MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION C)
Name:
This test has 7 pages and 7 problems.
No calculators are allowed. You are permitted only a pencil or pen.
Work out everything as far as you can before making decimal approximations.
The Cauchy–Riemann equations are
∂u
∂v
=
∂x
∂y
∂u
∂v
=−
∂y
∂x
1. Find all solutions z of the equation
tan z = 1.
Date: June 11, 2001.
1
2
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION C)
2. Calculate
in the form x + iy.
1−i
2+i
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION C)
3. Show that
u = 2x(1 − y)
is harmonic and find a harmonic conjugate v for it.
3
4
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION C)
4. At which points z does the function
f (z) = sin x2 (1 + i)
satisfy the Cauchy–Riemann equations? Explain your answer.
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION C)
5. Show that the derivative of
f (z) = 3z̄ + 2z
does not exist at any point of the plane.
5
6
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION C)
6. (a) Find all of the values of
sin−1 (z)
√
at z = 1/ 2.
(b) Find all of the values of
d
sin−1 (z)
dz
√
at z = 1/ 2.
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION C)
7
7. BONUS: State Stokes’ theorem in the plane (also known as
Green’s theorem, Gauß’s theorem, or the divergence theorem).