MA 445 University of Alabama at Birmingham Math and The Quadratic Formula Exam Practice Please make sure the answers are accurate and clear because I have

MA 445 University of Alabama at Birmingham Math and The Quadratic Formula Exam Practice Please make sure the answers are accurate and clear because I have to study for me upcoming exam. Complex Analysis, MA-445/545 July 24, 2020
Test 2
Submission deadline: Saturday, July 25, 11:59 pm.
Submission type: PDF file upload on Canvas.
Each problem is 13 points
1. Solve exp(z 2 ) = exp(|z|2 ).
?
3
1
i.
2. Let z = +
2
2
(a) Find the set of values A of log(z).
(b) Find the set of values B of log(z 3 ).
(c) Is it true that B = 3A?
1
2
3. We know that all values of |(1 + i)z | are the same. What are the possible values of z?
4. Solve sin(z) = cosh(1).
3
5. Let P be the polygon whose vertices are the solutions of the equation z 8 + 1 = 0, and C be the
boundary of P , as a closed contour traversed counterclockwise. Find
Z
zdz.
C
6. Let
3z 2 + 1
,
z3 + z
f (z) =
traversed counterclockwise,

C? = z |z ? ?| = 4
F (?) =
Z
C?
Find F (1).
f (z)dz.
4
7. Find the set of all real numbers ?, so that F (?) is well defined (from the previous problem) and
F (?) = 6?i.
1
, and CR be the circle centered at 0 with radius R, traversed counterclockwise,
z4 + 1
and R > 1. Calculate
Z
f (z)dz,
8. Let f (z) =
CR
and explain what you do.

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