Geometry of complex functions
1. The open mapping theorem says that if f : D → C is holomorphic and nonconstant, and if D is open in C, then f (D) is open, i.e. Given ε > 0 and b ∈ f (D), ∃ δ > 0, and a ∈ D, such that b = f (a),
D(a; δ) = {z ∈ D | |z − a| ≤ δ} ⊂ f¯ −1(D(b; ε)) =⇒ f ( ¯D(a; δ)) ⊂ D(b; ε), and
∀ w ∈ D(b, ε), ∃ z ∈ D(a; δ) ⊂ D such that w = f (z) ∈ f (D) =⇒ D(b; ε) ⊂ f (D).
This proves that f (D) is an open subset of C.
2. Schwarz Lemma says that If f : D → D is holomorphic from the unit disk D to D (extending continuously to the boundary) and f (0) = 0, then
(|f (z)| ≤ |z| ∀ z ∈ D
|f0(0)| ≤ 1 with equality in either if and only if f (z) = eiθz.
3. If f : D → D is holomorphic (extending continuously to the boundary) and f (a) = 0 for some |a| < 1, then
|f (z)| ≤ |Ba(z)| ∀ z ∈ D
|f0(a)| ≤ |Ba0(a)| = 1 1 − |a|2 with equality in either if and only if f (z) = eiθBa(z).
4. Morera’s Theorem says that if f is continuous on a region D and Z
Γ
f (z)dz = 0 whenever Γ is the boundary of a closed triangle ¯T in D, then f is holomorphic in D.
5. A Corollary to the Morera’s Theorem: suppose {fn} is holomorphic in an open set D, and fn → f uniformly on comapct subsets of D. Then f = lim
n→∞fn is holomorphic in D.
Singularities of functions
1. Remember the definition of isolated singularity, removable singularity, pole of order k, essential singularity.
2. Suppose f (z) is holomorphic in a deleted neighborhood D of α and if
z→αlimf (z)(z − α) = 0, then f (z) has a removable singularity at α.
3. Suppose f (z) is holomorphic in a deleted neighborhood D of α and ∃ n ∈ N such that
z→αlimf (z)(z − α)n = 0.
If k + 1 is the least such n, then f (z) has a pole of order k at α.
4. Casorati-Weierstrass Theorem says that if f has an essential singularity at α and if D is a deleted neighborhood, then f (D) = {f (z) | z ∈ D} is dense in C, i.e. for any ε > 0 and for any w ∈ C,
Bε(w) ∩ f (D) 6= ∅.
5. Remember the definition of Laurent series of a function about an isolated singularity, and the principal part and the analytic part of the Laurent series.
6. f is meromorphic on D if f (z) is holomorphic on D except at isolated singularities at which f has poles.
7. If f (z) =
∞
X
k=−∞
ck(z − α)k is holomorphic function in a deleted neighborhood of α, then the residue of f (z) at the point α, is defined as
Res (f ; α) = c−1.
8. If f (z) = A(z)
B(z) for A a nonzero function, and B having a simple zero, then Res (f ; α) = A(α)
B0(α).
9. Remember the definition of the winding number of γ about α. If γ : [0, 1] → C is a closed contour curve with γ(t) 6= α for all t ∈ [0, 1]. Then the winding number of γ about α, denoted n(γ, α), is defined by
n(γ, α) := 1 2πi
Z
γ
1
z − αdz = 1 2πi
Z 1 0
γ0(t) γ(t) − αdt.
10. n(γ, α) ∈ Z.
11. γ is called a regular closed curve if γ is a simple closed curve with n(γ, α) = 0 or 1 for all α /∈ γ.
12. Cauchy’s Residue Theorem says that if f is analytic in a simply connected domain D except for isolated singularities at α1, α2, . . . , αm, and if γ is a closed curve not intersecting any of the singularities, then
Z
γ
f = 2πi
m
X
k=1
n(γ, αk) Res (f, αk).
Applications of Cauchy’s Residue Theorem:
• If f is meromorphic inside and on a regular closed curve γ which contains no zeroes or poles of f, and if
Z = number of zeroes of f inside γ (a zero of order k being counted k times), P = number of poles of f inside γ (again with multiplicity),
then 1
2πi Z
γ
f0
f = Z − P.
• (Argument Principle) If f is analytic inside and on a regular closed curve γ (and is nonzero on γ) then
Z(f ) = number of zeroes of f inside γ
= 1
2πi Z
γ
f0 f
= winding number of f (γ) about 0
= n(f (γ); 0).
• (Rouch´e’s Theorem) Suppose that f and g are analytic inside and on a regular closed curve γ and that
|f (z)| > |g(z)| for all z ∈ γ.
Then
Z(f + g) = Z(f ) inside γ.
