Analytic Function
: A function f(z) is analytic if it
is complexly differentiable & continuous at z0 (y
en su vecindad ). = it can be represented by a
convergent power series:
Entire function
:
A function f(z) is entire if it is
analytic everywhere in the complex plane. (ex.
sinz, cosz, e^z)
Singular point
: a point where f(z) is
not analytic
inside D.
Cauchy
'
s Theorem
:
If f(z) is analytic in D, then the
integral of f(z) along any closed curve = 0
Integral and derivative in Cauchy:
Liouville
'
s Theorem
:
if f(z) is an entire function
and bounded ( exists M where ∣f(z)∣≤M| pt x) then
f(z) must be constant.
Laurent Series
:
extension of Taylor series for
functions with singularities. It expresses f(z) as a
series with both positive and negative powers of
(z−z0 ):
The Laurent series converges in regions
defined by r1 < |z - z0| < r2
A function f(z) is C^n if it has c
ontinuous
derivatives
up to the n-th order
Riemann Mapping
:
Any simply connected,
proper open subset on Complex can be
conformally mapped onto the unit disk.
Fourier Transform
: series that decomposes
a function f(t) into its frequency:
Convolution
:
operation of f(t) * g(t) =
means to multiply in frequency domain
Cauchy Riemann
:
f(z) is differentiable if:
Residue at z
0
:
The coefficient a−1 in the
Laurent series of f(z)) around z0 . if f(z) =
analytic except and zn singularities:
