Chaper 6: Residues and Poles

Question 1

Isolated singularity and singular point

Answer 1

point z0 is a singular point of f if f is not analytic at z0, but f is analytic at some point in every neighbourhood of z0.

The point z0 is an isolated singularity of f if f is not analytic at z0, but f is analytic in some deleted neighbourhood of z0.

Question 2

Residue

Answer 2

Residue of f at z0 is
Res$(f,z_0) = b_1$
where b1 is the coefficient of the first negative power term $\frac{1}{z-z_0}$in the LS expansion of f

Question 3

Residues and integrals

Answer 3

$$\int_C f(z)dz = 2 \pi i Res(f,z_0)$$

z0 is interior to C,
sing at z0
C is SCC
f is ana on and inside C\z0

Question 4

Cauchy’s residue theorem

Answer 4

Let C be a + SCC and f is analytic inside and on C except for a finite number of (isolated) singularities zk (k = 1, 2, . . . , n) inside C. Then
$$\int_C f(z)dz = 2 \pi i \sum_{k=1}^n Res(f,z_k)$$

Question 5

def of residue (w/ integral)

Answer 5

$$ Res(f,z_0) = \frac{1}{2 \pi i} \int_C f(z)dz $$

Question 6

Principal part of LS representation of a function

Answer 6

the - powers part
i.e.
$\sum_{n=1}^\infty \frac{b_n}{(z-z_0)^n}$
is the principal part of f at $z_0$

Question 7

three types of isolated singularities

Answer 7

Removable
essential
pole of order m

Question 8

z0 is removable

singularities OF F

Answer 8

if bn = 0 for all n ∈ N,

note the residue at a removable singularity is zero

Question 9

z0 is an essential singularity OF F

Answer 9

if bn is nonzero for an infinite number of n,

Question 10

z0 is a pole of order m of f

simple pole realtion here

Answer 10

if m ∈ N and
$$b_m \not= 0, \qquad b_{m+1} = b_{m+2} = · · · = 0$$

If m = 1, then z0 is called a simple (enkelvoudige)
pole of f.

Question 11

Thm pg 243,

Let z0 be an isolated singularity of a function
f. If m ∈ N, then TFAE

Answer 11

(a) z0 is a pole of order m of f,
(b) $\exists R > 0$ st $f(z) = \frac{\phi(z)}{ (z − z_0)^m}$ on D’(z0, R), where $\phi$ is analytic and nonzero at z0.

Moreover, if these statements are true, then
Res(f, z0) = $\frac{\phi ^{(m−1)}(z_0)}{(m − 1)!} $.

Note that if m = 1, then Res(f, z0) = $\phi (z_0)$

Question 12

zero of order m at z0

Answer 12

Suppose that a function f is analytic at z0.
If $f(z_0) = 0$ and $ \exists m \in N$ st $f(z_0) = f’(z_0) = f’‘(z_0) = . . . = f^{(m-1)}(z_0) = 0$
and $f ^{(m)}(z_0) \not = 0$ , then f is said to have a zero of order m at z0

Question 13

thm pg 248 TFAE for zeroes

Answer 13

Let f be a function that is analytic at a point z0. If m ∈ N, then TFAE:

(a) f has a zero of order m at z0,

(b) f can be written in the form
$$f(z) = (z − z0)^m g(z)$$
where g is analytic and nonzero at z0.

Question 14

analyticicties and zeroes and diskss (thm 2 pg 249)

Answer 14

if f is analytic and not identically zero on any open disk, then f has only isolated zeros.

Question 15

relate zeroes and poles

Answer 15

Let p and q be functions that are analytic at a point z0 st $$ p(z_0) \not = 0$$ and q has a zero of order m at z0. Then $\frac{p}{q}$ has a pole of order m at z0.

Hence if q is analytic at z0 with a zero of
order m at z0, then $\frac{1}{q}$ has a pole of order m at z0.

Question 16

thm 2 pg 251, simple poles trick

Answer 16

Let p and q be analytic at a pt z0. If $$p(z_0) \not = 0, \qquad q(z_0) = 0 \ and \ q’(z_0) \not = 0$$ then z0 is a simple pole of $\frac{p}{q}$ and $$ Res \left( \frac{p}{q} , z_0 \right) = \frac{p(z_0)}{q’(z_0)}.$$

Question 17

If z0 is a removable singularity of a function

f, what can we say about neighbourhood

Answer 17

f is bounded and analytic in some

deleted neighbourhood of z0

Question 18

Suppose that a function f is bounded and analytic in some deleted neighbourhood of z0. what can we say about sing at z0

Answer 18

f has at most a removable singularity

at z0.

Question 19

limits and isolated singularity (removable specifially)

Answer 19

If f has an isolated singularity at z0, then z0 is a removable singularity of f if and only if
$\lim_{z \rightarrow z_0} f(z)$ exists.

Question 20

poles and limits

Answer 20

If z0 is a pole of a function f, then

$$\lim_{z \rightarrow z_0} f(z) = \infty$$