Chap 4 integrals and intro to contours

Question 1

PG 126 def of contour itnegral
when to use

Use when:

• not closed C (rather use CG)
• no antiderivative exists (rather use EFTC)

Answer 1

$$\int_C f(z)dz = \int_a^b f(z(t)) \ z’(t) dt$$

or even $$\int_C f(z)dz = \int_a^b w(t) dz = \int_a^b u(t)dt \ + \ i\int_a^b v(t)dt $$

Question 2

Is the value of a lien integral 0 for a closed curve

Answer 2

No not always zero, it may or may not depend on the path

Question 3

Modulus and integrals inequality

Answer 3

$$| \int_a^b w(t) dt | \leq \int_a^b |w(t)| dt $$

Question 4

Contour upper bound on modulus of integral

Answer 4

Let C be a contour of length L, f a function
which is piecewise continuous on C and M a
constant such that |f(z)| ≤ M for all z on C
at which f is defined. Then

$$|\int_C f(z) dz | \ \leq \ ML$$

Question 5

simple closed curve

Answer 5

arc with no self crossings

Question 6

Differentiable arc

Answer 6

an arc z st z’ is continuous on [a,b]

Question 7

smooth arc

Answer 7

an arc z such that z’ is continuous on [a, b] and nonzero on (a, b).

Question 8

length of an arc

a differntiable arc

Answer 8

the length L of a differentiable arc
z : [a, b] → C is
$$L = \int_a^b |z’(t)| dt$$