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課程名稱︰複變函數論
課程性質︰必修
課程教師︰余正道
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/01/11
考試時限（分鐘）：12:50 - 15:20
試題 :
There are six problems (I) - (VI) in total; some problems contain
sub-problems, indexed by (1), (2), etc.
(I) [30%] Let f(z) = cos(πz).
(1) Show that f(z) is of growth order 1.
(2) Find the Hadamard product of f(z).
(3) Prove that
2˙2 6˙6 (4m+2)˙(4m+2)
√2 = ───˙───˙…˙────────˙….
1˙3 5˙7 (4m+1)˙(4m+3)
(II) [15%] Find the Hadamard product for e^z - 1.
(III) [20%] Prove that for |z|<1, we have
∞
(1) the product Π[1+z^(2^k)] = (1+z)(1+z^2)(1+z^4)(1+z^8)…converges, and
k=0
(2)
∞ 1
Π[1+z^(2^k)] = ───.
k=0 1 - z
(IV) [10%] Show that the equation e^z - z has infinitely many solutions in C.
(V) [15%] Let Γ(z) be the gamma function, which is a meromorphic function
on C. The Gauss multiplication formula is the following equation
n-1 z+k
Π Γ(───) = [√(2π)]^n-1˙(√n)^1-2z˙Γ(z) (n∈N). (◎)
k=0 n
For n=2, it reduces to the Legendre duplication formula. In this
problem, we will prove the case n=3.
Let F(z) = Γ(z/3)Γ([z+1]/3)Γ([z+2]/3).
F(z)
(1) Show that ── is an entire function (i.e. this meromorphic
Γ(z)
function can be extended to a holomorphic function on C) and is
nonzero everywhere.
(2) Show that
d ╭ F'(z)╮ d ╭Γ'(z)╮
──│───│ = ──│───│ for z≠0,-1,-2,….
dz ╰ F(z) ╯ dz ╰ Γ(z)╯
(3) Prove the formula (◎) for n=3. [Hint: (1) shows that
F(z)=e^g(z)˙Γ(z). Use (2) to find out g(z).]
(VI) [10%] Let Γ(s) and ζ(s) be the gamma function and the Riemann zeta
function, respectively; they are meromorphic functions on C.
(1) Show that for m∈Z, we have
╭ 1˙3˙5…(2m-1)√π
│ ────────── if m>0
│ 2^m
│
Γ(1/2 + m) = ＜ √π if m=0
│
│ 2^(-m)
│(-1)^m ─────────√π if m<0.
╰ 1˙3˙5…(-2m-1)
ζ(2n) ζ(2)
(2) Show that ───,n∈N, are rational numbers (e.g., ─── = 1/6∈Q).
π^2n π^2
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