課程名稱︰複變函數論
課程性質︰數學系大三必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2007/12/27
考試時限（分鐘）：10：20～12：10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Read me: This exam contains six problem sets, however, the maximal grade is
100 points. Your answers should include all the necessary calculations and
reasoning. You are allowed to use the following fomulae:
π^2 ∞ 1
---------- = Σ --------
sin^2(πz) -∞ (z-n)^2 ,
1 1 1
πcot(πz) = ---- + Σ (----- + ----)
z n≠0 z-n n ,
π
Γ(z)Γ(1-z) = --------
sin(πz) ,
d Γ'(z) 1
----( ------- ) = Σ --------
dz Γ(z) n=0 (z+n)^2 .
Problems:
(1) Prove the following statements:
(a)(10points) If f(z) = A_0 + A_1(z) + ...+ A_n(z^n) +...,for |z|< R where
(n)
f (0)
R is a positive real number, then A_n = -------- .
n!
(b)(15points) If R_1 < R_2 are two positive real numbers and we have
1 1
A_0 +A_1(z) + ...+ A_n(z^n) +... = B_1 ----+...+B_n -----+...,
z z^n
for R_1 <|z|< R_2, then A_n = B_m = 0 for every n and m.
(2)(15points) The Cibonaffi numbers are defined by F_0 =1, F_1 =0,
F_n = F_n-1 - F_n-2 .
Show that if F(z) = F_0 + F_1(z) + ... +F_n(z^n)+..., then F(z) is a
rational function.
1
(3)(10points) Show that the Laurent development of ------------ at the
( e^z - 1 )
1 1 ∞ k-1 2k-1
origin is of the form --- - --- + Σ(-1) * A_k z .
z 2 k=1
∞ 1
(4)(10points) Compute the value of Σ --------- .
n=1 n^4
(5) Prove the following statements:
(a) (10points) If a is not an integer, then the product
∞ z
Π ( 1 + ------- ) e^{-z/(n+a)}
n=-∞ n+a
converges absolutely and uniformly on every compect set.
(b) (5points) We have
γ(z) ∞ z
sin π(z+a) = e Π ( 1 + ------- ) e^{-z/(n+a)}
n=-∞ n+a
where γ(z) is a entire function.
(c) (10points) We have γ(z) = πz cot(πa) + B
(d) (5points) In (c), we have e^B = sin(πa)
(6) Prove the following statements:
(a) (10points) We have
d Γ'(t) |
9 --- (-------- ) |
dt Γ(t) | t=3z
d Γ'(z) d Γ'(z + 1/3) d Γ'(z + 2/3)
= --- (-------- ) + --- (------------- ) + --- (------------- )
dt Γ(z) dt Γ(z + 1/3) dt Γ(z + 2/3)
(b)(10points) We have
Az+B
Γ(3z) = e *Γ(z)Γ(z+1/3)Γ(z+2/3)
Az+B 3z-1/2 1
(c)(10points) In (b), we have e = 3 * -----
2π
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