Math 621 Solution of the Midterm Spring 2006
1. (10 points) Find the power series expansion of f(z) = 1
z+i, centered at z= 1.
What is its radius of convergence?
Answer: 1
z+i=1
(z−1)+1+i=1
1+i
1
(z−1
1+i)+1 =1
1+iP∞
n=0(−1)nz−1
1+in=
P∞
n=0
(−1)n
(1+i)n+1 (z−1)n.The radius of convergence is |1 + i|=√2.
2. (18 points) Compute the following integrals. Justify all your steps.
(a) ZC
e2iz +z2+ 3
(z−2)3dz, where C={z:|z|= 3}transversed counterclockwise.
Answer: Set f(z) = e2iz +z2+ 3. Then
ZC
e2iz +z2+ 3
(z−2)3dz = 2πi ·f(2) (2)
2! =πi((−4)e4i+ 2) = π[sin(4) + i(2 −4 cos(4))].
(b) I:= ZC
dz
z(1 −z),along the path Cconsisting of the line segments C1,C2,
C3, where C1goes from −2 to −i,C2goes from −ito 1 + i, and C3goes from
1 + ito 2.
Answer: Write 1
z(1−z)=1
z−1
z−1. We find primitives, for each summand, defined
in an open set containing C, by setting
f(z) = log(z) with arg(z)∈(−3π/2, π /2),
g(z) = log(z−1) with arg(z−1) ∈(−π/2,3π/2).
Thus, I=RCf0(z)−g0(z)dz = [f−g]2
−2={ln(2) −ln(1)} − {(ln(2) −iπ)−
(ln(3) + iπ)}= ln(3) + 2πi.
3. (18 points) Determine if the following statements are true or false. If true, prove
the statement. If false, give a counter example.
(a) Let f(z) and g(z) be two entire functions, such that |f(z)| ≤ |g(z)|, for every
complex number z. Then f(z) = cg(z), for some constant c.
Answer: True. If fis identically 0, take c= 0. Otherwise, g(z) is not
identically zero. The function h(z) := f(z)
g(z)has isolated singularities, since
the zeros of g(z) are isolated. Furthermore, h(z) is bounded, by assumption.
Hence, all the singularities of h(z) are removeable and h(z) extends to a
bounded entire function, which must be constant, by Liouville’s Theorem.
(b) Let fand gbe holomorphic functions on the upper-half-plane
H:= {x+iy :y > 0}. If f(i
n) = g(i
n), for all integers n≥1, then f=g.
Answer: False. Take f(z) = 1 and g(z) = e2π/z .
(c) Let U⊂Cbe an open connected subset. If both f(z) and f(z) are holomor-
phic functions on U, then fis constant in U.
Answer: True. Everybody did this part correctly.
4. (18 points) Find all linear fractional transformations, which map the upper half of
the unit disk U:= {z:|z|<1 and Im(z)>0}onto itself, and satisfy f(1) = −1.
Justify your answer. Hint: There are infinitely many such maps.
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