complex analysis

complex number

C = [a + bi, | a,b ∈ R]

Re(x + iy) & Im(x + iy)

x & y

addition: (x + iy) + (a + ib) = (x + a) + i(y + b)

multiplication: (x + iy) ⋅ (a + bi) = (xa - yb) + i(ay + bx)

norm (absolute value)

√x^2 + y^2

conjugate of z (z̄)

let z = a + bi ∈ C, z̄ ≔ a - bi

polar form

z = r(cosθ + isinθ)

argument

θ

e^z

e^x (cos y + i sin y) where z = x + iy

log z

ln |z| + i arg (z) where arg (z) is chosen to satisfy C ≤ arg (z) < + 2pi

a^b

e^(b log (a)) where a & b are complex numbers, a ≠ 0, and log is some branch of the logarithm function

r-neighborhood / D(z0;r)

{z ∈ C | |z-z0| < r}

interior point (z0)

there exists ∃r > 0 s.t. D(z0;r) ⊆ U

open

every point in U is an interior point of U

closed

its complement C - U is open

limit

f(z) = L if ∀ε > 0, ∃δ > 0 s.t. |z - z0| < δ, then |f(z) - L| < ε

continuous at a point

lim z → z0 f(z) = f(z0)

differentiable

lim z → z0 f(z) - f(z0) exists

continuous

f is continuous at z0 for all z0 in A

analytic

f is differentiable at z0 for all z0 in A

entire

f is analytic at z0 for all z0 in A