課程名稱︰應用數學方法
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選修
課程教師︰薛克民
開課學院：數學系
開課系所︰理學院
考試日期︰2014年04月24日(五)，15:30-17:30
考試時限：120分鐘
是否需發放獎勵金：是
試題 :
National Taiwan University Spring Semester, 2014
MATH 7421 Method of Applied Mathematics
Midterm
Date: 15:30-17:30, April 24th, 2014
．Open Books
1. (40 points) Consider an algebraic equation of the form
4
x - εx - 1 = 0
for x, where ε in R is a parameter.
(a) (20 points) Suppose that ε<<1, find approximate expressions, correct
to terms of O(ε), for each of the four solutions of the equation.
(b) (20 points) Suppose that ε>>1, find the leading order (non-zero)
approximations for all four of the solutions. In addition, find a more
accurate approximation to the smallest root in this case.
2. (20 points) Verify that
1
∫exp[-x*cosh(t)]dt ～ (2π/x)^(1/2) * exp(-x)
-1
as x →∞.
3. (40 points) Consider an integral of the form
∞ ixt 2 -x
I(x) = ∫ e (1 + t ) dt
-∞
for x in R.
(a) (10 points) Find the function ψ(t) so that I(x) can be rewritten in
the following form
∞ xψ(t)
I(x) = ∫ e dt.
-∞
(b) (10 points) Determine the steepest descent path.
(c) (20 points) Find asymptotic approximation of I(x) as x → ∞.
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