MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION D)
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 = sin z.
Date: February 19, 2002.
1
2
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION D)
2. Calculate
in the form x + iy.
3−i
3+i
MATH 3160: APPLIED COMPLEX VARIABLES
3. Show that
TEST #1 (VERSION D)
y
+ y2
is harmonic and find a harmonic conjugate v for it.
u=
x2
3
4
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION D)
4. At which points z does the function
f (z) = cos x cosh y − i tan x sinh y
satisfy the Cauchy–Riemann equations? Explain your answer.
MATH 3160: APPLIED COMPLEX VARIABLES
TEST #1 (VERSION D)
5
5. What is the region Re(1/z) < 1 in rectangular coordinates?
Describe it algebraically, and sketch it.
6
MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION D)
6. (a) Find all of the values of
sin−1 (z)
(the inverse
of the sine function, not the reciprocal) 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 D)
7
7. BONUS: State Stokes’ theorem in the plane (also known as
Green’s theorem, Gauß’s theorem, or the divergence theorem).