MATH 3160-1
NAM E
PRACTICE FINAL EXAM
1 + cos(2θ)
1 − cos(2θ)
, sin2 (θ) =
.
2
2
√
√
2/2,
cos(π/3)
=
1/2,
sin(π/3)
=
3/2, cos(π/6) =
cos(π/4)
=
sin(π/4)
=
√
3/2, sin(π/6) = 1/2, cos(π/2) = 0, sin(π) = 0, sin(π/2) = 1, sin(0) = 0,
cos(0) = 1.
cos2 (x) + sin2 (x) = 1, cos2 (θ) =
If f (z) is differentiable and f = u + iv then ux = vy , and uy = −vx ; in polar
form rur = vθ , and uθ = −rvr .
If f ∈ A(DR (z0 )), then f (z) =
∞
X
f (n) (z0 )
n=0
If f ∈ A(D0,R (z0 )) then f (z) =
1
bn =
2πi
n!
1
X
(z − z0 )n , for all z ∈ DR (z0 ).
∞
X
bn
+
an (z − z0 )n , where:
n
(z
−
z
)
0
n=−∞
n=0
Z
f (w)
dw;
−n+1
C (w − z0 )
Z
1
f (w)
an =
dw;
2πi C (w − z0 )n+1
and the series converges to f (z) for all z ∈ D0,R (z0 ).
∞
X
w2n+1
sin(w) =
(−1)
; for all w ∈ C.
(2n + 1)!
n=0
cos(w) =
∞
X
n
(−1)n
n=0
w
e =
∞
X
wn
n=0
n!
w2n
; for all w ∈ C.
(2n)!
; for all w ∈ C.
log(1 − w) = −
∞
X
wn
n=1
n
; for all |w| < 1.
∞
X
1
=
wn ; for all |w| < 1.
1 − w n=0
2
1. Find the image of D = {z ∈ C : |z| > 2 } under the transformation w = z 2 − 1.
Hint: do this in two stages: first consider the image of D under w1 = z 2 ; call it
D1 . Then find the image of D1 under w2 = z − 1.
2. Find Log([1 +
√
3i]−20 ) where Log denotes the principal logarithm.
3. Show that u(x, y) = 2x(1 − y) is harmonic and find the harmonic conjugate
v(x, y) that satisfies v(0, 0) = −1.
4. Evaluate
|z| = 1.
R
C
ez − z̄ dz where C is the simple closed positively oriented contour
e(z−iπ) − 1
5.a. Determine the value of f (iπ) that makes f (z) =
, z 6= πi, continuous
z − iπ
at πi.
Z
5.b. Evaluate
f (z) dz where f (z) is as in part(a) and C is the positively oriented
C
simple closed contour |z| = 6.
z
valid on the
+ 1)(z + 1)
annular region 1 < |z| < ∞. DO NOT simplify your answer!
6. Find the the Laurent series of the function f (z) =
7. Find the Laurent series of f (z) = z
2
sin( z12 )
(z 2
Z
then calculate
2
z sin
|z|=1
1
z2
dz.
Explain your answer!
8. Determine the type of the singularity of f (z) =
Z
9. Evaluate the integral
|z|=3
1 − cos(z 5 )
at z = 0.
sin(z 3 )
(z − 1)(z − 3 + 4i)
dz if the circle is positively oriz 2 (z + 2i)
ented.
I
10. Evaluate
|z|=2
Z
∞
11. Evaluate
−∞
e2z
dz, if the circle is positively oriented.
cos(πz)
cos(x) dx
.
(x + 1)2 + 4