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Home , Meromorphic function, Rational function

Math 207 - Spring ’17 - Fran¸coisMonard 1

Lecture 8 - The extended Cˆ, rational functions, M¨obius transformations

Material: [G]. [SS, Ch.3 Sec. 3]

1 The purpose of this lecture is to “compactify” C by adjoining to it a point at infinity , and to extend to concept of analyticity there. Let us first define: a neighborhood of infinity U is the complement of a closed, bounded set. A “basis of neighborhoods” is given by complements of closed disks of the form

Uz0,ρ = C − Dρ(z0) = {|z − z0| > ρ}, z0 ∈ C, ρ > 0.

Definition 1. For U a nbhd of ∞, the f : U → C has a limit at infinity iff there exists L ∈ C such that for every ε > 0, there exists R > 0 such that for any |z| > R, we have |f(z)−L| < ε. 1  We write limz→∞ f(z) = L. Equivalently, limz→∞ f(z) = L if and only if limz→0 f z = L.

With this concept, the algebraic limit rules hold in the same way that they hold at finite points when limits are finite.

1 Example 1. • limz→∞ z = 0.

z2+1 1 • limz→∞ (z−1)(3z+7) = 3 .

z 1 • limz→∞ e does not exist (this is because e z has an at z = 0). A way 1 0 1 to prove this is that both sequences zn = 2nπi and zn = 2πi(n+1/2) converge to zero, while the 1 1 0 sequences e zn and e zn converge to different limits, 1 and 0 respectively.

Definition 2. limz→z0 f(z) = ∞ if for every M > 0, there exists ρ > 0 such that |z − z0| < ρ 1 implies |f(z)| > M. Equivalently, limz→z0 f(z) = ∞ if and only if limz→z0 f(z) = 0.

Combining both definition above, we can say that limz→∞ f(z) = ∞ iff for every M > 0, there exists R > 0 such that |z| > R implies |f(z)| > M.

g(z) Example 2. • If f has a pole of order k > 0 at z0, f may be written as f(z) = k with (z−z0) k 1 (z−z0) 1 g(z0) 6= 0 and g analytic near z0. Then f(z) = g(z) so limz→z0 f(z) = 0, i.e., limz→z0 f(z) = ∞.

• For f any nonconstant , limz→∞ f(z) = ∞.

Definition 3. The extended complex plane (a.k.a. ) is the topological space Cˆ = C ∪ {∞}, where open sets are generated either by open disks Dρ(z0) or neighborhoods of infinity.

∞ is now an abstract point which is no different than the other ones. We may study the 1 behavior of a function f : C → C at ∞ by studying the function f( z ) near z = 0. The question now is: when can such a function be extended as an analytic function f˜ : Cˆ → Cˆ ?

1 Adjoining one point to C is not the only way to compactify it, this is why this process is called one-point compactification. Math 207 - Spring ’17 - Fran¸coisMonard 2

1 ˆ Definition 4. f : C → C is analytic at ∞ iff the function w 7→ f( w ) has a limit in C at w = 0 and

1 (i) if this limit is finite, f( w ) is analytic at w = 0, 1 (ii) if this limit is ∞, 1 is analytic at w = 0. f( w ) 1−z Example 3. Show that f(z) = 1+z is analytic at ∞ upon extending it with the value −1 there. 1 Definition 5. f : C → C has a pole of order n at ∞ if f( w ) has a pole of order n at w = 0 (we allow n = 0, which covers the case of a finite limit).

ˆ Another way of saying that f has a pole at ∞ is to say that limz→∞ f(z) exists in C (i.e. either as a finite number, or as ∞).

Theorem 1. f can be extended into an analytic function f˜ : Cˆ → Cˆ if and only if f is meromorphic on C and limz→∞ f(z) exists, either as a or ∞.

This extension is done as follows : when f is a on C such that limz→∞ f(z) exists in Cˆ, we define f˜ : Cˆ → Cˆ:

• At z ∈ C where f is analytic, define f˜(z) = f(z).

• At z ∈ C where f has a pole, define f˜(z) = ∞.

• At ∞, set f˜(∞) = limz→∞ f(z).

