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Complex Analysis Qual Sheet
Robert Won
“Tricks and traps. Basically all complex analysis qualifying exams are collections of tricks and traps.”
1 Useful facts
- ez^ =
∑^ ∞
n=
zn n!
- sin z =
∑^ ∞
n=
(−1)n^
z^2 n+ (2n + 1)!
2 i (eiz^ − e−iz^ )
- cos z =
∑^ ∞
n=
(−1)n^
z^2 n 2 n!
(eiz^ + e−iz^ )
- If g is a branch of f −^1 on G, then for a ∈ G, g′(a) =
f ′(g(a))
- |z ± a|^2 = |z|^2 ± 2Reaz + |a|^2
- If f has a pole of order m at z = a and g(z) = (z − a)mf (z), then
Res(f ; a) =
(m − 1)! g(m−1)(a).
- The elementary factors are defined as
Ep(z) = (1 − z) exp
z +
z^2 2
zp p
Note that elementary factors are entire and Ep(z/a) has a simple zero at z = a.
- The factorization of sin is given by
sin πz = πz
∏^ ∞
n=
z^2 n^2
- If f (z) = (z − a)mg(z) where g(a) 6 = 0, then f ′(z) f (z)
m z − a
g′(z) g(z)
2 Tricks
- If f (z) nonzero, try dividing by f (z). Otherwise, if the region is simply connected, try writing f (z) = eg(z).
- Remember that |ez^ | = eRez^ and argez^ = Imz. If you see a Rez anywhere, try manipulating to get ez^.
- On a similar note, for a branch of the log, log reiθ^ = log |r| + iθ.
- Let z = eiθ.
- To show something is analytic use Morera or find a primitive.
- If f and g agree on a set that contains a limit point, subtract them to show they’re equal.
- Tait: “Expand by power series.”
- If you want to count zeros, either Argument Principle or Rouch´e.
- Know these M¨obius transformations:
(a) To map the right half-plane to the unit disk (or back), 1 − z 1 + z
(b) To map from the unit disk to the unit disk, remember ϕa(z) = z − a 1 − az
. This is a bijective
map with inverse ϕ−a(z). Also, ϕa(a) = 0, ϕ′ a(z) = 1 − |a|^2 (1 − az)^2 , ϕ′ a(0) = 1 − |a|^2 , and
ϕ′ a(a) =
1 − |a|^2
- If f (z) is analytic, then f (z) is analytic (by Cauchy-Riemann). So if, for example, f (z) is real on the real axis, then f (z) = f (z).
- To prove that a function defined by an integral is analytic, try Morera and reversing the integral. (e.g.
e −ttz− (^1) dt is analytic since ∫ T
e −ttz− (^1) dtdz = ∫^ ∞
T e −ttz− (^1) dzdt = 0.)
- If given a point of∫ f (say f (0) = a) and some condition on f ′^ on a simply connected set, try [0,z] f^
′ (^) = f (z) − f (0).
- To create a non-vanishing function, consider exponentiating.
3 Theorems
- Cauchy Integral Formula: Let G be region and f : G → C be analytic. If γ 1 ,... , γm are closed rectifiable curves in G with
∑m k=0 n(γk;^ w) = 0 for all^ w^ ∈^ C^ ^ G, then for^ a^ ∈ G \ (∪mk=1{γk}),
f (n)(a) ·
∑^ m
k=
n(γk; a) =
n! 2 πi
∑^ m
k=
γk
f (z) (z − a)n+^ dz.
- Winding Number: To compute the index of a closed curve about a point a,
n(γ; a) =
2 πi
γ
dz z − a
∈ Z.
- Open Mapping Theorem: Let G be a region, f a non-constant analytic function. If U is an open subset of G, then f (U ) is open.
- Zero-Counting Theorem: Let G be a region, f : G → C analytic with roots a 1 ,... am. If {γ} ⊆ G and ak 6 ∈ {γ} for all k, and γ ≈ 0 in G, then
1 2 πi
γ
f ′(z) f (z) dz =
∑^ m
k=
n(γ; ak)
Corollary: If f (a) = α, then f (z) − α has a root at a. So if f (ak) = α, then
1 2 πi
γ
f ′(z) f (z) − α
dz =
∑^ m
k=
n(γ; ak)
Corollary 2: If σ = f ◦ γ and α 6 ∈ {σ} and ak are the points where f (ak) = α, then
n(σ; α) =
∑^ m
k=
n(γ; ak) or
n(f ◦ γ; f (a)) =
∑^ m
k=
n(γ; ak)
- Roots of analytic functions: Suppose f is analytic on B(a; R) and let f (a) = α. If f (z)−α has a zero of order m at z = a, then there exist > 0 and δ > 0 such that if 0 < |ζ − α| < δ, the equation f (z) = ζ has exactly m simple roots in B(a, ).
