United Arab Emirates University
College of Sciences
Department of Mathematical Sciences
EXAM 1
Complex Analysis I
MATH 315 SECTION 01 CRN 23516
9:30 – 10:45 on Monday & Wednesday
Due Date: Monday, October 19, 2009
ID No: 200
Name:
Score: /70
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Compute - Complex Analysis - Exam, Exams of Mathematics
These are the notes of Exam of Complex Analysis and its key important points are: Compute, Express, Real, Exponential Form, Formula, True, False, Origin, Unit Circle, Possible Arguments
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United Arab Emirates University College of Sciences Department of Mathematical Sciences
EXAM 1
Complex Analysis I MATH 315 SECTION 01 CRN 23516 9:30 – 10:45 on Monday & Wednesday Due Date: Monday, October 19, 2009
ID No: 200
Name:
Score: /
1. (Total 10 points) Let z =
3 + i and w = 1 + i
(1.1) (5 points) Compute w − z, zw, and | z |. Express w/z in the form x + iy with real x and y.
(1.2) (5 points) Express z =
3 + i in the exponential form reiθ.
2. (5 points) Write down Euler’s formula and de Moivre’s formula.
3. (Total 5 points) Determine whether the following is true (T) or false (F).
(3.1) (3 points) If Im(z) > 0, then | z − i | > | z + i |.......................................
(3.2) (2 points) If z 6 = 0 lies inside the unit circle centered at the origin, then 1/¯z lies outside the circle................................................................................
7. (5 points) Find all 3rd^ roots of 1 + i, i.e., (1 + i)^1 /^3.
8. (Total 10 points) Solve the following conversion problems.
(8.1) (5 points) Write f (z) = xy + iy^2 in terms of z and ¯z, where z = x + iy.
(8.2) (5 points) Write f (z) = z^2 − ¯z^2 in the form u(r, θ) + iv(r, θ), where r and θ are the modulus and the principal argument of z, respectively.
9. (5 points) Sketch and find the image S′^ of the semi–infinite strip S given below under the trans-
formation w = f (z) = ez^ :
S = { z = (x, y) ∈ C | − 1 ≤ x ≤ 1 , 0 ≤ y ≤ π }.
United Arab Emirates University College of Sciences Department of Mathematical Sciences
EXAM 2
Complex Analysis I MATH 315 SECTION 01 CRN 23516 9:30 { 10:45 on Monday & Wednesday Due Date: Wednesday, November 25, 2009
ID No: 200
Name:
Score: /
1. (5 points) Show that f^0 (z) does not exist at any point for f(z) = ez^.
2. (5 points) Show that the function f(z) = e ^ cos (ln r) + ie ^ sin (ln r) is di erentiable in the
domain of de nition r > 0, 0 < < 2 , and also to nd f^0 (z).
3. (5 points) Show that u(x; y) = 2x x^3 + 3xy^2 is harmonic in some domain and nd a harmonic
conjugate v(x; y).
7. (5 points) Evaluate log
p 3 + i
.
8. (5 points) Find all values of z such that Log
z^2 i
= i 2 , where the Log represents the prin- cipal value of log.
9. (5 points) Find the principal value of the complex exponent
(^) e 2
1 i
p 3
^3 i , i.e.,
P: V:
(^) e 2
1 i
p 3
3 i :
10. (5 points) Use the de nition zc^ = ec^ log^ z^ to show that
1 + i
p 3
3 = 2 = 2
p