Compex Analysis

Question 1

Define (complex) differentiable

Answer 1

Let a∈C, U⊂C be an open set containing a, and f:U → C be a function defined on some B(a,r). Then f is complex differentiable at a with derivative L if lim(z→a) [f(z)−f(a)]/[z-a] exists and equals L.

Question 2

Define a holomorphic function

Answer 2

Let U ⊆ C be an open set and f: U → C a function. If f is complex differentiable at every a ∈ U, f is holomorphic on U.

Question 3

Give the Cauchy Riemann equations

Answer 3

u_x = v_y
v_x = -u_y

Question 4

Give the Wirtinger derivatives

Answer 4

∂_(z)f = 1/2(∂_x − i∂_y) u + i/2(∂_x − i∂y)v
∂(z*)f = 1/2(∂_x + i∂_y) u + i/2(∂_x + i∂_y)v

Question 5

Define the Laplacian of a function

Answer 5

If u: R2 → R is a function on an open set U ⊆ R2 which is twice differentiable, then the Laplacian ∆u = ∂(xx)u + ∂(yy)u.

Question 6

Define a harmonic function

Answer 6

If u: R2 → R is a function on an open set U ⊆ R2 which is twice differentiable, then u is harmonic if ∆u = 0.

Question 7

Define the radius of convergence of a power series

Answer 7

Let the sum of (a_n)(z^n) be a power series, and S be the set of z ∈ C at which it converges. The radius of convergence of the power series is sup{|z| : z ∈ S}, or ∞ if the set S is unbounded.

Question 8

Define the exponential function

Answer 8

exp(z) = sum of (z^n)/n!

Question 9

Define the cosine function

Answer 9

cos(z) = sum of ((-1)^n)(z^(2n))/(2n)!

Question 10

Define the sine function

Answer 10

sin(z) = sum of ((-1)^n)(z^(2n+1))/(2n+1)!

Question 11

Define the logarithm function

Answer 11

Let D = C(−∞, 0]. Then Log : D → C is defined as: if z = |z|e^iθ with θ ∈ (−π, π] then Log(z)= log |z| + iθ.

Question 12

Define the principle value of an argument

Answer 12

The values θ ∈ (−π, π] such that z = |z|e^iθ is called the principal value of the argument.

Question 13

Define a multifunction

Answer 13

A multifunction on a subset U ⊆ C is a map f: U → P(C) assigning to each point in U a subset of the complex numbers (where P(X) is the power set of X).

Question 14

Define a branch

Answer 14

Let f: U → P(C) be a multifunction. A branch of f on a subset V ⊆ U is a function g: V → C such that g(z) ∈ f(z), for all z ∈ V.

Question 15

Define a branch point

Answer 15

Suppose that f: U → P(C) is a multifunction. We say that z0 ∈ U is a branch point of f if there does not exist an open ball B ⊆ U containing z0 such that there is a holomorphic branch of f defined on B{z0}.

Question 16

Define a branch cut

Answer 16

A branch cut of a multifunction is a curve containing all the branch points of the multifunction.

Question 17

Define a path

Answer 17

A continuous function γ: [a, b] → C.

Question 18

Define a closed path

Answer 18

A path where γ(a) = γ(b).

Question 19

Define the image of a path

Answer 19

Given a path γ, the image is γ* = {z ∈ C : z = γ(t) for some t ∈ [a, b]}.

Question 20

Define a differentiable path

Answer 20

A path γ: [a, b] → C is differentiable if its real and imaginary parts are differentiable as real-valued functions.

Question 21

Define C^k

Answer 21

A function where the first k partial derivatives exist and are continuous.

Question 22

Define an opposite path

Answer 22

For any path γ: [a, b] → C, the opposite path is given by γ−: [0, b−a] → C, γ−(t) = γ(b − t).

