1 1. Expand f (z) = z(z − 1)(z − 2) in a Laurent series for the domain {z ∈ C | 1 < |z| < 2}. 1 2. Expand f (z) = z2 + 1 in a Taylor series at z = 2i. Compute f (2013)(2i). 3. (a) State Cauchy’s integral formula, (b) Evaluate C ez z2 − 3z + 2 dz, where C is the circle {z ∈ C | |z| = 32}, described in the counterclockwise direction. 4. Prove the identity cos(3z) = 4 cos3(z) − 3 cos(z). 5. Prove that the function f (z) = z z 2 is differentiable at z = 0 but it
Trang 1| |
∞
0
}
{ ∈ | | | }
∫
Final Exam
(3 hours) Applied Complex Varibles, Fall 2013
1
1 Expand f (z) = 1 < |z| < z (z − 1)(z − 2) in a Laurent series for the domain {z ∈ C |
2}
1
2 Expand f (z) = z2
+ 1
in a Taylor series at z = 2i Compute f (2013)(2i).
3. (a) State Cauchy’s integral formula,
(b) Evaluate
C
e z
z2 − 3z + 2 dz , where C is the circle {z ∈ C | |z| = 3/2},
described
in the counterclockwise direction
4 Prove the identity cos(3z) = 4 cos
3(z) − 3 cos(z).
5 Prove that the function f (z) = z z
2 is differentiable at z = 0 but it is not
analytic at
z = 0
6 Suppose that z0 is a pole of order m (m ∈ N
∗) of function f (z).
(a) Prove that there exists an analytic function ϕ defined on a
neighborhood of z0
satisfying ϕ(z0) ƒ= 0 such that
ϕ (z)
f (z) =
(z − z ) m
(b) Show that lim f (z) =
z→z0
7 Let C ρ be the semi-circle z C z 1 = ρ, Im(z) 0
(ρ > 0), described in the clockwise direction Show that
lim
ρ→0
+
∫
C
ρ
1 +
z − 1
e z
z + 1
Σ
dz = −πi
8 Let C ρ be the circle z C z = ρ (ρ > 0), described in the
counterclockwise direction Find the values of the following integerals:
C2 1
Trang 2z2013 +
1 dz
Trang 3−
(b)
(c)
∫
C2
e z
(z2 + 1)(z + 3) dz
∫
(1 + z + z2)(e 1
+ e1
)dz.
C2
9. Calculate the following
integrals: (a)
+∞ x4 + x1 2 + 1 dx (b)
0 +∞ x2 + 1
x4 + 1 dx (c)
0 +∞ sin( x )
dx
(d
)
(e
)
, where −1 < a <
1
+∞ x a
x2 + 4 dx
+∞ ln( x )
x2 + 4 dx (f)
0
+∞ cos( x )
x2 + 9 dx (g)
0 +∞
sin( x )
x2 + x + 1 dx
(h)
−∞
+∞ x sin( x )
x2 + 16 dx (i)
0 +∞ 1 cos( x )
0
10.Find the poles and residues of the following functions:
∫
∫
∫
∫
∫
∫
∫
∫
0
Trang 4(b)
(c)
(d)
(e
)
1
f1(z) =
(z2 + 1)2 1
f2(z) =
cos(z)
z
f2(z) =
sin(z)
1
f3(z) =
cos2(z)
f4(z) =
e z − 1 − z .
(f)
f (z) = tan(z)
11.Find the values of the following integerals:
(a)
2π 2 + cos( x )
dx
0 2 + sin(x)
(b)
∫ 2π
dx
0 2 + sin(x)
(c)
∫ 2π
dx
0 2 − cos(x)
dx
1 + a cos(x) (−1 < a < 1)
(e
)
0
π
dx
a2 + sin2(x) (a > 0)
(f)
0
∫ π
dx
12.State Rouch´e’s
theorem
0 4 + sin 2(x)
∫
∫
∫
Trang 5{ ∈ | }
13.Determine the number of zeros, counting multiplicities, of the following
polynomials:
(a) z2013 + 2015z2015 + 2011z2011 + 2 in {z ∈ C | |z| < 1}
(b) z3 + 4z2 − 2z + 3 in {z ∈ C | |z| < 2}.
14.Determine the number of roots, counting multiplicities, of the following
equations:
(a) z2013 + 6z3 + z + 3 = 0 in the annulus {z ∈ C | 1 < |z| < 2}
(b) 0.9e −z − 1 = 2z in the domain {z ∈ C | |z| < 1, Rez < 0}.
15 Find the linear fractional transformation that maps the points z = 0, z 1 = 1, z2
3 = i
onto the points w1 = 0, w2 = i, w3 = 1
16 Find the image of the upper half plane H = {z ∈ C | Im(z) > 0} under the
transfor- mation w = i 1 + z 1 − z
17 Find the image of the semi-infinite trip D = z C 0 < Re(z) < π,
Im(z) > 0 under the map w = cos(z).
18 Find a one to one mapping w = f (z) that maps the domain D = {z ∈ C
| Im(z) >
0, |z| < 1} \ [i/2, i] onto the upper half plane H = {w ∈ C | Im(w) > 0}.
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