7. Complex Analysis

Question 1

Definition 7.5
Complex differentiable.
Cauchy-Reimann equations.

Question 2

Definition 7.6
Analytic (or holomorphic)
Entire.

Question 3

Theorem 7.7
f is complex diff. at z \in \Omega open iff.

Question 4

Theorem 7.11
Ratio Test

Question 5

Theorem 7.12
Root Test

Question 6

Theorem 7.14
Formula for radius of convergence

Question 7

Theorem 7.15
For all |z| < R.O.C., f(z) is…

Question 8

Corolary 7.16
if R.O.C. > 0 then f(z) = …

Question 9

Theorem 7.17
for all r < R with R > 0 f_k = \sum a_n z^n does what

Question 10

Definition 7.18
exponential, hyperbolic and trig complex sums.

Question 11

Proposition 7.19
exponential form of trig and hyperbolic functions

Question 12

Theorem 7.20
Propoerties of the complex exponential

Question 13

mod-arg form

Question 14

Proposition 7.21
Propoerties of the argument

Question 15

Principal value

Question 16

complex log

Question 17

properties of complex log

Question 18

Principal breanch of the logarithm
branch cut
complex power

Question 19

complex integration

Question 20

Definition 7.22
Curve, integral over curve.
piece-wise curves.

Question 21

Lemma 7.24
The intregral over a curve depends only on orientation of parameterisation.

Question 22

Length of curve

Question 23

|dz|

Question 24

Definition 7.25
d conjugate(z) …

Question 25

Theorem 7.28 If F analytic on open set and f = dF/dz...

Question 26

THeorem 7.29 Cauchy's Theorem

Question 27

Definiton 7.30 Connected Simply connected

Question 28

Theorem 7.31 Continuous deformation of contour theorem

Question 29

Fundamental contour integral.

Question 30

Definition 7.32 Interior and exterior.

Question 31

Theorem 7.33 (Cauchy's formula) positive oriented simple closed curve and f is analytic on interitor then f(z) = ... proof

Question 32

Theorem 7.35 positive oriented simple closed curve and f is analytic on interitor then f^(n)...

Question 33

Theorem 7.36 Taylor Series Expansion

Question 34

Theorem 7.38 Liouville's Theorem Proof

Question 35

Theorem 7.39 Fundamental Theorem of Algebra

Question 36

Theorem 7.40 If f_n ->-> f is a series of analytic functions...