chp 4 integrals

Question 1

Cauchys inequality

Answer 1

for func f analytic on and inside circle $C(z_0, R)$

$|f’^(n)(z)| \leq \frac{n! M_R}{R}$
for n = 1, 2, 3
where
$M_R = max{||f(z)|: z on Cr}$

Question 2

Louiville thm

Answer 2

If f is entire and bpounded on C then it is constant

Question 3

Proof of Louiville thm guide

Answer 3

• Establish conditions givena dn link to conditions for Cauchys inequality
• Create M which is arb max
• use Cauchy i.e. estibliish link $M_R$ c(z,R) which bounds f using Cauchy inequality
• Take limit of both sides of Cauchy st YOUR M is not the same adn the ARB M which bounds
• Limit makes RHS = 0
• abs says LHS = 0

Question 4

PDP

Answer 4

$\int_c_1 = \int_c_2$ if f is analytic interior to and between curves c1 and c2

Question 5

Multiply connected domains curve theorem (pokemon)

Answer 5

$\int_c = \sum_{k=1}^n \int_{ck} f(z) dz$

Question 6

3 equiv (OG, Cauchy- Gorsat, NO holes)

Answer 6

f is CONT=> 3 equiv

Cauchy Gorsat
f i analytic (on and interior to C (a simple closed curve)) => 3 equiv

No holes
f is ANY FUNCTION then => 3 equiv

Question 7

Conditions for CIF and GCIF

Answer 7

f be analytic interior to and onn curve c that is siimple, closed, + oriented