Appendix 1 Riemann, Schottky, and Schwarz on Conformal Representation In his Thesis [1851], Riemann sought to prove that if a function u satisfies cer- tain conditions on the boundary of a surface T, then it is the real part of a unique complex function which can be defined on the whole of T. He was then able to show that any two simply connected surfaces (other than the complex plane itself Riemann was considering bounded surfaces) can be mapped conformally onto one another, and that the map is unique once the images of one boundary point and one interior point are specified; he claimed analogous results for any two surfaces of the same connectivity [w To prove that any two simply connected surfaces are conformally equivalent, he observed that it is enough to take for one surface the unit disc K = {z : Izl _< l} and he gave [w an account of how this result could be proved by means of what he later [1857c, 103] called Dirichlet's principle. He considered [w (i) the class of functions, ~., defined on a surface T and vanishing on the bound- ary of T, which are continuous, except at some isolated points of T, for which the integral t= rx + ry dt is finite; and (ii) functions ct + ~. = o9, say, satisfying dt=f2 < oo Ox Oy + for fixed but arbitrary continuous functions a and ft. 224 Appendix 1. Riemann, Schottky, and Schwarz on Conformal Representation He claimed that f2 and L vary continuously with varying ,~. but cannot be zero, and so f2 takes a minimum value for some o9. The claim that this value is attained for some ,k in the first class of functions had earlier been made by Green [1883, first pub. 1835] and Gauss [1839/40] but it was to be questioned in another context by Weierstrass [1870], who showed by means of a counter-example that there is no general theorem of the kind: "a set of functions bounded below attains its bound". The use of Dirichlet's principle became contentious, and several mathematicians, notably Schwarz, sought to avoid it (Schwarz's work is described below). It was persuasively defended under slightly restricted conditions, by Hilbert [1900a] in a beautifully simple paper. In his next major paper on complex function theory, Theorie der Abel'schen Functionen [1857c], published just after his paper on the hypergeometric equation, Riemann recapitulated most of the above analysis. A complex function o9 of z = x + iy is one which satisfies 0o9 0o9 i Ox Oy and hence can be written uniquely as a power series in z-a where a is any point near which o9 is continuous and single-valued. If o9 is defined on a subset of the complex plane it may be continued analytically along strips of finite width. The continuation is unique at each stage, but if the path crosses itself the function may take different values on the overlap. If it does not, it is said to be single-valued or monodromic (einwerthig), otherwise multivalued (mehrwerthig). Multivalued functions possess branch points; e.g., log(z - a) has a branch point at a. The different determinations of the function according to the path of the continuation he now called its branches or leaves [Zweige, Bli~ttern]. Riemann extended his analysis of surfaces to include those without boundary by the simple trick of calling an arbitrary point of an unbounded surface its bound- ary. He concluded the elementary part of this paper by repeating his argument, based on what he now explicitly called Dirichlet's principle, that is, that a complex function can be defined on a connected surface T, which, on the simply connected surface T' obtainable from T by boundary cuts, (i) has arbitrary singularities, in the sense of becoming infinite at finitely many points in the manner of a rational function; and (ii) has a real part which takes arbitrary values on the boundary. The conformal representation of multiply connected regions was studied by Schot- tky [1877]. He considered the integrals of rational functions defined on such a region, A, and showed (w by looking at their periods that every single-valued function on A, which behaves like a rational function on the interior of A and is real and finite on the boundary of A, can be written as a rational function of functions like 1 ~ a) i U -- +UO+ } iui(x-- x--a i=1 Appendix 1. Riemann, Schottky, and Schwarz on Conformal Representation 225 i (X) v= + vo + ) i vi (x - a) i . x--a i-1 He defined the genus of A (w as Weierstrass had clone (Schottky referred to Weierstrass' lectures of 1873-74), noted that it agreed with Riemann's definition and that genus was preserved by conformal maps. He concluded the paper (w by taking the special case