- Existence of Logarithm: Let f (z) be analytic and f (z) 6 = 0 on G, a simply connected region. Then there is analytic function g(z) on G such that f (z) = eg(z)^ for all z ∈ G.
- Existence of Primitive: Let f (z) be analytic on G, a simply connected region. Then f has a primitive.
- Laurent Series: Let f be analytic on R 1 < |z − a| < R 2 , then there exists a sequence {an}∞ n=−∞ and
f (z) =
∑^ ∞
n=−∞
an(z − a)n
with absolute convergence in the open annulus and uniform convergence on every compact subset of the annulus. This series is called a Laurent series, and if γ is a closed curve in the annulus, then an =
2 πi
γ
f (z) (z − a)n+^
dw.
(Note that this is just the same as number 11).
- Classification of Singularities: Suppose f analytic on B(a; R) \ {a} and f has an isolated singularity at a. Then a is
(a) Removable singularity if there is a function g analytic on B(a; R) such that f (z) = g(z) for all z ∈ B(a; R) \ {a}. The singularity is removable if and only if lim z→a (z − a)f (z) = 0. Also, the singularity is removable if and only if the Laurent series of f has no coefficients an for n < 0. (b) Pole if lim z→a |f (z)| = ∞. If a is a pole, then there is a unique m ≥ 1 and an analytic function g such that f (z) = g(z) (z − a)m^
for all z ∈ B(a; R) \ {a} and g(a) 6 = 0. The singularity is a pole if and only if the Laurent series of f has only finitely many coefficients an for n < 0. The partial series for these coefficients is called the singular part of f. (c) Essential singularity if a is not removable and not a pole. The singularity is essential if and only if the Laurent series of f has infinitely many coefficients an for n < 0.
- Casorati-Weierstrass: If f has an essential singularity at a, then for all δ > 0, f ({z | 0 < |z − a| < δ}) is dense in C.
- Residues: If f has an isolated singularity at a, then the residue of f at a, Res(f ; a) = a− 1. We can calculate the residue using the formula for Laurent coefficients:
Res(f ; a) =
2 πi
γ
f (z)dz.
If a is a pole of order m, then if g(z) = (z − a)mf (z)
Res(f ; a) = g(m−1)(a) (m − 1)!
- Residue Theorem: Let f be analytic on a region G except for singularities at a 1 ,... , am. Let γ ≈ 0 be a closed curve in G with a 1 ,... , am ∈ {/ γ}. Then
1 2 πi
γ
f (z)dz =
∑^ m
k=
n(γ; ak) · Res(f ; ak).
- Argument Principle: Let f be meromorphic with roots z 1 ,... , zm and poles p 1 ,... , pn with z 1 ,... , zm, p 1 ,... , pn ∈ {/ γ}. Then
1 2 πi
γ
f ′ f
∑^ m
k=
n(γ; zm) −
∑^ n
j=
n(γ; pn).
(i) |f ′(a)| ≤ 1 − |f (a)|^2 1 − |a|^2
(ii) if equality, then f (z) = ϕ−a(cϕa(z)).
- Logarithmic Convexity: Let a < b, G = {z ∈ C | a < Re z < b}, and f : G → C. If f is continuous on G, analytic on G and bounded, then M (x) = sup y∈R
|f (x + iy)| is logarithmically convex.
- Phragm`en-Lindel¨of : Let G be simply connected, f : G → C analytic, and suppose there exists ϕ : G → C analytic, bounded, and nonzero on G. Suppose further that ∂∞G = A ∪ B and
(i) for all a ∈ A, lim sup z→a
|f (z)| ≤ M
(ii) for all b ∈ B, for all η > 0, lim sup z→b
|f (z)||ϕ(a)|η^ ≤ M ,
then |f (z)| ≤ M on G.
