R-- C,t-C+y'(t;)t, tER
1- Ihc(z)12 = iI2
ftc,(z) =
1 _ 1 z 1 2I1-zI2
These equations imply thatfor z
-i,
forz34 1.
1-Ihc(z)I2>0
for Q3z>0;3`hc'(z)>0 forizl<1
and consequently hc(H) C E and hc'(E) C H. Together with the above descriptions of the composites of hc, hc, with one another, this says that hC maps H biholomorphically onto E with hc1 = hc,. This discussion has proved the
Theorem. The mapping he : H Z E, z
z + i is biholomorphic with inverse mapping hC, : E Z H, z i + z
-z
For historical reasons the mappings hC, hc, are called the Cayley map- pings of H onto E and E onto H, respectively.
3. Remarks on the Cayley mapping. A critical reader might ask: "How would the function hc(z) = 1+ ever occur to one as a candidate for a biholo- morphic mapping H Z E?"
The general question: "Are H and E biholomorphically equivalent?" offers no hint of the role of this function. But once it is known, the rest is routine verifications; the real mathematical contribution consists just in writing down this function.
With hindsight we can discern certain simple heuristic considerations that could have led to the function hc. First suppose one already recognizes the
fractional linear transformations as an interesting class of functions (which itself requires some mathematical experience) and is prescient enough to look for a biholomorphic mapping Ill Z E among them. Then it isn't wild to speculate that the sought-for function h will map the boundary of H, i.e., the real axis, into the boundary of E, i.e., into the unit circle. (Or is that plausible?: the circle is compact but the line is notl) So we try to map some pre-assigned points of R to certain points of the circle, say (here we have considerable freedom),
h(O) := -1 ; h(l) := -i ; h(oo) := 1 , that is, lim h(x) = 1.
IxIpp
For a function of the form h(z) = +d these requirements translate into the equations
-1 = h(0) = d ; - i = h(l) = cc+d ; 1 = h(oo) = c+ d/oo
= c'
From which one sees that b = -d, a = c and so (via the second equation) a - d = -ia - id, that is, d = ia. With these specifications the function h turns out to be the Cayley mapping.4*. Bijective holomorphic mappings of H and E onto the slit plane. It is quite surprising what can be done with just the Cayley mapping he and the squaring function z2. Let C- denote the complex plane slit along the negative real axis, i.e.,
C :=C\{zEC: tz<0,clz=0}.
First we claim that
The mapping q : H -, C-, z -z2 is holomorphic and bijective.
(Were we to have slit C along the positive real axis, then we would consider z2 here instead of -z2. The reason for slitting along the negative real axis has to do with the complex logarithm function to be introduced later (in 5.4.4): this can only be done in a slit plane and we are reluctant to discard the positive real axis where the classical real logarithm has lived all along.)
Proof. There is no c E H such that t := q(c) is real and non-positive, since from q(c) = -c2 = t E H follows c2 = -t > 0 and therewith c E E, c H. It follows
that q(H) C C-. For c,c E H, q(c) = q(c') occurs if and only if c' = ±c and since not both c and -c can lie in HH, it must be true that c' = c; thus q : H - C- is injective.
Every point w E C has a q-pre-image in H, because the quadratic equation z2 = -w has two distinct roots, one of which lies in H - cf. the explicit square- root formula at the end of 0.1.3.
As a simple consequence we note that
The mapping p : E -+ C-, z (,+,)2 is holomorphic and bijective.
Proof. Via q o hc, the region E is mapped holomorphically and bijectively onto C , and q(hc'(z))
(i+ )a = ()2, that is, qo hc, = p.
Scarcely anyone would trust that such a seemingly simple function as p could really map the bounded unit disc bijectively and conformally onto the whole plane minus its negative real axis (cf. the figure).
Onmegative
real axis
It is appropriate to mention here that the mappings q : H - C- and p : E -*
C- are even biholomorphic. This is because the inverse mapping q-' : C- -. H is automatically holomorphic, as will be proved in 9.4.1.
Remark There is no biholomorphic mapping of the unit disc E onto the whole plane C, a fact which follows from LIOUVILLE'S theorem (cf. 8.3.3). However, the famous theorem announced by R!EMANN in 1851 ((R], p. 40) says that every simply connected region other than C can be biholomorphically mapped onto E.
But we won't reach this remarkable theorem until the second volume.
