Answers to Final Exam Math 403(02), Spring 2020, Dr. Z.

NAME: Jason Luo

EMAIL: jl2362@scarletmail.rutgers.edu

MEMBER: SCC: No (pick one)

1(a) (2 points): R is not open, because it contains the boundary {z: |z|=5}

1(b) (2 points): R is not closed, because it does not contain the boundary {z: |z|=7}

1(c) (2 points): R is connected, because there exists a path between any 2 points in R
that still lies in R.

1(d) (2 points): R is not simply connected, because its complement is not connected 
(see the images on the handwritten work)

1(e) (2 points): The interior is {z: 5 < |z| < 7}. It is obtained by replacing
all the inequalities with strict inequalities (<, >).

1(f) (2 points): The closure is {z: 5 <= |z| <= 7}. It is obtained by replacing
all the inequalities with weak inequalities (<=, >=).

1(g) (3 points): The boundary is the union of {z: |z|=5} and {z: |z|=7}.
It is obtained by subtracting the interior from the closure (set subtraction).

I checked for nonsense, and could not find any.
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2(a) (8 points) f is not analytic because the answer for the derivative is different
depending on the direction of approach (imaginary axis -> -2z' vs real axis -> 2z') 

2(b) (7 points) f is not analytic because the Cauchy-Riemann equation u_y = -v_x
fails to hold for y=\=0.

I checked for nonsense, and could not find any.
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3. (15 points): The root(s) is (are): z = 1+i

I checked for nonsense, and could not find any.
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4. (15 points): The change in argument is: 2pi

I checked for nonsense, and could not find any.
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5(a): (12 points): Max. value= 64       ; Location= -3i

5(b): (3 points):  Min value=  0       ; Location= i

Explanations are in the handwritten work

I checked for nonsense, and could not find any.
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6(a) (3 points): 1/2

6(b): (8 points): -1

6(c): (3 points): 1/2

6(d): (1 point): 1/2 + 1/2 - 1 = 0, so Cauchy's Theorem is verified for this special case.

I checked for nonsense, and could not find any.
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7 (15 points): -10pi - 4pi*i

I checked for nonsense, and could not find any.
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8a) (8 points): pi

8b) (7 points): Not Applicable (tanz not analytic on |z-pi/4|<=3, ex z = pi/2)

I checked for nonsense, and could not find any.
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9a) (5 points): Converges by Ratio test (see handwritten work)

9b) (10 points): i

I checked for nonsense, and could not find any.
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10a) (5 points): f(D) = {z : |z|<1}

10b) (10 points): NOT one-to-one; consider z_1 = 1/2, z_2 = -1/2

I checked for nonsense, and could not find any.
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11. (15 points) -8i - 12(1/10) + 6i(1/10)^2 = -6/5 - i(397/50)

I checked for nonsense, and could not find any.
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12. (15 points) z = 2^{1/6} * (cos(pi/12) + i*sin(pi/12))
only one zero of f lying in the first quadrant

I checked for nonsense, and could not find any.
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13a) ( 4 points): cos(z) doesn't work because cos(z) is not bounded over the upper half plane

13b): (16 points): Use integral of e^{iz}/(z^2+100). Applying the Residue Theorem,
the value of the original integral is then pi/(10 * e^{10})

I checked for nonsense, and could not find any.
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