- Logic of the ρ metric: For all > 0, there exist δ > 0 and K ⊆ G compact such that
ρK (f, g) < δ =⇒ ρ(f, g) <
and for all δ > 0, K compact, there exists an such that
ρ(f, g) < =⇒ ρK (f, g) < δ
- Spaces of Continuous Functions: If Ω is complete, then C(G, Ω) is complete.
- Normal Families: F ⊆ C(G, Ω). F is normal if all sequences have a convergent subse- quence. F is normal iff F is compact iff F is totally bounded (i.e. for all K, δ > 0, there exist f 1 ,... , fn ∈ F such that F ⊆
⋃n i=1{g^ ∈^ C(G,^ Ω)^ |^ ρK^ (f^ ;^ g)^ < δ}.
- Arzela-Ascoli: F is normal iff
(i) for all z ∈ G, {f (z) | f ∈ F } has compact closure in Ω, and (ii) for all z ∈ G, F is equicontinuous at z (for all > 0, there exists δ > 0 such that |z − w| < δ ⇒ d(f (z), f (w)) < for all f ∈ F ).
- The Space of Holomorphic Functions: Some useful facts:
(a) fn → f ⇐⇒ for all compact K ⊆ G, fn → f uniformly on K. (b) {fn} in H(G), f ∈ C(G, C), then fn → f =⇒ f ∈ H(G) (If fn converges, it will converge to an analytic function). (c) fn → f in H(G) =⇒ f (^) n(k )→ f (k)^ (If f converges, its derivatives converge). (d) H(G) is complete (Since H(G) is closed and C(G, C) is complete).
- Hurwitz’s Theorem: Let {fn} ∈ H(G), fn → f , f 6 ≡ 0. Let B(a; r) ⊆ G such that f 6 = 0 on |z − a| = r. Then there exists an N such that n ≥ N =⇒ fn and f have the same number of zeros in B(a; r).
Corollary: If fn → f and fn 6 = 0, then either f (z) ≡ 0 or f (z) 6 = 0.
- Local Boundedness: A set F in H(G) is locally bounded iff for each compact set K ⊂ G there is a constant M such that |f (z)| ≤ M for all f ∈ F and z ∈ K. (Also, F is locally bounded if for each point in G, there is a disk on which F is uniformly bounded.)
- Montel’s Theorem: F ⊆ H(G), then F is normal ⇐⇒ F is locally bounded (for all K compact, there exists M such that f ∈ F ⇒ |f (z)| ≤ M for all z ∈ K).
Corollary: F is compact iff F is closed and locally bounded.
- Meromorphic/Holomorphic Functions: If {fn} in M (G) (or H(G)) and fn → f in C(G, C∞), then either f ∈ M (G) (or H(G)) or f ≡ ∞.
- Riemann Mapping Theorem: G simply connected region which is not C. Let a ∈ G, then there is a unique analytic function such that:
(a) f (a) = 0 and f ′(a) > 0; (b) f is one-to-one; (c) f (G) = D.
- Infinite Products: Some propositions for convergence of infinite products:
(a) Re zn > 0. Then
zn converges to a nonzero number iff
log zn converges. (b) Re zn > −1. Then
log(1 + zn) converges absolutely iff
zn converges absolutely. (c) Re zn > 0. Then
zn converges absolutely iff
(zn − 1) converges absolutely.
- Products Defining Analytic Functions: G a region and {fn} in H(G) such that fn 6 ≡ 0. If
[fn(z) − 1] converges absolutely uniformly on compact subsets of G then
fn converges in H(G) to an analytic function f (z). The zeros of f (z) correspond to the zeros of the fn’s.
- Entire Functions with Prescribed Zeros: Let {an} be a sequence with lim |an| = ∞ and an 6 = 0. If {pn} is a sequence of integers such that for all r > 0 ∑^ ∞
n=
r |an|
)pn+ < ∞,
then f (z) =
Epn (z/an) converges in H(C) and f is an entire function with the correct zeros. (Note that you can choose pn = n − 1 and it will always converge).
- The (Boss) Weierstrass Factorization Theorem: Let f be an entire function with non- zero zeros {an} with a zero of order m at z = 0. Then there is an entire function g and a sequence of integers {pn} such that
f (z) = zmeg(z)
∏^ ∞
n=
Epn
z an
In fact, for z ∈ B(0; r),
u(z) =
2 π
∫ (^2) π
0
Re
reiθ^ + z reiθ^ − z
u(reiθ) dθ.
