7. Complex Analysis Flashcards
Definition 7.5
Complex differentiable.
Cauchy-Reimann equations.
Definition 7.6
Analytic (or holomorphic)
Entire.
Theorem 7.7
f is complex diff. at z \in \Omega open iff.
Theorem 7.11
Ratio Test
Theorem 7.12
Root Test
Theorem 7.14
Formula for radius of convergence
Theorem 7.15
For all |z| < R.O.C., f(z) is…
Corolary 7.16
if R.O.C. > 0 then f(z) = …
Theorem 7.17
for all r < R with R > 0 f_k = \sum a_n z^n does what
Definition 7.18
exponential, hyperbolic and trig complex sums.
Proposition 7.19
exponential form of trig and hyperbolic functions
Theorem 7.20
Propoerties of the complex exponential
mod-arg form
Proposition 7.21
Propoerties of the argument
Principal value
complex log
properties of complex log
Principal breanch of the logarithm
branch cut
complex power
complex integration
Definition 7.22
Curve, integral over curve.
piece-wise curves.
Lemma 7.24
The intregral over a curve depends only on orientation of parameterisation.
Length of curve
|dz|
Definition 7.25
d conjugate(z) …
Theorem 7.28
If F analytic on open set and f = dF/dz…
THeorem 7.29
Cauchy’s Theorem
Definiton 7.30
Connected
Simply connected
Theorem 7.31
Continuous deformation of contour theorem
Fundamental contour integral.
Definition 7.32
Interior and exterior.
Theorem 7.33 (Cauchy’s formula)
positive oriented simple closed curve and f is analytic on interitor then f(z) = …
proof
Theorem 7.35
positive oriented simple closed curve and f is analytic on interitor then f^(n)…
Theorem 7.36
Taylor Series Expansion
Theorem 7.38
Liouville’s Theorem
Proof
Theorem 7.39
Fundamental Theorem of Algebra
Theorem 7.40
If f_n ->-> f is a series of analytic functions…
