課程名稱︰應用分析一
課程性質︰數學系選修、數學研究所選修、應用數學科學研究所必修
課程教師︰薛克民
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰11/21, 2014
考試時限（分鐘）：10:10~11:40
試題 :
Instructions:
・Total points 60
・Open books, notes, and laptops
・Answer the questions thoroughly and justify all your answers.
Show all your work to maximize partial credit.
1. (15 points)
Let f(z) be simple within and on a simple closed contour C. Let the
interior of C be D and let D map onto D* and C onto C*, the boundary
of D*. If C is a circle of radius R centered at the origin, prove
that the area of D* ≧ π(|f'(0)|)^2 R^2
2. (20 points) Solve the integral equation
-t^2 / 2 -1/2 ∞ -|t-τ|
e = (2π) ∫ e u(τ) dτ
-∞
3. (25 points)
Consider the Cauchy problem of the partial differential equation
u_tt + 2αu_t = u_xx, t > 0, x ∈R
with initial data
u(x, 0) = 0, u_t(x, 0) = f(x),
where α ∈ R is a constant. Show that the solution u(x, t) of this
problem takes the form
∞
u(x, t) = ∫ K(x-τ,t)f(τ)dτ,
-∞
where
-αt -1 -1/2
K(x, t) = 1/2 e L [(s^2 -α^2) exp{-|x|sqrt(s^2-α^2)};t ]