when A is bounded by a circle L0 which encloses n - 1 circles that are disjoint and do not enclose one another. This region has genus n - 1 and there is a group of conformal self-transformations of the disc bounded by L0 ob- tained by inverting A in each L 1..... L n-1 and then in the circle which bounds the images, and so on indefinitely. (The limit point set can be quite dramatic; Fricke in Fricke-Klein Automorphe Functionen, I, 104, gave an example when A is bounded by three circles and the limit set is a Cantor set). Such regions had already been con- sidered by Riemann, and the fragmentary notes he left on the question edited into a brief coherent text by Weber (Riemann [1953f], and published in 1876). Like Schot- tky, Riemann showed that a conformal map of the region can always be found which maps the boundary circles onto the real axis and takes every value in the upper half plane, H, n times. Repeated inversion then produces a function, invariant under the group, and this function is the inverse of the quotient of a second-order differential equation with algebraic coefficients, that is, the differential equation is defined on the above covering of H. Conversely, a conformal representation of this covering on A is obtained by solving such a differential equation, provided the coefficients take conjugate imaginary values at conjugate points. Schottky's work is essentially his thesis of 1875, and consequently was done independently of Riemann. Schwarz wrote several papers on the conformal representation of one region upon another between 1868-1870 which should be mentioned, although they do not immediately relate to differential equations. In the first of them, [1869a], Schwarz remarked that Mertens, a fellow student of his in Weierstrass' 1863-64 class on analytic function theory, had observed to him that Riemann's conformal representation of a rectilineal triangle on a circle raised a problem: the precise determination of such a function seemed to be beyond one's power to analyse because of the discontinuities at the corners. Discontinuity at that time meant any form of singularity; in this case the map is not analytic. That had set Schwarz thinking, for he knew no examples of the conformal representation of prescribed regions, and in this paper be proposed to map a square onto a circle. He said [ 1869a, in Abhandlungen, II, 66] "For this and many other representation problems this fruitful theorem leads to the solution: "If, to a continuous succession of real values of a complex argument of an analytic function, there corresponds a continuous succession of real values of the function, then to any two conjugate values of the argument correspond conjugate values of the function". This result is now called the Schwarz reflection principle. In symbols, it asserts that if f is analytic and f(x) is real for real x, then f(x + yi) = f(x - yi). To 226 Appendix 1. Riemann, Schottky, and Schwarz on Conformal Representation represent the square on a circle by a function t = f (u) Schwarz said it was simpler to replace the circle by a half-plane, which can be done by inversion ("transforma- tion by reciprocal radii"). The singular points would be t = c~, -1, 0, and 1 and the upper half plane is to correspond to the inside of the square. The boundary of the square will be mapped by t onto the real axis. As the variable u crosses a side of the square t (u) will cross into the lower half plane, and by the reflection principle t is thus defined on the four squares adjacent to the original one; so on iterating this process t is seen to be a doubly periodic function defined on a square lattice, thus a lemniscatic function. Indeed, Schwarz went on, the function v = u 2 converts the wedge-shaped region within an angle of rr/2 to a half plane, and any analytic function with non- vanishing derivative at v = 0 will then produce a conformal copy of the wedge. One might speak of such a function straightening out the corner. But even so, this u is not sufficiently general, for an everywhere analytic map sending lines to lines, namely u' -- Clu + C2, where C1 and C2 are constants, is surely equivalent to it. To eliminate the constants Schwarz put forward the equation d du' d du dt log dt = dt log dt' which he said was an important step, for d du dt log dt du is infinite whenever ~ is zero or infinite, which happens at each singular point. So d log du 1 -1 dt -~ = - -~ t +dl + d Et +..., and by considering the vertices t = -1, 0 and 1, Schwarz showed that the expres- sion dlogdU 1(1 1 1 ) dt -~-+2 /+1 +-+t t-1 =* 1 is to be analytic for all finite t.