課程名稱︰複分析
課程性質︰數學系大三必修
課程教師︰蔡忠潤
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/01/14
考試時限（分鐘）：150
試題 :
[Total: 37 points] In what follows, Ω is always assumed to be an open and
connected proper subset of €.
(1) [7 points] True/False questions, no justifications needed.
￣￣￣￣￣￣￣￣￣
(a) Given any three distinct complex numbers of unit length, ζ , ζ , and
1 2
ζ , define
3
z 1
f(z) = ∫ ───────────────── dw
0 2/3
((w - ζ ) (w - ζ ) (w - ζ ))
1 2 3
for any z in the open unit disk. (Assume that the branch of 2/3-power is
suitably chosen.) Then, we can find ζ , ζ , and ζ such that the
1 2 3
1
image of f(z) is a right triangle .
(b) Suppose that f(z) is a function on ∂Ω, which is non-negative but not
necessarily continuous. Let u(z) be the harmonic function on Ω produced
2
from the Perron's method with f(z). Then, u(z) ≧ 0 for any z ∈ Ω.
(c) Suppose that ∂Ω is compact, and f(z) is a continuous function on ∂Ω.
Then, there exists a harmonic function u(z) on Ω, which extends
continuously to ∂Ω and which is equal to f(z) on ∂Ω.
︿
(d) For any α < -1, there exists an automorphism of € = € ∪ {∞}, f(z),
such that f(0) = 0, f(1) = 1, and f(-1) = α.
(e) Let Ω be the region enclosed by the heart curve, Ω = {(x, y) ∈ €|
2 2 3 2 3
(x + y - 1) - x y < 0}. Suppose that ψ(z) is an automorphism of Ω
with ψ(0) = 0. Then, ψ(z) must be the identity map.
1
(f) Consider Ｆ = {f (z) =───} on €. It is normal, in the sense of
n z + n n∈|N
︿
€.
(g) There exists a non-constant entire function f(z), whose image does not
contain the negative real axis.
∞ -2
(2) [5 points] Evaluate Σ n . Show your work.
n=1
Hint: The zeros of sin(πz) are exactly Ｚ. Since
￣￣
sin(πz) = sin(πx) cosh(πy) + i cos(πx) sinh(πy)
and
cos(πz) = cos(πx) cosh(πy) - i sin(πx) sinh(πy),
it is not hard to show that
1
─|sin(πz)| ≦ |cos(πz)| ≦ 2|sin(πz)|
2
when |y| ≧ 1000. Another formula you might need is that
∞ 2 2 -1 π
∫ (a + s ) ds = ─.
-∞ a
(3) [2 + 5 points] Suppose that Ω is simply-connected. Fix a point z ∈ Ω.
0
Let Ｆ = {f(z): injective and analytic on Ω| f(z ) = 0, f'(z ) > 0, |f(z)|
0 0
< 1 for any z ∈ Ω}.
(a) Why is Ｆ a normal family, in the sense of €?
(b) Suppose that there exists a function g(z) ∈ Ｆ such that g'(z ) ≧
0
f'(z ) for any f ∈ Ｆ. Prove that the image of g(z) is the (open) unit
0
disk. Namely, prove the surjectivity part of the Riemann mapping
theorem.
z - a 2
Hint: Given a ∈ D, let φ (z) = ───. Then, φ'(0) = 1 - |a| and
￣￣ a ─ a
1 - az
1 1 + |a|
φ'(a) = ────. Another fact is that ────> 1 for any a ∈ D\{0}.
a 2 ＿＿
1 - |a| 2 √|a|
(4) [5 points] Construct a harmonic function u(z) on the first quadrant, {z = x
+ iy ∈ €| x > 0 and y > 0} with the following property:
‧ u(z) extends continuously to the boundary except at the points 0 and 1;
‧ u(z) = 1 on the half-lines {y = 0, x > 1} and {x = 0, y > 0};
‧ u(z) = 0 on the segment {0 < x < 1, y = 0}.
↑...........
│...........
u=1│...........
│...........
└──┴──→
0 u=0 1 u=1
1
Hint: On the upper half space, ─ arg(z) is harmonic, and extends
￣￣ π
continuously to the boundary except at the origin. Its value is 0 on the
positive real axis, and is 1 on the negative real axis.
(5) [6 points] Let
Ｆ = {f(z): injective and analytic on Ω| f(z) ≠ 0 for any z ∈ Ω}.
︿
Prove that Ｆ is a normal family, in the sense of €.
(6) [3 + 4 points] Let f(z) be an entire function.
(a) Suppose that 0 is a pole of g(w) = f(1/w). Show that f(z) must be a
polynomial.
(b) Suppose that f(z) is not a polynomial. Then, given any w ∈ €, how
0
many roots does the equation f(z) = w have? Explain your reason.
0
Hint: You can think about what happens for f(z) = exp(z).
￣￣
────
1
That is to say, one angle of the triangle is π/2.
2
Namely, consider the supremum of certain subharmonic functions, u(z) =
sup v(z).
v∈Ｂ