This means the following. Let a curve starting at z = a be given, i.e., to each real λ in the interval 0 ≤ λ ≤ 1 there corresponds in continuous fashion a point zλ of the complex zplane; in particular, to λ = 0 there corresponds the point z0 = a, and to λ = 1 a point z = c. Also, to every value of the parameter λ let there correspond a function element Bλ with center zλ; let B0 be the given function element (1.1) and also let the following condition hold: if λ0 is any λvalue, then there exists a positive number ε such the points zλ which correspond to values λ in the interval
max(0, λ0  ε) ≤ λ [less than or equal to] min(1,λ0 + ε)
all lie in the convergence disc of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the associated function elements Bλ are all immediate analytic continuations of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then we say that we have continued the given element (1.1) analytically along c, and we call B1 the function element with center c which results from the analytic continuation of (1.1) along c. Conversely, the analytic continuation of B1 backwards along c results in B0.
The analytic continuation along a given curve c is unique if it is possible at all The proof is based on a theorem which belongs to the foundations of analysis and which we will discuss in a more systematic context in § 4. It is the following: if to each λ in the unit interval 0 ≤ λ ≤ 1 there corresponds not only the point zλ, in continuous fashion, but also a disc z  zλ < rλ about zλ with positive radius rλ, then the curve defined by zλ can be partitioned into a finite number of arcs, λi ≤ λ ≤ λi+1 (0 = λ0< λ1< ... < λn = 1), and a point μi (λi< μi< λi+1) may be chosen on each arc, so that the ith arc is contained in the disc about zμi = zi of radius rμi = ri (i = 0,1, ..., n  1). If now Bλ and B*λ are two analytic continuations along the curve zλ, choose the positive number rλ so that Bλ and B*λ both converge in the disc about zλ with radius 3rλ. Then not only do Bμi and B*μi converge in the disc of radius 3ri about zμi, but Bλi and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converge in the disc of radius 2ri about zλi; the complete arc zλ for λi ≤ λ ≤ λi+1 is contained in this disc, and all the associated Bλ and B*λ arise by immediate analytic continuation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus the equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] entails [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By proceeding from division point to division point, one obtains from B0 = B*0 equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the whole unit interval and in particular for λ = 1; thus the last element is uniquely determined by the initial element and the curve along which the continuation takes place.
These considerations also show that if continuation along a given curve is possible, then one can get from the initial element to the last element by a finite number of applications of immediate analytic continuation. If the continuation of the initial element along is impossible, then there exists a definite point on the curve, the "critical point," at which the process finds its necessary end. More precisely: there exists a threshold λ = Λ0 of the following sort. If λ0< Λ0, then analytic continuation along the subcurve z = zλ (0 ≤ λ ≤ λ0) can be carried out; but not if λ0 = Λ0. Naturally Λ0 = 1 is possible; i.e., the endpoint of is the critical point; on the other hand, we always have Λ0 > 0.
Still another theorem on analytic continuation is important. If
z = z1(λ), z = z2(λ)
are two curves, from the same point a{ = z1(0) = z2(0)} to the same endpoint c, which remain sufficiently close together, then if the analytic continuation is possible along the first curve, it is possible along the second and yields the same end element. The condition that the curves remain sufficiently close together is: there exists a positive number δ such that if z1(λ)  z2(λ) < δ for all λ, then the conclusion of the theorem is valid. The proof follows immediately from the fact that one can obtain the end element from the initial element by a finite chain of immediate analytic continuations.
Now we are in a position to state the general Weierstrass definition of an analytic function as follows: ananalytic functionis the totality G of all those function elements which can arise from a given function element by analytic continuation.
Every function element of G may be obtained from any other by analytic continuation.
It can be shown that if the two analytic functions G1 and G2 have a single function element in common, then they are identical; i.e., every element of G1 is an element of G2 and conversely.
If
B(z  a) = A0 + A1(z  a) + A2 (z  a)2 + ...
is an element of G, then A0 is called a value of the analytic function G at the point z = a.
Certainly, at first glance, there is something artificial about Weierstrass' concept of a manyvalued analytic function as a collection of function elements. When one talks of [square root of z] or log z, one hardly envisages the totality of power series which represent pieces of these manyvalued functions. Nevertheless, Weierstrass' definition, whose simplicity and precision cannot be denied, has the advantage of being a solid starting point for analytic function theory. By gradual reworking of Weierstrass' formulation we will arrive at Riemann's formulation, in which the independent variable z as well as the dependent variable u, which up to now is represented by a totality G of function elements, appear as uniform analytic functions of a parameter; a parameter, to be sure, which in general takes values not in the complex plane but on a certain twodimensional manifold, the socalled Riemann surface.
