课程名称︰应用分析一
课程性质︰数学系选修、数学研究所选修、应用数学科学研究所必修
课程教师︰薛克民
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰11/21, 2014
考试时限（分钟）：10:10~11:40
试题 :
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 ]