chp 4 integrals Flashcards

1
Q

Cauchys inequality

A

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
$M_R = max{||f(z)|: z on Cr}$

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2
Q

Louiville thm

A

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

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3
Q

Proof of Louiville thm guide

A
  • 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
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4
Q

PDP

A

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

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5
Q

Multiply connected domains curve theorem (pokemon)

A

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

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6
Q

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

A

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

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7
Q

Conditions for CIF and GCIF

A

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

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