Chapter 3: Analytic Functions

Question 1

What’s the radius of convergence of a

power series?

Answer 1

( lim sup n√|an| )^-1

Question 2

Cauchy-Riemann equations say…

Answer 2

If f’ exists,

u_x = v_y

u_y = -v_x

Question 3

Is the converse of the Cauchy-Riemann true?

Answer 3

No,

need partials to be cont. and satisfy equations

to get f’ exists

Question 4

f’ exists at a means what in apostol land…

Answer 4

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))

Question 5

a path is PCD if …

Answer 5

path is cont. diff

on some partition

(remember a path is by def continous on all [a, b])

Question 6

Goursat’s Theorem says…

Answer 6

if f’ exists, then ∫f = 0 over any triangle.

Question 7

How does FTC generalize?

Answer 7

• f cont. on open set
• path is PCD (call it y)

Then,

∫ f = F( y(b)) - F(y(b))

where F’ = f

Question 8

What do we need to write down a complex integral?

Answer 8

• cont f
• PCD path

then

∫ f = ∫ f(path) (path)’ dt as t = 0 to 1

(or domain of path)

Question 9

If f’ exists and f is defined on a

star-like set then…

Answer 9

F (the primitive)

exists!

Question 10

∫ f dz on a closed, pcd path

then …

Answer 10

= 0

b/c closed means path(a) = path(b)

then 0 by FTC

Question 11

Answer 11

Question 12

For f continous on open, as radius -> 0⁺,

what happens to

Answer 12

it goes to f(a)

“integral around circle converge to value at the center!”

Question 13

What does the Cauchy Integral Formula say…

Answer 13

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!”

Question 14

What more can we conclude from Cauchy Integral Formula?

Answer 14

f^(k) exists and equals…

Question 15

If diff fn –> f and

fn are locally bounded, then…

Answer 15

f’, f’’, exist and equal limit as expected

Question 16

Inside R, power series converges to…

Answer 16

analytic function

and can be differentiated term by term

Question 17

trick

log(1 + x)

for large x is bounded by what?

Answer 17

1 < log( 1 + x) < x

b/c exponential growth is faster than linear

Question 18

What’s the set of points in [z1, z2] ?

Answer 18

(1-t) z1 + t z2

for 0 ≤ t ≤ 1

Question 19

What’s the Weirstrauss M-test?

Answer 19

∑ Mn converges

==>

∑ fn converges uniformly

to a func f

bonus: if each fn is cont, then so is f

fn | ≤ Mn

Question 20

If differentiable, locally bounded fn —> f , then …

Answer 20

f^(k)_n —> f^(k)

happens uniformly on

• compact set
• disk (a, r/2) if bounded (not just locally)

Question 21

f analytic (= differentiable) on disk means…

Answer 21

f = taylor series

Question 22

Does f infinitely differentiable in R mean f = taylor?

Answer 22

NO!

e.g., e^{-1/x^2}…no series around 0

Question 23

What’s Monera’s Theorem?

Answer 23

f is analytic if it’s

continuous

∫ is 0 around any triangle

Question 24

Cauchy Estimate says

Answer 24

If |f| ≤ M for f analytic,

|fⁿ(a) | ≤ M n! / R^n

Question 25

What’s Liouville’s Theorem?

Answer 25

bounded & entire means constant!