0 Analytic functions on Cˆ define conformal maps near every point where f 6= 0. ˆ ˆ 1 Theorem 2. The map s : C → C defined by s(z) = z with s(0) = ∞ and s(∞) = 0 is a conformal equivalence of Cˆ onto itself.

Proof. Check that s is one to one and onto: this is because s ◦ s = Id. s is analytic at every 1 1 z 6= 0, ∞. At z = ∞, s( w ) = w is analytic at w = 0 so s is analytic at z = ∞. At z = 0, s(z) = z is analytic at z = 0 so s is analytic at z = 0.

Theorem 3. f : Cˆ → Cˆ is analytic (or meromorphic) if and only if f is a .

Proof. We only prove ( =⇒ ), the converse is left to the reader. The first claim is that f has finitely ˆ many poles z1, . . . , zn on C (denote (d1, . . . , dn) their multiplicities). Indeed, since f has a limit in C 1 ˙ as z → ∞, then there exists r > 0 such that w 7→ f( w ) has not poles on Dr(0) (you may split cases 1 according to whether ∞ is a pole of f or not). In particular, f has no poles on {|z| > r , z ∈ C}. Then since D 1 (0) is compact, f has finitely many poles there, so the first claim is proved. We r d dn then define g(z) := (z − z1) 1 ··· (z − zn) f(z), and we now claim that g is a polynomial. g has removable singularities at z1, . . . , zn and is analytic elsewhere, so it’s an entire function. Moreover 1 d since f has a pole at ∞ (of possible order 0), there exists d ≥ 0 such that f( w )w is bounded near zero, or equivalently, f(z)z−d is bounded near ∞. This implies that g(z)z−(d+d1+···+dn) is bounded near ∞ and from a previous exercise, since g is entire, this implies that g is a polynomial of degree at most d + d + ··· + d . As a conclusion, f(z) = g(z) is a rational function. 1 n Qn dj j=1(z−zj ) Math 207 - Spring ’17 - Fran¸coisMonard 3

2 2 2 3 The Riemann sphere Consider the sphere in S = {x1 + x2 + (x3 − 1/2) = 1/4} ⊂ R centered at (0, 0, 1/2), with North Pole N = (0, 0, 1).

S is sitting on the plane {x3 = 0} ≡ C, and we can define the stereographic projection map, for M ∈ S − {N} to be the unique intersection point between the line NM and the plane {x3 = 0}. This map is a homeomorphism: it is bijective, continuous with continuous inverse ρ−1. As the neighborhoods of ∞ we defined earlier are mapped via ρ−1 to neighborhoods of the North Pole N, we may extend ρ : S − {N} → C into a homeomorphismρ ˜ : S → Cˆ by declaringρ ˜(N) = ∞ and ρ˜(M) = ρ(M) for any M ∈ S − {N}. N

M

0 ρ(M)

{x3 = 0}

Figure 1: Stereographic projection

In that sense, we can assimilate Cˆ as a two-dimensional sphere. We will get other (visual) opportunities to see why in the future. An important property of Cˆ is that it is now a compact topological space, unlike C. In particular, every sequence in Cˆ has at least one accumulation point. This will imply that discrete subsets of Cˆ can only be finite, with consequences on analytic functions Cˆ → Cˆ (e.g. the “identity” theorem), namely:

Theorem 4 (Identity theorem on Cˆ). If f, g : Cˆ → Cˆ are analytic and agree on an infinite set, then they are identically equal.

This implies in particular that such non-trivial functions can only have finitely many .

Linear Fractional (a.k.a. M¨obius)transformations

Definition 6. A Linear Fractional Transformation is a transformation of the form h(z) = az+b cz+d , with a, b, c, d ∈ C such that ad − bc 6= 0.

a b  Each non-singular 2×2 complex matrix A = c d determines a linear fractional transformation az+b −1 φA(z) = cz+d . A straighforward calculation shows that φA ◦ φB = φAB and that φA = φA−1 , −1 1 d −b  where AB is matrix multiplication and A = ad−bc −c a . In other words, the mapping A 7→ φA is a “group homomorphism”. Most importantly, this property makes it easier to compute the composition of two LFT’s on the fly, by computing the product of the corresponding matrices. Math 207 - Spring ’17 - Fran¸coisMonard 4

Example 4. • Affine transformations L(z) = az + b (c = 0, d = 1) requires a 6= 0.

