[試題] 105-1 蔡宜洵 複分析導論 期末考
課程名稱︰複分析導論
課程性質︰系必修
課程教師︰蔡宜洵
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/1/12
考試時限（分鐘）：120
試題 :
Total points: 120
1. (20 points) Let α ∈ R - Z. Show that
ｚ z
Π ( 1 + －－－ ) exp ( - －－－ )
n∈Z n + α n + α
converges for all z ∈ C.
2. (20 points) Find the linear transformation which carries the line
{ z : Re z = Im z } into the circle | z + i | = 2, the point i into
the origin, and the origin into i.
2
3. (20 points) Let Ω be a domain in C (identified with R ) and u(x,y)
be a harmonic function defined on Ω. Show the following.
a) If Ω is simply connected, then a conjugate harmonic function of u,
defined on Ω, necessarily exists.
b) Suppose Ω is not simply connected. Give an example, including a proof,
in which a conjugate harmonic function defined on Ω does not exist.
2
4. (20 points) Let u(x,y) be a harmonic function on the entire R identified
with C. Suppose that |u| is bounded, i.e. sup |u(x,y)| is finite.
C
Show that u is a constant function. (hint: try Schwarz formula and ordinarly
Liouville theorem for bounded holomorphic functions).
s+1
c+i∞ x ζ'(s)
5. (15 points) Let ψ (x) = ∫ --------( - ------ ) ds, x, c > 1.
1 c-i∞ s(s+1) ζ(s)
By evaluating this integral by deforming the contours, show that ψ (x)
1
2
is asymptotically equal to x / 2 (as x → ∞). Note that the estimates
on the norms of the Riemann zeta function ζ(s), ζ'(s) and 1/ζ(s)
ε
(all ≦ C |β| where β = Im s ≧ 1 or ≦ -1, 1 ≦ α ≡ Re s ≦ c,
ε
with any given ε > 0) can be used without proof. The fact that ζ(s) has
no zeros along Re s = 1, can also be assumed without proof.
-s/2
6. (15 points) Let ξ(s) = π Γ(s/2)ζ(s) where Γ(s) and ζ(s) are
∞ -x s-1
Gamma function ( = ∫ e x dx for Re s > 0) and Riemann zeta function
0
2 2
∞ -πn x 1 ∞ - πn /x
respectively. Assume the identity Σ e = ----- Σ e , x > 0.
n=-∞ √x n=-∞
Assume also the interchangeability of the summation and integration.
Show that the functional equation ξ(s) = ξ(1-s) holds, Re s > 1.
7. (10 points) Let f(z) be a holomorphic function defined on the domain
2
Ω = { 0 < |z| < 1+ε }, ε > 0 small. Suppose ∫∫ |f(z)| dx dy < ∞.
Ω
Show that
a) f(z) can be extended as a holomorphic function across z = 0
(i.e. z = 0 is removable). (hint: try Laurent series and polar coordinates
in the double integral.)
b) Instead of f(z), consider a harmonic function u(x,y) satisfying similar
conditions. In this case, does the similar conclusion hold for u? If not,
give a counterexample.
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