Exercises
In what follows QI denotes the first quadrant {z E H : H?z > 0} of the complex plane.
Exercise 1. Show that the Cayley transformation he : HH -+ E, z maps QI biholomorphically onto {w E E : £1,w < 0}.
Exercise 2. Supply holomorphic, bijective and angle-preserving mappings of Qt onto E \ (-1, 0] and of QI onto E.
Exercise 3. Let f (z) := "Z+d with c 96 0 and ad - be 96 0. Further, let L be a circle or a (real) line in C. With the aid of Exercise 1 of §1 determine the images f (L), respectively, f (L \ {-d/c}). (Case distinctions have to be made!) Hint. First prove that f can be written as
f(z) = be -ad
(z + d/c)-1 + a
Exercise 4. For M > R > 0 form the punctured upper half-plane H \ BR(iM). With the aid of a fractional linear transformation map it biholo- morphically onto an annulus of the form {w E C : p < jwi < 1}. Hint.
First look for a c < 0 such that z ' - (z - ic)-1 maps the boundary of BR(iM) and the real axis onto concentric circles. Use Exercise 1 of §1.
Exercise 5.
Find a map of E onto {w E C :
13'w > (Rw)2} which is holomorphic, bijective and angle-preserving.§3 Automorphisms of the upper half-plane and the unit disc
A biholomorphic mapping h : D Z D of a domain D onto itself is called an automorphism of D. The set of all automorphisms of D will be denoted by Aut D. Pursuant to the remarks in the introduction of §2, it is clear that
Aut D is a group with respect to the composition of mappings and the identity mapping id is its neutral element.
The group Aut C contains, for example, all "affine linear" mappings z + az + b, a E C', b E C and because of this it is not commutative.
Aut C is actually rather simple, for these affine linear maps exhaust it; but we will only see that via the theorem of CASORATI and WEIERSTRASS in
10.2.2.
In this section we will study the groups AutH and Aut E exclusively.
First we consider the upper half-plane H (in subsection 1) and then (in subsection 2) transfer the results about H to E via Cayley mappings. In subsection 3 we give a somewhat different representation of the automor- phisms of E; finally (subsection 4) we show that H and E are homogeneous with respect to their automorphisms.
1. Automorphisms of H. The sets GL+(2, R) and SL(2, R) of all real 2 x 2 matrices with positive determinant, respectively, with determinant 1, are each groups with respect to matrix multiplication (proof!). We write
A = (ry I for the typical such matrix and designate with hA(z) _°yi+6, as we did in 2.1, the fractional linear transformation associated with A.
Then
(1) QrhA(z) = det A
Iyz+61221x for every A
= \7 I E GL+(2,If).
Proof. Because A is real
2i3*A(z)
= hA(z) - hA(z)
- az + Q a2 + /3 ab -'3-Y(x - z) ryz+b ryz+b Iryz+b12
= 2i det A Zlz.
Iyz+612 From (1) follows immediately
Theorem. For every matrix A E GL+(2, R) the mapping hA : H -* H is an automorphism of the upper half-plane and hA-I is its inverse mapping.
Proof Since A, A-1 E GL+(2, Ht), hA and hA-I are holomorphic in H.
On account of (1), hA (H) C H and hA-I (HH) C H. Finally, hA o hA-I = hA-I o hA = id puts hA in Aut H.
0
Furthermore (with the help of the rules in 1.2) we now get:
The mapping GL+(2, R) -+ Aut H given by A - * hA is a group homo- morphism whose kernel consists of the matrices AE, ,1 E R \ {0}. The restriction of this homomorphism to the subgroup SL(2, R) has the same image group and its kernel consists of just the two matrices ±E.
2. Automorphisms of E. If f : D Z D' is biholomorphic, then the
mapping h i-+ f o h o f -1 effects a group isomorphism of Aut D onto Aut D' (proof!). Thus, knowing f, f -I and automorphisms of D, we can construct automorphisms of D'. Applying this process to the Cayley mapping he : H --+ E together with its inverse hC, shows, in view of theorem 1, that all the functions
he o hA o hc, = hCAC' with A E SL(2, H) are automorphisms of E. This observation leads to the following
Theorem. The set M := {B :_ (6 a)
:
a, b E C , det B = 1} is a
subgroup of SL(2, C) and the mapping from SL(2, R) into M defined by A -. z, CAC' is a group isomorphism.The mapping B ' hB(z) = ax b of M into Aut E is a group homo- morphism whose kernel consists of the two matrices ±E.