1 • s(z) = z , i.e. (a, b, c, d) = (0, 1, 1, 0).

Note that for every λ 6= 0, the transformations ha,b,c,d and hλa,λb,λc,λd define the same trans- formation. One could add the normalizing condition ad − bc = 1, though it is sometimes more readable to not normalize.

Let us first notice that every LFT can be written as either (i) h = L1 for L1 some affine transformation, or (ii) as h = L1 ◦ s ◦ L2 for L1,L2 two affine transformations. Indeed, if c = 0, then (i) is satisfied, and if c 6= 0, we write

az + b a (cz + d) + b − ad a bc − ad 1 h(z) = = c c = + , cz + d cz + d c c cz + d

bc−ad a so that, upon defining L1(z) = c z + c and L2(z) = cz + d, then h = L1 ◦ s ◦ L2.

Definition 7. For U ⊂ Cˆ an open set, h is a conformal automorphism of U (i.e. h ∈ Aut (U)) iff h : U → U is conformal and bijective.

Proposition 5. For U ⊂ Cˆ an open set, Aut (U) is a group under composition.

Proof. The composition of conformal, bijective maps is clearly conformal and bijective. Compo- sition is always associative ((f ◦ g) ◦ h = f ◦ (g ◦ h)), the identity element is Id : U → U and if f ∈ Aut (U), then so it f −1.

Theorem 6 (Characterization of Aut (Cˆ)). h ∈ Aut (Cˆ) if and only if h is a linear fractional transformation.

Proof. ( ⇐= ) Suppose h is an LFT. h is bijective: indeed, if h is an affine map L(z) = az + b, then −1 1 it is bijective with inverse L (w) = a (w − b); if h is of the form L1 ◦ s ◦ L2, i.e. a coumpound of −1 −1 −1 three invertible transformations, then h is bijective with inverse h = L2 ◦ s ◦ L1 . Moreover, 0 h is conformal: if h(z) = L(z) = az + b, then h (z) = a 6= 0 for every z ∈ C so h is conformal; if h = L2 ◦ s ◦ L1, h is conformal as a composition of conformal functions. ( =⇒ ) Suppose f ∈ Aut (Cˆ) (so f is one to one, onto and conformal). 0 First case: if f(∞) = ∞, then q(z) = s ◦ f ◦ s ∈ Aut (Cˆ) is such that q(0) = 0. But q (0) 6= 0 since q is conformal, so f has a pole of order 1 at ∞. Since f is one to one on Cˆ, ∞ has no other −1 preimage by f, so f has no pole on C, i.e. the function z (f(z) − f(0)) is analytic and finite everywhere on Cˆ: by Liouville’s theorem, it is constant on C, then there exists c ∈ C such that z−1(f(z) − f(0)) = c, i.e. f(z) = cz + f(0). 1 ˆ Second case: If f(∞) = k 6= ∞. Let p(z) = z−k an LFT so p ∈ Aut (C). By the group property, ˆ 1 p ◦ f ∈ Aut (C) and p ◦ f(z) = f(z)−k is such that p ◦ f(∞) = ∞. By the first case, there exists a, b 1 1 such that az + b = p ◦ f(z) = f(z)−k so that f(z) = az+b + k is an LFT.

As Aut (Cˆ) can be viewed as the “conformal motions” on the sphere, one may view the action of an LFT on the Riemann sphere dynamically via stereographic projection. See [AR] for a visual example of this. Math 207 - Spring ’17 - Fran¸coisMonard 5

References

[AR] M¨obiustransformations revealed, by D. Arnold and J. Rogness, Youtube video

[G] , Theodore W. Gamelin. Undergraduate Texts in , Springer.4

[SS] Complex Analysis, Elias M. Stein and Rami Shakarchi, Princeton Lectures in Analysis II.1

[T] Complex Variables, Joseph L. Taylor. Pure and Applied Undergraduate Texts Vol. 16, 2011.

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