But first we must extend, with Weierstrass, the concept of analytic function to that of analytic form.
§ 2. The concept of an analytic form
The concept of the analytic form arises from that of an analytic function when one considers not merely the points where the function is regular, as has been the case up to now, but adds those points at which it has a branch point of finite order or a pole (or both at once). If we suitably generalize the previous concept of function element, then we obtain the precise formulation of the concept of an analytic form.
With the aid of a complex parameter t we can represent the function element (1.1): u = B(z  a), as follows:
Z = a + 1, u = B(t) = A0 + A1t + A2t2 + ???.
If we abandon the distinguished role played by z and also allow a finite number of negative powers of t, we obtain the more general formulation. Let
z = P(t), u = Q(t)
be any two series in integral powers of t which contain only a finite number of terms with negative powers of t and which are such that in some neighborhood t < r (r a positive constant) of the origin: (1) both series converge, and (2) no two different values of t in this neighborhood give the same pair of values (z, u). Then we say that this pair of power series defines a function element. We add the condition that P(t) is not a mere constant.
It is not our intention that the pair of power series P(t), Q(t) is understood to be the "function element"; rather we regard the two series only as a representation of the intended function element, which has infinitely many other representations with equal claims. Concerning the transformation of one representation to another, we make the following agreement.
If one substitutes the power series
t(τ) = c1τ + c2τ2 + ???
for the parameter t in both P(t) and Q(t) then P(t) turns into a power series Π(τ), Q(t) into Κ(τ). We assume that t(τ) converges in some neighborhood of τ = 0 and that the first coefficient c1 ≠ 0; then there is a positive constant p such that in τ < ρ, t(τ) (1) converges and has modulus < r, and (2) assumes different values at any two distinct points τ. Then, in this neighborhood τ < ρ, Π and Κ converge, and for any two distinct points τ1, τ2 of this neighborhood, not both equations Π(τ1) = Π(τ2), (τ1) = (τ2) hold. We say the pair Π, Κ is equivalent to the original pair P, Q, no matter what the coefficients c1, c2, ... in the series t(τ) are, provided only that t(τ) converges and c1 [not equal to] 0. The last assumption has the consequence that conversely P(t), Q(t) may be obtained from Π(τ), (τ) by substituting for τ a certain power series in t:
τ = λ1t + λ2t2 + ... (λ1 = 1/c1 ≠ 0).
The relation of equivalence is thus symmetric. Also it is obvious that any pair of power series is equivalent to itself, and that if two pairs of power series are equivalent to a third, then they are equivalent to each other. These facts justify us in regarding equivalent pairs of power series as representations of the same, and nonequivalent pairs as representations of different, function elements. Or, to restate it: two pairs of power series, each representing a function element, define the same function element if and only if they are equivalent.
We depend here on a method of definition which one must use frequently in mathematics and which has its psychological roots in our minds' capability for abstraction. This kind of definition rests on the following general principle. If between the objects of any domain of operation there is specified a relation ~ which has the character of equivalences i.e., a relation satisfying the laws
(1) a ~ a, (2) from a ~ b follows b ~ a, (3) from a ~ c, b ~ c follows a ~ b;
then it is possible to regard each object a of that original domain of operation as a representative of an object α such that two objects a, b are representatives of the same object α if and only if they are equivalent in the sense of the relation ~. Precisely this principle is always to be used when we are interested only in those properties of the objects a, b, ... which are invariant under the relation ~. Its application has the advantage that a cumbersome terminology is replaced by a shorter one which suits the center of interest of the investigation in that it automatically strips the objects of what is inessential relative to this center. I mention here two examples of such "Definition by Abstraction"
(1) One says that two parallel lines have the same direction; two nonparallel lines have different directions. The original objects (a) are the lines; the relation with the character of equivalence is parallelism. One wishes to associate with each line a "something," its "direction," so that parallelism of fines corresponds to the identity of the associated "directions."
(2) A "motion" (of a point) is specified if the position of the moving point p is given at each instant λ of a certain time interval λ0 ≤ λ ≤ λ1: p = p(λ). If one has two such motions, p = p(λ), q = q(μ), then one says these motions travel the same "path" if and only if λ, the time parameter of the first motion, can be expressed as a continuous monotone increasing function of the time parameter μ of the second motion, λ = λ(μ), such that thereby the first motion becomes the second: p(λ(μ)) [equivalent] q(μ). Here it is the concept of "path" which is to be defined.
Excerpted from The Concept of a Riemann Surface by Hermann Weyl, Gerald R. MacLane. Copyright © 2009 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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