chp 4 integrals Flashcards
Cauchys inequality
for func f analytic on and inside circle $C(z_0, R)$
$|fâ^(n)(z)| \leq \frac{n! M_R}{R}$
for n = 1, 2, 3
where
$M_R = max{||f(z)|: z on Cr}$
Louiville thm
If f is entire and bpounded on C then it is constant
Proof of Louiville thm guide
- 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
PDP
$\int_c_1 = \int_c_2$ if f is analytic interior to and between curves c1 and c2
Multiply connected domains curve theorem (pokemon)
$\int_c = \sum_{k=1}^n \int_{ck} f(z) dz$
3 equiv (OG, Cauchy- Gorsat, NO holes)
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
Conditions for CIF and GCIF
f be analytic interior to and onn curve c that is siimple, closed, + oriented