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MTH 365

Fall 2018

Final

Name:

Section: 21

Instructions

1. Read the directions carefully.

2. Please write neatly in pencil and show all your work.

3. Use the appropriate notation.

4. There are 10 problems, each worth 20 points. Please box your final answer.

5. Please turn off your cell phone and all other electronic devices.

6. Please do not share your calculator during the exam.

7. Do not use decimals on any intermediate step unless stated otherwise.

8. If you have the trouble during the exam, feel free to ask me for help.

Pg 1

Pg 2

Pg 3

Pg 4

Pg 5

Pg 6

Pg 7

Pg 8

Pg 9

Pg 10

Total

1. Let f (z) = z 2 − 2z.

a. Express f as f (z) = u(x, y) + iv(x, y).

b. Express f as f (z) = u(r, θ) + iv(r, θ).

c. Plot z1 = 1 + 2i in the xy-plane and w1 = f (z1 ) in the uv-plane.

2. Evaluate the following.

a. Evaluate lim (4 cos(x) + 3y 2 + i2ex ), if it exists. If the limit does not exist, state why.

z→2i

b. Evaluate lim Arg(z), if it exists. If the limit does not exist, state why.

z→1

c. Determine if f (z) =

z

is continuous at the given point z0 = 4. If not, state why.

(z − 4)3

3. Evaluate the following.

a. Show that f (z) = |z|2 is not differentiable at any point z 6= 0.

z7 + i

, if it exists. If the limit does not exist, state why.

z→i z 14 + 1

b. Evaluate lim

c. Determine the points at which f (z) =

2 cos(z)

is not analytic.

z 2 + 4iz

4. Let f (z) = x3 + 3xy 2 + i(y 3 + 3×2 y).

a. Determine the values where the Cauchy-Riemann equations are satisfied. If f is analytic, find the appropriate domain. If not, state why.

b. Use the Cauchy-Riemann equations to determine where f is differentiable and find a

formula for f 0 .

5. Let u(x, y) = xy 3 − x3 y.

a. Show that u is harmonic on an appropriate domain D

b. Find the harmonic conjugate of u.

c. Find the analytic function f (z) = u(x, y) + iv(x, y).

6. Evaluate the following integrals.

I

1

√ dz, where C is the circle |z| = 1

a.

z

C

I

b.

C

I

c.

C

z2

dz, where C is the circle |z| = 2

z+4

sin(πz)

dz, where C is the circle |z| = 5

2z − 3

7. Express the given function in terms of a power series centered at the indicated point z0 .

Give the radius of convergence R of each series.

a. f (z) =Ln(1 + 3z), z0 = 0

b. f (z) = ez , z0 = 2

8. Evaluate the following.

a. Rewrite f (z) =

1

as a Laurent series on 0 < |z − 1| < 2.
(z − 1)2 (z − 3)
b. Determine the order of the poles of f (z) =
sinh(z)
.
z 4 (z − 6)
Z
9. Evaluate
0
2π
1
dθ.
(cos(θ) + 2)2
Z
∞
10. Evaluate the Cauchy principal value of
−∞
x
dx.
(x + 2)(x2 + 1)
...
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