Question 23

Define concatenation of paths

Answer 23

If γ1: [a, b] → C and γ2: [c, d] → C are two paths such that γ1(b) = γ2(c) then the concatenation is γ1⋆γ2: [a, b + d − c] → C where γ1⋆γ2(t) = γ1(t) for t ≤ b and γ2(t − b + c) for t ≥ b.

Question 24

Define a piecewise C^k path

Answer 24

A finite concatenation of C^k paths.

Question 25

Define a reparameterisation of paths

Answer 25

Let ϕ: [a, b] → [c, d] be continuously differentiable with ϕ(a) = c and ϕ(b) = d and let γ: [c, d] → C be a path. Letting γ˜ = γ◦ϕ, means that γ˜: [a, b] → C is a reparameterization of γ.

Question 26

Define equivalent paths

Answer 26

Two parametrized paths γ1: [a, b] → C and γ2: [c, d] → C are equivalent if there is a continuously differentiable bijective function s: [a, b] → [c, d] such that s′(t) > 0 for all t ∈ [a, b] and γ1 = γ2◦s.

Question 27

Define a (complex) integral

Answer 27

Suppose f(t) is a function and write it in terms of its real and complex components, so f(t) = u(t) +iv(t) where u, v are real functions. Then we say that f is Riemann integrable if both u and v are and we define the integral from a to b of f(t) = the integral from a to b of u(t) + i times the integral from a to b of v(t).

Question 28

Define the length of a path

Answer 28

If γ: [a, b] → C is a C1 path, then we define the length of γ to be l(γ) = integral from a to b of |γ′(t)|.

Question 29

Define a primitive

Answer 29

Let U ⊆ C be an open set and let f: U → C be a continuous function. If there exists a differentiable function F: U → C with F’(z) = f(z) then we say F is a primitive for f on U.

Question 30

Give the (complex) fundamental theorem of calculus

Answer 30

Let U ⊆ C be open and let f: U → C be a continuous function. If F: U → C is a primitive for f and γ: [a, b] → U is a piecewise C1 path in U then we have the integral of f(z) along γ = F(γ(b)) − F(γ(a)).

Question 31

Give Cauchy’s Theorem

Answer 31

Let U ⊂ C be a domain and γ be a closed curve such that it and all bounded components of C\γ∗ are inside U. Let f be a function holomorphic in U. Then the integral of f(z) along γ = 0.

Question 32

Define convex

Answer 32

Let X be a subset of C. We say that X is convex if for each z, w ∈ X the line segment between z and w is contained in X.

Question 33

Define star-like

Answer 33

Let X be a subset of C. We say that X is star-like if there is a point z0 ∈ X such that for every w ∈ X the line segment [z0, w] joining z0 and w lies in X.

Question 34

Define simply connected

Answer 34

A domain U in C is simply connected if for every a, b ∈ U, any two paths from a to b are homotopic.

Question 35

Give the deformation theorem

Answer 35

Let U be a domain in C and suppose that γ1 and γ2 are two homotopic paths in U. Then if f : U → C is a holomorphic function, the integral of f(z) along γ1 equals the integral of f(z) along γ2.

Question 36

Define the winding number

Answer 36

If γ: [0, 1] → C{0} is a closed path and γ(t) = |γ(t)|e^(2πia(t)), then since γ(0) = γ(1) we must have a(1)−a(0) ∈ Z. This integer is called the winding of γ around 0.

Question 37

Give the formula for the winding number

Answer 37

Let γ be a piecewise C1 closed path and z0 ∈ C a point not in the image of γ. Then the winding number I(γ, z0) of γ around z0 is given by I(γ, z0) = 1/2πi times the integral of 1/(z-z0) along γ.

Question 38

Define analytic

Answer 38

If f: U → C is a function on an open subset U of C, then we say that f is analytic on U if for every z0 ∈ U there is an r > 0 with B(z0, r) ⊆ U such that f(z) = the power series a_k(z − z0)^k with radius of convergence at least r.

