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

∂x_u = ∂y_v
∂x_v = −∂y_u

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.