Complex Analysis Flashcards
(20 cards)
Modulus of a complex number
√ x^2 + y^2
Argument of a complex number
arg(z) is any angle θ such that x = |z|cosθ, y = |z|sinθ
Principal value of the argument
Arg(z) ∈(−π,π]
Neighbourhood of z0
z0 : { z: |z - z0| < r }
Punctured Neighbourhood
{z : 0 < |z - z0| < r }
Annulus
{ z { r1 < |z - z0| < r2 }
Limit of a function
lim z->z0 f(z) = w0 if for every ε > 0 there exists δ > 0 such that | f(z) - w0 | < ε whenever | z - z0| < δ
Continuity
A function f is continuous at z0 if f(z0) is defined lim z->z0 f(z) exists and lim z->z0 f(z) = f(z0)
Complex Differentiability
f is differentiable at z if f’(z) = lim Δz→0 [f(z+Δz) − f(z)] / Δz exists and is the same for all directions Δz→0
Analytic Function
A function is analytic or holomorphic at z0 if it is differentiable at every point in some neighbourhood of z0
Cauchy- Riemann Equations
If f(z) = u(x,y) + iv(x,y) is analytic then it satisfies: ∂u / ∂x = ∂v / ∂y, ∂u / ∂y = −∂v / ∂x
Contour
A piecewise smooth, simple, closed, positively oriented curve
Contour Integral
∫ C f(z)dz = b ∫ a f(z(t))z′(t)dt
Cauchy Theorem
If f is analytic on an insde a simple closed contour C, then ∮C f(z)dz=0
Isolated Singularity
A point z0 where f is not analytic but is analytic in some punctured neighbourhood 0 < |z - z0| < r
Laurent Series
If f is analytic in an annulus around z0 then: f(z) = n=∞ ∑ -∞ an(z−z0)^n, an=1/2πi ∮C (f(w) / (w−z0)^n+) dw
Pole of Order m
An isolated singularity z0 is a pole of order m if the Laurent series has finitely many negaitve powers down to (z - z0)^-m
Essential Singularity
A singularity where the Laurent series has infinitely many negative powers
Residue at an Isolated Singularity
The coefficient a-1 in the Laurent series of f about z0. If f has a simple pole at z0: Res(f, z0) = lim z -> z0 (z - z0)f(z)
Cauchy Reside Theorem
If f is analytic inside and on a closed contour C except for isolated singularities zk, then: ∮C f(z)dz = 2πi ∑ k Res(f, zk)
