課程名稱︰應用數學三
課程性質︰系必修
課程教師︰賀培銘 教授
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2009/04/16
考試時限（分鐘）：180
是否需發放獎勵金：是！
試題 :
(total grades = 110)
[如果答案正確，沒有計算過程不會扣分；但答案未化約到最簡形式可能會酌量扣分]
∞ -ax
1.(10%) Let f(a) = ∫ dx e . Which one (ones) of the following can be (and
0
only be) defined by analytic continuation?
(a) f(1); (b) f(-1); (c) f(0); (d) f(-1-i); (e) f(1+i).
2.(5%) Afunction f(z) with n simple poles at z (a = 1,…,n) is analytic
a
everywhere else on the complex plane. (Let the residue at z bedenoted R .)
a a
f(z) approaches to a constant C as z → ∞. Find the explicit expression of
f(z).
3.(20%) Consider the following functions on the complex plane:
(A) (z+1)/z, (B) (z+1)/z^(1/2), (C) cosh(z), (D) log(sinh(z)), (E) e^(1/z).
(a) Which one(s) of the above has (have) branch cut(s)?
(b) Which one(s) of the above has (have) simple pole(s)?
4.(15%)
∞ x
A ≡ ∫ dx ──── = ? (1)
0 x^3 + 1
(Recall: sin(π/3) = sin(2π/3) = (√3)/2. )
5.(15%)
∞ x^(1/3)
A ≡ ∫ dx ──── = ? (2)
0 (x+1)^2
6.(15%) For large n, using saddle point approximation, find the lowest order
approximation of
∞ -n
A ≡ ∫ dx cosh (x). (3)
0
7.(15%) For large n, using saddle point approximation, find the lowest order
approximation of
2π 2n
A ≡ ∫ dx sin (nx). (4)
0
8.(15%) For small g, using saddle point approximation, find the leading order
term Ｏ(1) and the next-to-leading order term Ｏ(g) of the integral
∞ -z0[(x-z0)^2/2 + g(x-z0)^4/4]
A ≡ ∫ dx e , (5)
-∞
where z0 = 3e^(iπ/4). (Warning: The final answer can not involve ambiguous
expressions such as √z0.)
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.102.7