課程名稱︰複變數函數論
課程性質︰數學系大三必修
課程教師︰王金龍
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/01/08
考試時限（分鐘）：12:30 ~ 3:20
試題 :
There are four problems, each problem deserves 25 points.
1. Define the gamma function Γ(s) for Re(s) > 0 and prove its analytic
continuation to s∈C and functional equation Γ(s)Γ(1-s) = π/sinπs. Find
its zeros and poles as well as the corresponding residues. Hint: Prove first
that a-1
∞ v π
∫ ———dv = ————.
0 1+v sinπa
2. (a) Define the Riemann zeta function ζ(s) for Re(s) > 1 and discuss its
analytic continuation to s∈C with the only simple pole at s = 1. (b) Show
that ζ(s)≠0 for Re(s) > 1, and all zeros of ζ(s) for Re(s) < 0 are
precisely s = -2n, n∈N. (Hint: On way to do it is to consider ξ(s) :=
-s/2
π Γ(s/2)ζ(s) for Re(s) > 1 and prove its functional equations ξ(s) =
2
∞ -πn t
ξ(1-s) through the one for Θ(t) := Σ e where t > 0.)
n=-∞
3. (a) Prove the Schwarz Lemma for f : D→D with f(0) = 0 and use it to
determine the group Aut(D). (b) Assuming the Schwarz-Christoffel formula
from H to a polygon domain P. Duduce the formula for a conformal map f:D→P
by way of an explicit conformal map g:D→H.
2 3
4. (a) Show that (ρ') = 4ρ - g ρ - g for some g , g ∈C. (b) Let Ω⊂C be a
2 3 2 3
3
simply connected domain not containing any root of 4x - g x - g . Show that
2 3
ω
ds
I(ω) := ∫ ——————————, ω, ω∈Ω
3 0
ω √(4s - g s - g )
0 2 3
defines an inverse of ρ(z+a) for some a.
(Bonus) Write down something you have well prepared but not shown above.