Sec 52, Simply conencted domains

Question 1

Simply connected domain

Answer 1

domain with every simple closed contour within D encloses only points of D

‘No holes’

Question 2

Cauchy Goursat Theorem

Answer 2

If a functions is analytic everywhere on and interior to a simple closed contour C, then its lien integral along C is 0

Question 3

How does Cauchy Goursat theorem apply to simply connected domains

Answer 3

as long as we work on domains wihtout holes, the CG thoerem extends to any closed contour

so that if a function f is analytic thorughout a simply connected domain then $\int_c f(z) dz = 0 \forall C \in D$

Question 4

What kind of function always has an antiderivative

Answer 4

An entrie function.
SInce it will always be analytic interior to and on any simple closed contour C $\forall \ z \ in C $ and so by Cauchy-Goursat its line integral for closed C will be zero, then by that equiv theoerm (thm pg 141) it has an antiderivative

Question 5

Prinicple of deformation of paths

Answer 5

let C1 and C2 denote positively oriented simple closed contours, where C1 is interior to
C2.

If a function f is analytic in the closed
region consisting of those contours and all
points between them, then

$\int_{c1} f(z) dz = \int_{c2} f(z) dz $

Question 6

Thm page 156, for integrals of multiply connected domains

Answer 6

Suppose
1. C is a simple closed contour, + direction
2. Ck (k = 1, . . . , n) are simple closed contours interior to C, all described in the
counterclockwise direction, that are disjoint ,interiors have no points
in common.

f is analytic on all of these contours and throughout the multiply connected
domain consisting of the points inside C and
exterior to each Ck
, then
$\int_C f(z) dz = \sum_k=1 ^n \int_{c_k} f(z) dz$

Question 7

NB exercise result for circle integral

Answer 7

$\int_C (z-z_0)^n dz = $

• 0 iff $n \in Z not including -1$
• $2 \pi i$ if n = -1

Question 8

how does CIF related analytic functions and their derivatives

Answer 8

if f is analytic at a pt z

then the derivatives of all orders of f are analytic at z as well
So this means if $f(z) = u(x,y) +iv(x,y)$ is analytic at $x+iy$ then u and v have cont partial derivatives of all orders at $(x,y)$ as used in chp 1

Question 9

Morera’s theorem

Answer 9

If f is cont on domain D and $\int_C f(z) dz = 0\forall$ closed contour $C \ \in D$

then f is analytic in D

Question 10

Morera’s theorem is the converse of which theorem for the continuous functions on a simply connected domain

Answer 10

If a function f is analytic throughout a simply connected domain D, then

$\int_C f(z) dz = 0$

for every closed contour C lying in D.

Question 11

Cauchy’s inequality

Answer 11

if f analytic inside adn on a + circle $C_R$ centre $z_0$ radius R, and we let
$M_R = max{|f(z)|: z on C_R}$
then for n = 1, 2, …
$$|f^{(n)}(z_0)| \leq \frac{n! M_R}{R^n}$$