- Jensen’s Formula: Let f be analytic on B(0; r) and suppose a 1 ,... , an are the zeros of f in B(0; r) repeated according to multiplicity. If f (0) 6 = 0, then
log |f (0)| = −
∑^ n
k=
log
r |ak|
2 π
∫ (^2) π
0
log |f (reiθ)|dθ.
- Poisson-Jensen Formula: Let f be analytic on B(0; r) and suppose a 1 ,... , an are the zeros of f in B(0; r) repeated according to multiplicity. If f (z) 6 = 0, then
log |f (z)| = −
∑^ n
k=
log
r^2 − akz r(z − ak)
2 π
∫ (^2) π
0
Re
reiθ^ + z reiθ^ − z
log |f (reiθ)|dθ.
- Genus, Order, and Rank of Entire Functions:
- Rank : Let f be an entire function with zeros {ak} repeated according to multiplicity such that |a 1 | ≤ |a 2 | ≤.. .. Then f is of finite rank if there is a p ∈ Z such that ∑^ ∞
n=
|an|p+^
If p is the smallest integer such that this occurs, then f is of rank p. A function with only a finite number of zeros has rank 0.
- Standard Form: Let f be an entire function of rank p with zeros {ak}. Then the canonical product
f (z) = zmeg(z)
∏^ ∞
n=
Ep
z an
is the standard form for f.
- Genus: An entire function f has finite genus if f has finite rank and g(z) is a polynomial. If the rank is p and the degree of g is q, then the genus μ = max(p, q). If f has genus μ, then for each α > 0, there exists an r 0 such that |z| > r 0 implies
|f (z)| < eα|z|
μ+ .
- Order : An entire function f is of finite order if there exists a > 0 and r 0 > 0 such that |f (z)| < exp(|z|a) for |z| > r 0. The number λ = inf{a | |f (z)| < exp(|z|a) for |z| sufficiently large}
is called the order of f. If f has order λ and > 0, then |f (z)| < exp(|z|λ+) for all |z| sufficiently large, and a z can be found, with |z| as large as desired, such that |f (z)| ≥ exp(|z|λ−). If f is of genus μ, then f is of finite order λ ≤ μ + 1.
- Hadamard’s Factorization Theorem: If f is entire with finite order λ, then f has finite genus ≤ λ. Combined with above, we have that f has finite order if and only if f has finite genus. Corollary: If f is entire with finite order, then for all c ∈ C with one possible exception, we can always solve f (z) = c. Corollary: If f is entire with finite order λ /∈ Z, then f has an infinite number of zeros.
5 Special Functions
- The Riemann Zeta Function
ζ(s) =
∑^ ∞
n=
ns^
p prime
1 − p−s^
and ζ(s) = ζ(1 − s)
This function has a pole at s = 1, zeros at the negative even integers, and its remaining zeros are in the critical strip {z | 0 < Re z < 1 }. Riemann’s functional equation is
ζ(z) = 2(2π)z−^1 Γ(1 − z)ζ(1 − z) sin
πz
- The Gamma Function: The gamma function is the meromorphic function on C with simple poles at z = 0, − 1 , − 2 ,... defined by:
Γ(z) =
0
e−ttz−^1 dt
e−γz z
∏^ ∞
n=
z n
ez/n
= lim n→∞
n!nz z(z + 1) · · · (z + n)
Γ(z + n) z(z + 1) · · · (z + n − 1)
The residues at each of the poles is given by
Res(Γ, −n) = (−1)n n!
The functional equation holds for z 6 = 0, 1 ,...
Γ(z + 1) = zΓ(z).
Note further that
Γ(1 − z)Γ(z) = π sin(πz) and Γ(z) = Γ(z) and Γ(1/2) =
π.
- Components of the Sheaf of Germs:
- There is a path in S (G) from (a, [f ]a) to (b, [g]b) iff there is a path γ in G from a to b such that [g]b is the analytic continuation of [f ]a along γ.
- Let C ⊆ S (G) and (a, [f ]a) ∈ C. Then C is a component of S (G) iff
C = {(b, [g]b) | [g]b is the continuation of [f ]a along some curve in G}.
