課程名稱︰應用數學三
課程性質︰物理系大二必帶
課程教師︰蔡爾成
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2019年06月20日
考試時限（分鐘）：127分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
註：由於式子過於複雜，故難以使用圖像表達之處使用Latex語法輸入。
範例：A^{a}_{bc}表A帶上標a及下標bc
\vec{a} 表A向量
\hat{a} 表A單位向量
試題：
Applied Mathematics III 6-20-2019
Final Exam
1.[10%] Represent f(x)=sin(x), 0<x<π, as a Fourier cosine series.
2.[10%] The Bessel function J_ν(x) and satisfies the recurrencee relations
x^(-ν)d/dx[x^(-ν)J_ν(x)]=J_(ν-1)(x) and x^(ν)d/dx[x^(-ν)J_ν(x)]=
-J_(ν+1)(x). Show that, between two successive positive zeros of
J_(ν+1)(x) on the real axis, there is one and only one zero of J_ν.
3.[10%] The Rodrigues's formula for Hermite polynomials is
H_n(x)=(-1)^ne^(x^2)d^n/dx^n(e^(-x^2)). Prove that |H_n(x)|≦|H_n(ix)|.
4.The Rodrigues's form for the Laguerre Polynomial is
L_n(x)=e^x/n!d^n/dx^n(x^ne^(-x)).
(a)[5%] Show that L_n(x)=1/(2πi)∮(s^ne^(x-s))/(s-x)^(n+1)ds.
(b)[5%] Show that L_n(x)=1/(2πi)∮(e^(-xt/(1-t))/((1-t)t^(n+1))dt.
[Hint: Use t=1-x/s]
(c)[5%] Show that Σ_{n=0}^{∞}L_n(x)t^n=e^(-xt/(1-t))/(1-t).
(d)[5%] Show that L'_n(0)=-n, L''_n(0)=n(n-1)/2.
5.[10%] Show that ∫_{-1}^{1}P_m(x)P_n(x)dx=0 where P_m(x) and P_n(x) are
Legendre polynomials and both n,m are non-negative integers with n≠m.
6.[10%] The Legendre function P_n may be written in terms of contour integral.
as
P_n(t)=1/(2^n2πi)∮(z^2-1)^n/(z-t)^(n+1)dz (1)
which satisfies
[d/dt(1-t^2)d/dt+n(n+1)]P_n(t)
=(n+1)/(2^n2πi)∮d/dz[(z^2-1)^(n+1)/(z-t)^(n+2)]dz
Show that (1) is a solutinno of the Legendre equatinon even for non-integer
n by choosing the cut to be (-∞,-1) and a line joining 1 and t (t may be
complex as well).
7.The spherical harmonics
Y_{lm}(θ,φ)=((2l+1)(l-m)!/(4π(l+m)!))^{1/2}P^{m}_{l}(cosθ)e^{imψ}
satisfies L^2Y_{lm}=l(l+1)Y_{lm} where
L^2=-1/sinθ∂/∂θ(sinθ∂/∂θ)-1/sin^2(θ)∂^2/∂ψ^2.
(a)[10%] Show htat L^2 is Hermitian, <f|L^2|g>=<L^2f|g>, with respect to the
innner product
<f|g>≡∫f*(θ,φ)g(θ,φ)dΩ
where dΩ=sinθdθdφ.
(b)[10%] Show that
　　　　　　　　　　　∫Y*_{l'm'}(θ,φ)Y_{lm}(θ,φ)dΩ=0
if m≠m' or l≠l'.
8.[10%] For a dipole \vec{p} at the origin, the electric field is
\vec{E}_{\vec{p}}(\vec{x})=(3(\vec{p}‧\hat{r})\hat{r}-\vec{p})/(4πε_0r^3)
hwew r=|\vec{x}|≠0 and \hat{r}=\vec{x}/r. The average of \vec{E}_p over a
spherical surface of radius R≠0 is
<\vec{E}_{\vec{p}}>=1/4π∮_{|\vec{x}|=R}\vec{E}_{\vec{p}}(\vec{x})dΩ.
Show that <\vec{E}_{\vec{p}}>=0.