Chapter 3: Analytic Functions Flashcards
What’s the radius of convergence of a
power series?
( lim sup n√|an| )-1
Cauchy-Riemann equations say…
If f’ exists,
ux = vy
uy = -vx
Is the converse of the Cauchy-Riemann true?
No,
need partials to be cont. and satisfy equations
to get f’ exists
f’ exists at a means what in apostol land…
there exists a special function f* cont at p :
f(z) - f(p) = (z - p) f*(z)
for some z
(if f* exists, then f’(p) = f*(z))
a path is PCD if …
path is cont. diff
on some partition
(remember a path is by def continous on all [a, b])
Goursat’s Theorem says…
if f’ exists, then ∫f = 0 over any triangle.
How does FTC generalize?
- f cont. on open set
- path is PCD (call it y)
Then,
∫ f = F( y(b)) - F(y(b))
where F’ = f
What do we need to write down a complex integral?
- cont f
- PCD path
then
∫ f = ∫ f(path) (path)’ dt as t = 0 to 1
(or domain of path)
If f’ exists and f is defined on a
star-like set then…
F (the primitive)
exists!
∫ f dz on a closed, pcd path
then …
= 0
b/c closed means path(a) = path(b)
then 0 by FTC
For f continous on open, as radius -> 0+,
what happens to
it goes to f(a)
“integral around circle converge to value at the center!”
What does the Cauchy Integral Formula say…
If f is differentiable on open, then for any z in B(a; r)
(where closure of B is in open set)
“if diff, don’t even need to take limit, just take any z in ball!”
What more can we conclude from Cauchy Integral Formula?
f(k) exists and equals…
If diff fn –> f and
fn are locally bounded, then…
f’, f’’, exist and equal limit as expected
Inside R, power series converges to…
analytic function
and can be differentiated term by term
trick
log(1 + x)
for large x is bounded by what?
1 < log( 1 + x) < x
b/c exponential growth is faster than linear
What’s the set of points in [z1, z2] ?
(1-t) z1 + t z2
for 0 ≤ t ≤ 1
What’s the Weirstrauss M-test?
∑ Mn converges
==>
∑ fn converges uniformly
to a func f
bonus: if each fn is cont, then so is f
fn | ≤ Mn
If differentiable, locally bounded fn —> f , then …
f(k)n —> f(k)
happens uniformly on
- compact set
- disk (a, r/2) if bounded (not just locally)
f analytic (= differentiable) on disk means…
f = taylor series
Does f infinitely differentiable in R mean f = taylor?
NO!
e.g., e^{-1/x^2}…no series around 0
What’s Monera’s Theorem?
f is analytic if it’s
continuous
∫ is 0 around any triangle
Cauchy Estimate says
If |f| ≤ M for f analytic,
|fn(a) | ≤ M n! / R^n
What’s Liouville’s Theorem?
bounded & entire means constant!