• (Generalized Cauchy’s Theorem) Suppose that f is analytic in a simply connected domain D and that γ is a regular closed curve contained in D. Then for each z inside γ and k = 0, 1, 2, . . . ,
(∗) f(k)(z) = k!
2πi Z
γ
f (w)
(w − z)k+1 dw.
• (Hurwitz’s Theorem) Let {fn} be a sequence of non-vanshing analytic functions in a region D and suppose fn→ f uniformly on compacta of D. Then either
f ≡ 0 in D or f (z) 6= 0 for all z ∈ D.
• Evaluation of integrals and sums using the Residue Theorem.
Conformal functions
1. Suppose f is defined in a neighborhood of z0. f is said to be conformal at z0 if f pre- serves angles there. That is, for each pair of smooth curves C1 and C2 intersecting at z0,
∠(C1, C2) = ∠(f (C1), f (C2)). Similarly, we say f is conformal in a region D if f is conformal at all points z ∈ D.
2. Remember the definition of a function being locally 1-1 at a point and 1-1 in a region.
3. Note that if f is analytic at z0 and f0(z0) 6= 0, then f is conformal and locally 1-1 at z0. 4. Suppose f is a 1-1 analytic function in a region D. Then
• f−1 : f (D) → D exists and is analytic in f (D),
• f and f−1 are conformal in D and f (D), respectively.
f is called a biholomorphic map from D to f (D) if f : D → f (D) is holomorphic and has a holomorphic inverse f−1 : f (D) → D.
5. Remember the following definitions.
• A 1-1 analytic mapping is called a conformal mapping.
• Two regions D1 and D2 are said to be conformally equivalent if there exists a conformal mapping of D1 onto D2.
• A conformal mapping of a region onto itself is called an automorphism of that region.
6. (Lemma) Suppose f : D1 → D2 is a conformal mapping. Then
• any other conformal mapping h : D1 → D2 is of the form g ◦ f, for some automorphism g of D2;
• any automorphism h of D1 is of the form f−1 ◦ g ◦ f, where g is an automorphism of D2.
If D1 and D2 are conformally equivalent, then there is an isomorphisim ϕ : Aut (D1) → Aut (D2) defined by
ϕ(h) = f ◦ h ◦ f−1 ∀ h ∈ Aut (D1) with ϕ−1(g) = f−1◦ g ◦ f ∀ g ∈ Aut (D2).
M¨obius transformations
1. Remember the denition of M¨obius transformation.
2. Know that there exists a unique M¨obius transformation f (z) sending sending z1, z2, z3 into
∞, 0, 1, respectively, and f (z) is given by
f (z) = (z − z2)(z3− z1) (z − z1)(z3− z2).
3. Remember the cross-ratio of the four complex numbers z1, z2, z3, z4, denoted (z1, z2, z3, z4), is the image of z4 under the bilinear map T which maps z1, z2, z3 into ∞, 0, 1, respectively, i.e.
(z1, z2, z3, z4) = T (z4) = (z4 − z2)(z3− z1) (z4 − z1)(z3− z2).
4. Know that there exists a unique M¨obius transformation w = f (z) mapping z1, z2, z3 into w1, w2, w3 respectively, is given by
(w − w2)(w3 − w1)
(w − w1)(w3 − w2) = (z − z2)(z3− z1) (z − z1)(z3− z2). 5. Applications of Schwarz Lemma:
• If f : D → D, f(0) = 0, and f is an automorphism of the disk (there exists a holomorphic inverse f−1 : D → D), then f (z) = eiθz for some θ ∈ R.
• The automorphisms of D are of the form
g(z) = eiθBα(z) = eiθ z − α 1 − ¯αz
, |α| < 1, i.e. the M¨obius transformations taking D to itself.
• If
H = {f : D → D | f is holomorphic (extending continuously to ∂D) with f (0) = α, f0(0) > 0}, then ∃! g ∈ H such that |g0(0)| = max
f ∈H |f0(0)| and g is bijective, i.e. the max
f ∈H |f0(0)| is achieved by an automorphism of D.
6. The conformal mappings h of H onto D of the form h(z) = eiθ z − α
z − ¯α
, Im α > 0.
7. The automorphisms of H are of the form f (z) = az + b
cz + d, a, b, c, d ∈ R and ad − bc > 0, i.e. the M¨obius transformations taking H to itself.
8. Every injective holomorphic function f : C → C is an automorphismof C and can be written in the form
z 7→ az + b for some a ∈ C×; b ∈ C.
9. C is not conformally equivalent to any proper subset U ( C.
10. (Riemann Mapping Theorem) Let U ( C be open and simply connected proper subset of C, and let α ∈ U. Then ∃! conformal mapping (a bijective analytic mapping) f : U → D such that f (α) = 0 and f0(α) > 0 is a positive real number, where D denotes the open unit disk.