bz+a
Proof. Since det C = det C' = 2i, the multiplicativity of the determinant gives det(-LCAC') = 1. Therefore the mapping
(p : SL(2, R) -SL(2, C) ,
A'-' I CAC'
is well defined and, thanks to the fact CC' = C'C = 2iE, a group
monomorphism. The first claim in the theorem therefore follows as soon as we have shown that the image of cp is M. To this end note that forA= (7 b)
CAC' = (1 i) (ry b)(-1 1)
(2) =
i( a+b+i(/3-y) a-b-i((3+ry) ) a-b+i(/+ry) a+b-i(/3-ry)
Upon setting a:= 1 [(a + b) + i(/3 - ry)] and b := z (a - b) - i(/+ y)], it follows that
B:=SP(A)=(6 b)
a ,and so image
The other inclusion M C image of V requires that for every B = (6 a) E M there be an A = ( b) E SL(2, R) with 2iB = CAC'. To realize such an A it suffices to set
a:= t(a+b), /3:=! (a-b), -y:=-!a(a+b), b:= t(a-b).
The matrix A so defined is then real and satisfies CAC' = 2iB, thanks to (2), which holds for all real 2 x 2 matrices A. Of course, det B = 1 means that det A = 1, so A E SL(2, Ht), as desired.
To verify the second claim, recall that by 2.1(1) hB = hC o hA o hC, whenever B = ICAC'. Therefore (1) says that for all B E M the functions hB are automorphisms of E. The homomorphic property of the mapping M AutE and the assertion about its kernel now follow from 2.1(2) and
2.1(1).
0
The subgroup M of SL(2, C), which according to our theorem is isomorphic to SL(2, R), is often designated by SU(1, 1).
3. The encryption slwz
1 for automorphisms of E. The automor- phisms of E furnished by theorem 2 can be written in another way; to do it we need a matrix-theoretic77
-
I, rl E 8E, w E E, corresponds a Lemma. To every matrix W = (Wmatrix B E M such that W = sBfor
some s E C'.
Proof. Since w E E, 1-1w12 > 0 and -,-w'Lrx E C". So by 0.1.3 there exists an aEC" with a2 = w For b:_-wa,then JaJ2-1b12=1n1 = 1, and
soB:= (a a) EM. If we sets:=rla-I=-a(1-Jw12),then sa=-1
(so s E C"!) and aB = W.
0
Bearing in mind that the functions hw and hB coincide when W, B are related as in the lemma, it follows directly from theorem 2 that
Theorem. Every function z H 0wz i with 17 E OE and w E E, defines an automorphism of E.
In case to = 0 the automorphism is z i- riz, a rotation about the origin.
A special role is played by the automorphisms
(1)
g:E. E, zH z-w , wEE.
wz-1
For them g(0) = to, g(w) = 0 and g o g = id. The latter equality following from 2.1(1) and the calculation
(w -1) (w -1) _ (1 - Iw12)E.
Because of the property g o g = id, the automorphisms g are called involu- tions of E.
4. Homogeneity of E and H. A domain D in C is called homogeneous with respect to a subgroup L of Aut D, if for every two points z, z E D there is an automorphism h E L with h(z) = I. It is also said in such cases that the group L acts transitively on D.
Lemma. If there is a point c E D whose orbit {g(c) : g E L} fills D, then D is homogeneous with respect to L.
Proof. Given z, z E D there exist g, g E L such that g(c) = z and g(c) = z.
Then h := g o g-1 E L satisfies h(z) = z.
Theorem. The unit disc E is homogeneous with respect to the group Aut E.
Proof. . The point 0 has full orbit: for each w E E the function 9O z
1
= wi-1 lies in Aut E (theorem 3) and satisfies g(0) = w. The theorem therefore follows from the lemma.If D is homogeneous with respect to Aut D and if D' is biholomorphi- cally equivalent to D, then D' is homogeneous with respect to Aut D' (the verification of which is entrusted to the reader). From this and the fact that the Cayley mapping hc, effects a biholomorphic equivalence of E with H, it is clear that
The upper half-plane H is homogeneous with respect to the group Aut H.
This also follows directly from the lemma, with c = i. Every w E H has
the form w = + ia2 with
pp,aElfandv0. ForA:= 0
0_1}-1 E SL(2, It), the associated function hA satisfies hA(i) = w./
A region G in C is generally not homogeneous; in fact Aut G most of the time is just {id}. But we won't become acquainted with any examples of this until 10.2.4.
