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MATH 316 Complex Variables - Coggle Diagram
MATH 316 Complex Variables
Ch1.Complex Numbers
z = a + bi
Polar form \( z = r e^{i\theta} = r cos(\theta) + i *r sin(\theta)\)
Subset: open/closed, connected/disconnected
A domain is an open, connected subset
Ch2.Analytic Funcs
Limit
Continuity: when z approach z0, f(z) approach f(z0)
Analytic: if f is differentiable at every point of domain S
Note: the complement function z_bar is not analytic
Cauchy-Riemann Equations: determine if a function is analytic
Ch3.Elementary Funcs
Rational Functions
Partial fraction decomposition
Trig Functions
Polynomials
Logarithm
Multivalued Functions
Power Functions
Inverse Trig Functions: consists of log function
Ch4.Complex Integral
Contour Integral
Parametrize and compute
A simple closed contour is a closed contour whose only repeated point is the initial/terminal point
Upper bound of integral
Antiderivative
(Cauchy-Gaursat) If f analytic on D and L is a contractible loop on D, then the contour integral of f(z) over L is equal to 0
1) f has an antiderivative in domain D
2) Integral of f(z) is 0 for all CLOSED contour in D
3) Contour integrals are independent of path in D
Parametrize & Contour
2). Smooth closed curve
A contour is a sequence of 1) and 2)
1). Smooth Arc
Orientation of a closed contour
Homotopic
Loops L1,L2 are "homotopic" if there exists a continuous function z(s,t) that deforms L1 to L2
A loop is "contractible" if it is homotopic to a point
A domain D is "simply connected" if all loops in D are contractible
The interior of a simple closed contour is "simply connected"
The integral of f(z) over two homotopic looks are equal
Conclusions
If D is a "simply connected" domain, then a function f with domain D has an antiderivative on D
(Cauchy Integral Thm): if f is analytic on a simply connected domain D, and L is a simple closed contour in D, and \(a \in interior(D) \), then \( \int_L \frac{f(z)}{(z-a)} dz = 2 \pi i f(a) \)
(Liouville's Thm) If f is entire and bounded, it is constant.
Ch5.Series Representation
Taylor Series
If f is analytic on a disk \( D = {|z-z_0| < R}\), then the Taylor expansion of f at z0 converges to f pointwise in D, uniformly in all closed subdisks \( D' = {|z-z_0| \leq R'} \), R' < R
Laurent Expansion
pole of order "N"
essential singularity
removable singularity
convergence
Pointwise: on open disk |z| < |a|
Uniformly: on closed subdisk |z| <= R, R < |a|
Great Picard Theorem about essential singularity
Ch6.Residue Theory
Def: residue is the coefficient of \(1/(z-z_0)\) in the Laurent Expansion, denoted as \( Res(f; z_0) \)