Question 39

Define positively oriented

Answer 39

Let γ be a simple closed curve. Then, we say that γ is positively oriented if the winding number around any point from the bounded component is 1.

Question 40

Give Cauchy’s formula

Answer 40

Suppose that f: U → C is a holomorphic function on an open set U, w ∈ U and γ is a positively oriented closed curve such that γ∗ and the interior of γ are inside of U. Then for any w inside of γ we have f(w) = 1/2πi times the integral along γ of f(z)/(z – w).

Question 41

Give the strong version of Taylor’s Theorem

Answer 41

If f: U → C is holomorphic on an open set U, then for any z0 ∈ U, f(z) is equal to its Taylor series at z0 and the Taylor series converges on any open disk centred at z0 lying in U. Moreover, the derivatives of f at z0 are given by f^(n)(z0) = n!/2πi times the integral along γ(z0,r) of f(z)/(z − z0)^(n+1) for any r < R where r0 is such that B(z0, R) ⊂ U.

Question 42

Define an entire function

Answer 42

A function which is holomorphic in C.

Question 43

Give Liouville’s Theorem

Answer 43

Let f: C → C be an entire function. If f is bounded, then it is constant.

Question 44

Give Morera’s Theorem

Answer 44

Suppose that f: U → C is a continuous function on an open subset U ⊆ C. If for any closed path γ: [a, b] → U we have that the integral along γ of f(z) is 0, then f is holomorphic.

Question 45

Define uniform on compacts

Answer 45

Let U be an open subset of C. If (fn) is a sequence of functions defined on U, we say fn → f uniformly on compacts if for every compact subset K of U, the sequence (fn|K) converges uniformly to f|K.

Question 46

Give the identity theorem

Answer 46

Let U be a domain and suppose that f1, f2 are holomorphic functions defined on U. Then if S = {z ∈ U: f1(z) = f2(z)} has a limit point in U, we must have S = U, that is f1(z) = f2(z) for all z ∈ U.

Question 47

Define a regular point

Answer 47

Let f: U → C be a function, where U is open. We say that z0 ∈ closure(U) is a regular point of f if f is differentiable at z0.

Question 48

Define a singularity

Answer 48

Let f: U → C be a function, where U is open. We say that z0 ∈ closure(U) is singularity of f if f is not differentiable at z0.

Question 49

Define an isolated singularity

Answer 49

Let f: U → C be a function, where U is open. If z0 is a singularity where f is holomorphic on B(z0, r){z0} for some r > 0, then this singularity is said to be isolated.

Question 50

Define a removable singularity

Answer 50

Let z0 be an isolated singularity of a function f. If there is a function g holomorphic in B(z0, r) for some r > 0 such that f(z) = g(z) in B(z0, r){z0}, the singularity is said to be removable.

Question 51

Define a pole

Answer 51

Let z0 be an isolated singularity of a function f. If there is a function g holomorphic in B(z0, r) for some r > 0 such that g(z0) ≠ 0 and f(z) = g(z)/(z-z0)^n in B(z0, r){z0}, the singularity is said to be a pole (of order n).

Question 52

Define an essential singularity

Answer 52

An isolated singularity which is neither removable nor a pole.

Question 53

Define a meromorphic function

Answer 53

A function on an open set U where all singularities are isolated and poles is said to be meromorphic on U.

Question 54

Give Laurent’s Theorem

Answer 54

Suppose that 0 < r < R and A = A(z0, r, R) = {z : r < |z − z0| < R}. If f: U → C is holomorphic on an open set U which contains the closure of A, then there exist cn ∈ C such that f(z) = the sum from -∞ to ∞ of c_n(z − z0)^n for all z ∈ A. The series converges for all z ∈ A and uniformly for all z ∈ A(z0, r′, R′) where r < r′ < R′ < R. Moreover, the cn are unique and are given by the following formulae: cn = 1/2πi times the integral along γs of f(z)/(z − z0)^(n+1), where s ∈ [r, R] and for any s > 0 we set γs(t) = z0 + se2πit.