Exercises
Exercise 1. Show that the unbounded regions C and C" are homogeneous with respect to their automorphism groups.
Exercise 2. Let L be a circle in C, a, b two points in C \ L. Show that there is a fractional linear transformation f whose domain includes L U {a} and which satisfies f (a) = b, f (L) = L.
Exercise 3 (cf. also 9.2.3).
Let g(z) := i7z i, q E 8E, w E E be an
automorphism of E. Show that if g is not the identity map, then g fixes at most one point in E.
Chapter 3
Modes of Convergence in Function Theory
Die Annaherung an eine Grenze durch Operationen, die nach bestimmten Gesetzen ohne Ende fortgesetzt werden - dies ist der eigentliche Boden, auf welchem die transscendenten Functionen erzeugt werden. (The approach to a limit via operations which proceed ac- cording to definite laws but without termination - this is the real ground on which the transcendental functions are generated.) - GAUSS 1812.
1. Outside of the polynomials and rational functions, which arise from ap- plying the four basic species of calculation finitely often, there really aren't any other interesting holomorphic functions available at first. Further func- tions have to be generated by (possibly multiple) limit processes; thus, for example, the exponential function exp z is the limit of its Taylor polyno- mials Eo z"/v! or, as well, the limit of the Euler sequence (1 + z/n)".
The technique of getting new functions via limit processes was described by GAUSS as follows (Werke 3, p.198): "Die transscendenten Functionen haben ihre wahre Quelle allemal, offen liegend oder versteckt, im Un- endlichen. Die Operationen des Integrirens, der Summationen unendlicher Reihen .. . oder iiberhaupt die Annaherung an eine Grenze durch Oper- ationen, die nach bestimmten Gesetzen ohne Ende fortgesetzt werden - dies ist der eigentliche Boden, auf welchem die transscendenten Functio- nen erzeugt werden ." (The transcendental functions all have their true source, overtly or covertly, in the infinite. The operation of integration, the summation of infinite series or generally the approach to a limit via operations which proceed according to definite laws but without termina- tion - this is the real ground on which the transcendental functions are generated ... )
The point of departure for all limit processes on functions is the concept
91
of pointwise convergence, which is as old as the infinitesimal calculus itself.
If X is any non-empty set and fn a sequence of complex-valued functions fn : X -+ C, then this sequence is said to be convergent at the point a E X if the sequence fn(a) of complex numbers converges in C. The sequence fn is called pointwise convergent in a subset A C X if it converges at every point of A: then via
f (x) := lim fn (X) , x E A , the limit function f : A -+ C
of the sequence in A is defined; we write, somewhat sloppily, f := lim fn.
This concept of (pointwise) convergence is the most naive one in analysis, and is sometimes also referred to as simple convergence.
2. Among real-valued functions simple examples show how pointwise convergent sequences can have bad properties: the continuous functions xn on the interval [0,1] converge pointwise there to a limit function which is discontinuous at the point 1. Such pathologies are eliminated by the in- troduction of the idea of locally uniform convergence. But it is well known that locally uniformly convergent sequences of real-valued functions have quirks too when it comes to differentiation: Limits of differentiable func- tions are not generally differentiable themselves. Thus, e.g., according to the Weierstrass approximation theorem every continuous f : I -+ R from any compact interval I C R is uniformly approximable on I by polynomials, in particular by differentiable functions. A further example of misbehavior is furnished by the functions n-1 sin(n!x), x E R; they converge uniformly on R to 0 but their derivatives (n - 1)! cos(n!x) don't converge at any point of R.
For function theory the concept of pointwise convergence is likewise un- suitable. Here however such compelling examples as those above cannot be adduced: We don't know any simple sequence of holomorphic functions in the unit disc E which is pointwise convergent to a limit function that is not holomorphic. Such sequences can be constructed with the help of Runge's approximation theorem, but we won't encounter that until the sec- ond volume. At that point we will also see, however, why explicit examples are difficult to come by: pointwise convergence of holomorphic functions is necessarily "almost everywhere" locally uniform. In spite of this somewhat pedagogically unsatisfactory situation, one is well advised to emphasize locally uniform convergence from the very beginning. For example, this will allow us later to extend rather effortlessly the various useful theorems from the real domain on the interchange of limit operations and orders of integration. It is nevertheless surprising how readily mathematicians accept this received view; perhaps it's because in our study of the infinites- imal calculus we became fixated so early on the concept of local uniform convergence that we react almost like Pavlov's dogs.
