Compex Analysis Flashcards
Define (complex) differentiable
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.
Define a holomorphic function
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.
Give the Cauchy Riemann equations
u_x = v_y
v_x = -u_y
Give the Wirtinger derivatives
∂_(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
Define the Laplacian of a function
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.
Define a harmonic function
If u: R2 → R is a function on an open set U ⊆ R2 which is twice differentiable, then u is harmonic if ∆u = 0.
Define the radius of convergence of a power series
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.
Define the exponential function
exp(z) = sum of (z^n)/n!
Define the cosine function
cos(z) = sum of ((-1)^n)(z^(2n))/(2n)!
Define the sine function
sin(z) = sum of ((-1)^n)(z^(2n+1))/(2n+1)!
Define the logarithm function
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.
Define the principle value of an argument
The values θ ∈ (−π, π] such that z = |z|e^iθ is called the principal value of the argument.
Define a multifunction
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).
Define a branch
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.
Define a branch point
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}.
Define a branch cut
A branch cut of a multifunction is a curve containing all the branch points of the multifunction.
Define a path
A continuous function γ: [a, b] → C.
Define a closed path
A path where γ(a) = γ(b).
Define the image of a path
Given a path γ, the image is γ* = {z ∈ C : z = γ(t) for some t ∈ [a, b]}.
Define a differentiable path
A path γ: [a, b] → C is differentiable if its real and imaginary parts are differentiable as real-valued functions.
Define C^k
A function where the first k partial derivatives exist and are continuous.
Define an opposite path
For any path γ: [a, b] → C, the opposite path is given by γ−: [0, b−a] → C, γ−(t) = γ(b − t).
Define concatenation of paths
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.
Define a piecewise C^k path
A finite concatenation of C^k paths.
Define a reparameterisation of paths
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.
Define equivalent paths
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.
Define a (complex) integral
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).
Define the length of a path
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)|.
Define a primitive
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.
Give the (complex) fundamental theorem of calculus
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)).
Give Cauchy’s Theorem
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.
Define convex
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.
Define star-like
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.
Define simply connected
A domain U in C is simply connected if for every a, b ∈ U, any two paths from a to b are homotopic.
Give the deformation theorem
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.
Define the winding number
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.
Give the formula for the winding number
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.
Define analytic
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.
Define positively oriented
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.
Give Cauchy’s formula
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).
Give the strong version of Taylor’s Theorem
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.
Define an entire function
A function which is holomorphic in C.
Give Liouville’s Theorem
Let f: C → C be an entire function. If f is bounded, then it is constant.
Give Morera’s Theorem
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.
Define uniform on compacts
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.
Give the identity theorem
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.
Define a regular point
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.
Define a singularity
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.
Define an isolated singularity
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.
Define a removable singularity
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.
Define a pole
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).
Define an essential singularity
An isolated singularity which is neither removable nor a pole.
Define a meromorphic function
A function on an open set U where all singularities are isolated and poles is said to be meromorphic on U.
Give Laurent’s Theorem
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.
Define a principle part
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.
Define residue
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.
Give the classification of isolated singularities theorem
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.
Give Riemann’s removable singularity theorem
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.
Give the alternative classification of a pole
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.
Give the Casorati-Weierstrass Theorem
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.
Give the Residue theorem
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.
Give the argument principle
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).
Give Roche’s Theorem
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.
State the open mapping theorem
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.
Give the Residue Formula
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).
State Jordan’s Lemma
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.