- Riemann Surfaces: Fix a function element (f, D). The complete analytic function F associated with (f, D) is the collection
F = {[g]z | [g]z is an analytic continuation of [f ]a for any a ∈ D}.
Then R = {(z, [g]z ) | [g]z ∈ F } is a component of S (C), and (R, ρ) is the Riemann Surface of F.
- Complex Manifolds: Let X be a topological space.
- A coordinate chart is a pair (U, ϕ) such that U ⊆ X is open and ϕ : U → V ⊆ C is a homeomorphism.
- A complex manifold is a pair (X, Φ) where X is connected, Hausdorff and Φ is a collection of coordinate patches on X such that (i) each point of X is contained in at least one member of Φ and (ii) if (Ua, ϕa), (Ub, ϕb) ∈ Φ with Ua ∩ Ub 6 = ∅, then ϕa ◦ ϕ− b 1 is analytic.
- Analytic Functions: Let (X, Φ) and (Ω, Ψ) be analytic manifolds, f : X → Ω continuous, a ∈ X, and (a) = α. Then f is analytic at a if for any patch (Λ, ψ) ∈ Ψ which contains α, there is a patch (U, ϕ) ∈ Φ which contains a such that
(i) f (U ) ⊆ Λ; (ii) ψ ◦ f ◦ ϕ−^1 is analytic on ϕ(U ) ⊆ C.
- Some Results on Analytic Functions:
- Let F be a complete analytic function with Riemann surface (R, ρ). If F : R → C is defined by F (z, [f ]z ) = f (z) then F is an analytic function.
- Compositions of analytic function are analytic
- (Limit Points) If f and g are analytic functions X → Ω and if {x ∈ X : f (x) = g(x)} has a limit point in X, then f = g.
- (Maximum Modulus) If f : X → C is analytic and there is a point a ∈ X and a neighborhood U of a such that |f (a)| ≥ |f (x)| for all x ∈ U , then f is constant.
- (Liouville) If (X, Φ) is a compact analytic manifold, then there is no non-constant ana- lytic function from X into C.
- (Open Mapping) Let f : X → Ω be a non-constant analytic function. If U is an open subset of X, then f (U ) is open in Ω.
- Mean Value Property: If u : G → R is a harmonic function and B(a; r) ⊂ G then
u(a) =
2 π
∫ (^2) π
0
u(a + reiθ) dθ.
- Maximum Principles:
I. Suppose u : G → R has the MVP. If there is a point a ∈ G such that u(a) ≥ u(z) for all z in G, then u is constant. (Analogously, there is a Minimum Principle). II. Let u, v : G → R be bounded and continuous functions with the MVP. If for each point a ∈ ∂∞G,
lim sup z→a
u(z) ≤ lim inf z→a v(z)
then u(z) < v(z) for all z in G or u = v. Corollary: If a continuous function satisfying the MVP is 0 on the boundary, then it is identically 0. III. If ϕ : G → R is a subharmonic function and there is a point a ∈ G with ϕ(a) ≥ ϕ(z) for all z in G, then ϕ is constant. IV. If ϕ, ψ : G → R are bounded functions such that ϕ is subharmonic and ψ is superhar- monic and for each point a ∈ ∂∞G,
lim sup z→a
ϕ(z) ≤ lim inf z→a ψ(z)
then ϕ(z) < ψ(z) for all z in G or ϕ = ψ is harmonic.
- The Poisson Kernel: For 0 ≤ r < 1 , −∞ < θ < ∞, the Poisson kernel is the following:
Pr(θ) =
∑^ ∞
n=−∞
r|n|einθ^ = Re
1 + reiθ 1 − reiθ
1 − r^2 1 − 2 r cos θ + r^2
- Dirichlet Problem in the Disk: If f : ∂D → R is a continuous function, then there is a continuous harmonic function u : D → R such that u(z) = f (z) for all z ∈ ∂D. Moreover, u is unique and defined by
u(reiθ) =
2 π
∫ (^) π
−π
Pr(θ − t)f (eit) dt.
- Harmonicity vs. MVP: If u : G → R is a continuous function which has the MVP, then u is harmonic.
- Harnack’s Inequality: If u : B(a; R) → R is continuous, harmonic in B(a; R), and u ≥ 0 then for 0 ≤ r < R and all θ R − r R + r u(a) ≤ u(a + reiθ) ≤
R + r R − r u(a).