As soon as one knows the Weierstrass convergence theorem, which among other things ensures the unrestricted validity of the relation lim fn = (lim fn)' for local uniform convergence, any residual doubt dissipates: no
undesired limit functions intrude; local uniform convergence is the optimal convergence mode for sequences of holomorphic functions.
3. Besides sequences we also have to consider series of holomorphic func- tions. But in calculating with even locally uniformly convergent series cau- tion has to be exercised: in general such series need not converge absolutely and so cannot, without further justification, be rearranged at will. WEIER- STRASS confronted these difficulties with his majorant criterion. Later a virtue was made of necessity and the series satisfying the majorant crite- rion were formally recognized with the name normally convergent (cf. 3.3).
Normally convergent series are in particular locally uniformly and abso- lutely convergent; every rearrangement of a normally convergent series is itself normally convergent. In 4.1.2 we will see that because of the classical Abel convergence criterion, power series always converge normally inside their discs of convergence. Normal convergence is the optimal convergence mode for series of holomorphic functions.
In this chapter we plan to discuss in some detail the concepts of lo- cally uniform, compact and normal convergence. X will always designate a metric space.
§1 Uniform, locally uniform, and compact convergence
1. Uniform convergence. A sequence of functions fn : X - C is said to be uniformly convergent in A C X to f : A - C if every e > 0 has an no = no(e) E N such that
Ifn(x)-f(x)I <e
for all n > no and all x E A;when this occurs the limit function f is uniquely determined.
A series E f of functions converges uniformly in A if the sequence 8n = E" f of partial sums converges uniformly in A; as with numerical series, the symbol E f is also used to denote the limit function.
Uniform convergence in A implies ordinary convergence. In uniform convergence there is associated with every e > 0 an index no(e) which is independent of the location of the point x in A, while in mere pointwise convergence in A this index generally also depends (perhaps quite strongly) on the individual x.
The theory of uniform convergence becomes especially transparent upon introduction of the supremum semi-norm
If 1A :=sup{If(x)I : x E Al
for subsets A C X and functions f : X -+ C. The set V :_ if : X -+ C : VIA < oo} of all complex-valued functions on X which are bounded on A is a C-vector space; the mapping f i VIA is a "semi-norm" on V; more precisely,
IflA=0pflA=0,
If +9IA <_ VIA + 191A
IefIA = ICIIfIA
f,9EV,CEC.
A sequence fn converges uniformly in A to f exactly when limlfn - fIA=O.
Without any effort at all we prove two important
Limit rules. Let fn, gn be sequences of functions in X which converge uniformly in A. Then
Ll For all a, b E C the sequence a fn + bgn converges uniformly in A and lim(a fn + bgn) = a lim fn + b lim gn (C-linearity).
L2 If the functions lim fn and limgn are both bounded in A, then the product sequence fngn also converges uniformly in A and
lim(fngn) = (lim fn)(limgn)
Of course there is a corresponding version of L1 for series E f,,, E g,,.
To see that the supplemental hypothesis in L2 is necessary, look at X :_
{x E I t : 0 < x < 1}
, fn(x) = gn(x) := x + n, f (x) = g(x) := s. Then
fn-fIX = I9n-91X = n, yet Ifngn-fgl X > fngn(n )-fg(n ) =2n+n
.2. Locally uniform convergence. The sequence of powers zn converges in every disc Br(0),r < 1, uniformly to the function 0, because IznIB,.(o) = rn. Nevertheless the convergence is not uniform in the unit disc E: for every 0 < e < 1 and every n > 1 there is a point c E E, for example, c:= V e-, with Icnl > e. This kind of convergence behavior is symptomatic of many sequences and series of functions. It is one of WEIERSTRASS'S significant contributions to have clearly recognized and high-lighted this convergence phenomenon: uniform convergence on the whole space X is usually not an issue; what's important is only that uniform convergence prevail "in the small".
A sequence of functions fn : X - C is called locally uniformly convergent in X if every point x E X lies in a neighborhood Ux in which the sequence fn converges uniformly.
A series E f is called locally uniformly convergent in X when its asso- ciated sequence of partial sums is locally uniformly convergent in X.