课程名称︰应用数学三 课程性质︰物理系大二必带 课程教师︰蔡尔成 开课学院：理学院 开课系所︰物理学系 考试日期（年月日）︰2019年06月20日 考试时限（分钟）：127分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 注：由於式子过於复杂，故难以使用图像表达之处使用Latex语法输入。 范例：A^{a}_{bc}表A带上标a及下标bc \vec{a} 表A向量 \hat{a} 表A单位向量 试题： 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. --

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