MATH 3160-1
PRACTICE EXAM 1-Sp12
cos2 (x) + sin2 (x) = 1, cos(θ ± φ) = cos(θ) cos(φ) ∓ sin(θ) sin(φ), sin(θ ± φ) =
sin(θ) cos(φ) ± cos(θ) sin(φ)
√
√
cos(π/4)
=
sin(π/4)
=
2/2,
cos(π/3)
=
1/2,
sin(π/3)
=
3/2, cos(π/6) =
√
3/2, sin(π/6) = 1/2, cos(π/2) = 0, sin(π) = 0, sin(π/2) = 1, sin(0) = 0,
cos(0) = 1.
If f (z) is differentiable and f = u + iv then ux = vy , and uy = −vx ; in polar
form rur = vθ , and uθ = −rvr .
1. Find the real and imaginary part of z =
z.
2. Find all fourth roots of
2i
1+i
4
, |z|, the exponential form of
1
√ . Write your answer in rectangular coordinates.
1+i 3
3. Find the image of the region |z| > 3 under the transformation w = iz + 1.
4. Find the following limits:.
a. lim
iz + 3
.
+1
z→i iz
iz 2 − z + 1
.
z→∞ z 2 + 2z + i
b. lim
5. Where (if anywhere) is the function f (z) =
|z|
;
|z| + 1
a. differentiable?
b. analytic?
6. Using the rules for derivatives find the derivative of f (z) = (z 2 + 1)10 (DO NOT
simplify you answer).
8. Determine where the function f (x + iy) = sin(x) cosh(y) + i cos(x) sinh(y) is
differentiable.
9. Determine where u(x, y) = (x − 1)3 − 3xy 2 + 3y 2 is harmonic and on the largest
domain contained in that set find a harmonic conjugate of u that satisfies the
condition v(0, 0) = 0.