Chapter 7: More on int over para curve of f

Question 1

DEF 7.1

PRIMITIVE

Answer 1

Let f be a complex-valued function defined on a region D. A function g analytic on D and such that g’ = f at all points of D is said to be a primitive of f on D
(i.e., a primitive is an indefinite integral).

NO NEED TO PARAMETRIZE

Question 2

Thus, if γ is ANY path in D, given by z = z(t) for a ≤ t ≤ b, then

Answer 2

∫_γ f(z).dz = ∫_γ g'(z).dz=∫_[a,b] g'(z(t))z'(t) .dt 
=  ∫_[a,b] d/dt  [g(z(t))] .dt 
=  [g(z(t))]a to b
= g(z(b)) - g(z(a))   
=[g(z)]_γ

d/dt[g(z))] = g’(z(t))dz/dt
and [g(z)]_γ is used for the value of g at the final point of γ minus the value of g at the initial point of γ

Question 3

if _γ is a contour and..

Answer 3

Note that if γ is a contour, then z(a) = z(b), and so
∫_γ f(z)dz = 0.

So if γ is a contour
in a region and f has a primitive in the region then

∫_γf(z)dz = 0.

Question 4

EXAMPLE: Evaluate ∫_γ z .dz where γ is the path consisting of a line segment from 0 to 1 and
then a line segment from 1 to 1 + i.

Answer 4

z^2/2 is a primitive for z on the whole of C and γ is a path

HENCE
∫_γ z .dz = [z^2/2]_γ = (1+i)^2 /2 - 0^2/2 =i

Question 5

EXAMPLE: Evaluate

∫_γsin z dz along the line segment from 1 to i.

Answer 5

-cosz is a primitive for sinz on C γ is a path

HENCE

∫_γsin z dz = [-coz]_γ =-cosi - (-cos1) = cos1-cosh1

by cosiz = coshz

Question 6

EXAMPLE:
Evaluate
∫_γ sin z dz where γ is the contour z = e^it (0 ≤ t ≤ 2π).

Answer 6

-cosz is a primitive for sinz on C γ is a CONTOUR

HENCE
∫_γsin z dz =[-cosz]_γ =0

shape of contour doesnt matter
∫_γsin z.dz=0 for all contours in C

Question 7

This works for

∫_γ f(z) dz whenever f has a primitive on C and γ is a contour.

Answer 7

∫_γ f(z) dz =0 for CONTOUR

if f has a primitive ON C

Question 8

Lemma 7.2

mod and ∫ for complex valued funct

Answer 8

Suppose that g : [a, b] → C is continuous (so g is a function of a real variable,
but g(t) is complex- valued). Then 
|∫_[a, b]   of  g(t)dt| ≤ ∫_[a, b]   of  |g(t)| dt

proof by mod arg form and swapping for real parts

Question 9

THM 7.3

M-L estimate

by the lemma

Answer 9

Suppose that f is continuous on a path γ. Let γ have length L and suppose that |f(z)| ≤ M on γ. Then
| ∫_γ f| ≤ ML .

Question 10

EXAMPLE:

Let γ be a line segment lying within D = {z ∈ C : |z| < 1}. Estimate
∫_γ (Re z + z^2)/ (3 + zbar ) dz.

Answer 10

Clearly by diagram length L of any straight line segment in the disc {z ∈ C : |z| < 1} is less than 2, diameter.

for all z in D

|Rez + z^2| ≤ |Rez| + |z^2| by the triangle inequality
≤ 1 + 1 = 2

|3 + zbar| ≥ | |3| -|zbar|| ≥ 3 -1 = 2

Hence for all z in D
|(Re z + z^2)/ (3 + zbar )| ≤ 2 * 0.5 =1

So we can take M=1 and by thm 7.3

|∫_γ (Re z + z^2)/ (3 + zbar ) | ≤ 2