Question 55

Define a principle part

Answer 55

Let z0 be an isolated singularity of f and the sum of c_n(z −z0)^n its Laurent’s expansion. The principal part of f at z0 is the sum of terms with negative powers and denoted P(z0)f.

Question 56

Define residue

Answer 56

Let z0 be an isolated singularity of f. Then the residue of f at z0 is defined as the coefficient c−1 of the Laurent expansion.

Question 57

Give the classification of isolated singularities theorem

Answer 57

Let z0 be an isolated singularity of f. and the sum of c_n(z − z0)^n its Laurent expansion. Then z0 is a removable singularity if c_n = 0 for all n < 0, a pole of order n if c_(−n) ≠ 0 and c_k = 0 for all k < −n and an essential singularity if there are arbitrary large n such that c_(−n) ≠ 0.

Question 58

Give Riemann’s removable singularity theorem

Answer 58

Suppose that U is an open subset of C, z0 ∈ U and suppose that f: U{z0} → C is holomorphic. Then z0 is a removable singularity if and only if f is bounded near z0.

Question 59

Give the alternative classification of a pole

Answer 59

Let f be a holomorphic function in a neighbourhood of z0. Then z0 is a pole if and only if |f(z)| → ∞ as z → z0.

Question 60

Give the Casorati-Weierstrass Theorem

Answer 60

Let U be an open subset of C and let a ∈ U. Suppose that f:U{a} → C is a holomorphic function with an isolated essential singularity at a. Then for all ρ > 0 with B(a, ρ) ⊆ U, the set f(B(a, ρ){a}) is dense in C, that is, the closure of f(B(a, ρ){a}) is all of C.

Question 61

Give the Residue theorem

Answer 61

Suppose that U is an open set in C and γ is a closed curve that is contained in U together with its inside. Suppose that f is holomorphic on U\S where S is a finite set of isolated singularities of f. We also assume that f has no S ∩ γ* is empty. Then 1/2πi times the integral along γ of f(z) = The sum of I(γ, a)Res_a(f) for all a∈S.

Question 62

Give the argument principle

Answer 62

Suppose that f is meromorphic on U and γ be a simple positively oriented contour such that the contour and its inside are contained in U. We assume that f has no zeroes or poles on γ*. If N is the number of zeros (counted with multiplicity) and P is the number of poles (again counted with multiplicity) of f inside γ then N − P = 1/2πi times the integral along γ of f’(z)/f(z).

Question 63

Give Roche’s Theorem

Answer 63

Suppose that f and g are holomorphic functions on an open set U in C and let γ be a simple contour that is contained inside of U together with its interior. If |f(z)|>|g(z)| for all z ∈ γ* then f and f+g have the same change in argument around γ, and hence the same number of zeros (counted with multiplicities) inside of γ.

Question 64

State the open mapping theorem

Answer 64

Suppose that f: U → C is holomorphic and non-constant on a domain U. Then for any open set V ⊂ U the set f(V) is also open.

Question 65

Give the Residue Formula

Answer 65

Suppose that f has a pole of order m at z0, then Resz0(f) = limz→z0 of 1/(m − 1)! times d^(m−1)/dz^(m−1) of (z − z0)^mf(z).

Question 66

State Jordan’s Lemma

Answer 66

Let f be a continuous function on γ*_R where γ_R(t) = Re^(it) with t ∈ [0, π]. Then for all positive α, |integral along γ_R of f(z)e^(iαz)| ≤ πM_R/α, where M_R = max t∈[0,π] of |f(Re^(it))|. In particular, suppose that f is holomorphic on H\S where H = {z ∈ C : Im (z) > 0} and S is a finite set of isolated singularities. Suppose that f(z) → 0 as z → ∞ in H. Then the integral along γ_R of f(z)e^(iαz) → 0 as R → ∞ for all α > 0.