1750 CHAPTER 55. COMPLEX MAPPINGS

It follows f ◦ h−1 can be analytically continued along every curve through 0. By Lemma55.4.4 there exists g analytic on B(0,1) which agrees with f ◦ h−1 on B(0,δ ). DefineG(z) ≡ g(h(z)) . For z = h−1 (w) , it follows G

(h−1 (w)

)= g(w) . If w ∈ B(0,δ ) , then

G(h−1 (w)

)= f ◦h−1 (w) and so G = f on h−1 (B(0,δ )) , an open set contained in B(a,r).

Therefore, G = f on B(a,r) because h−1 (B(0,δ )) has a limit point. This proves the theo-rem.

Actually, you sometimes want to consider the case where Ω =C. This requires a smallmodification to obtain from the above theorem.

Corollary 55.4.6 Suppose ( f ,B(a,r)) is a function element with B(a,r) ⊆ C. Supposealso that this function element can be analytically continued along every curve through a.Then there exists G analytic on C such that G agrees with f on B(a,r).

Proof: Let Ω1 ≡ {z ∈ C : a+ it : t > a} and Ω2 ≡ {z ∈ C : a− it : t > a} . Here is apicture of Ω1.

Ω1

a

A picture of Ω2 is similar except the line extends down from the boundary of B(a,r).Thus B(a,r) ⊆ Ωi and Ωi is simply connected and proper. By Theorem 55.4.5 there

exist analytic functions, Gi analytic on Ωi such that Gi = f on B(a,r). Thus G1 = G2 onB(a,r) , a set with a limit point. Therefore, G1 = G2 on Ω1 ∩Ω2. Now let G(z) = Gi (z)where z ∈Ωi. This is well defined and analytic on C. This proves the corollary.

55.5 The Picard TheoremsThe Picard theorem says that if f is an entire function and there are two complex numbersnot contained in f (C) , then f is constant. This is certainly one of the most amazing thingswhich could be imagined. However, this is only the little Picard theorem. The big Picardtheorem is even more incredible. This one asserts that to be non constant the entire functionmust take every value of C but two infinitely many times! I will begin with the little Picardtheorem. The method of proof I will use is the one found in Saks and Zygmund [115],Conway [32] and Hille [65]. This is not the way Picard did it in 1879. That approach isvery different and is presented at the end of the material on elliptic functions. This approachis much more recent dating it appears from around 1924.

Lemma 55.5.1 Let f be analytic on a region containing B(0,r) and suppose∣∣ f ′ (0)∣∣= b > 0, f (0) = 0,

and | f (z)| ≤M for all z ∈ B(0,r). Then f (B(0,r))⊇ B(

0, r2b2

6M

).