MATH 543: METHODS OF APPLIED MATHEMATICS I

First Midterm Exam: 31 October 2001 Wednesday 18.00-20.00 SAZ:13

QUESTIONS: (25)1. Generating function g(x, t) for the Legendre polynomials P_{n}(x) is given by

\frac{1}{\sqrt{t^{2} − 2xt + 1}} = ∑_{n=0}^{∞} t^{n}P_{n}(x)

(i) Find P_{i}(x) for i = 0 − 3, (ii) Prove that P′_{n+1} − xP′_{n} − (n + 1)P_{n} = 0, (iii) Find ||P_{n}(x)||, (iv) Expand \frac{1}{\sqrt{5−4x}} where x ∈ [−1, 1] interms of the Legendre polynomials. Find the first three terms and discuss the convergence of the series expansion.

(25)2. The Rodriguez formula for the orthogonal polynomials C_{n}(x) is given by

C_{n}(x) = \frac{1}{w} \frac{d^{n}}{dx^{n}} (ws^{n})

where

(1) C_{1}(x) is a first degree polynomial in x, (ii) s(x) is polynomial in x of degree ≤ 2 , with real roots, (iii) w(x) is real , postive and integrable in the interval [a, b] and satisfies the boundary conditions w(a)s(a) = w(b)s(b) = 0. Prove the following

(a) Let us denote p_{(≤k)}(x) as any arbitrary polynomial in x of degree ≤ k. Prove te following for all m , n and for all k

\frac{d^{m}}{dx^{m}} (ws^{n} p_{(≤k)}) = ws^{(n−m)}p_{(≤k+m})

(b) C_{n}(x) is a polynomial in x of degrre n which is orthogonal on [a, b] with weight w(x) to any polynomial p_{m}(x) of degree m less then n

(c) Prove that C_{n}(x) with x ∈ I = [a, b] has n zeros in I. (25)3. (a) If (e_{1}, e_{2}, ⋯, e_{n}) is a finite orthonormal family of functions in L_{w}^{2}(a, b) and f ∈ L_{w}^{2}(a, b), then

|||f > − ∑_{k=1}^{n} f^{k} |e_{k} > ||

has its minimum value when f^{k} =< f|e_{k} > and the Bessel’s inequality holds

∑_{k=1}^{n} | < f|e_{k} > |^{2} ≤ ||f||^{2}

b) Prove that the set of orthonormal vectors |e_{i} >, (i = 1, 2, ⋯) form a basis of L_{w}^{2}(a, b) if and only if the Fourier coefficients with respect to |e_{i} > satisfy the Parseval’s relation.

(25)4. (a) Prove that the sequence of functions f_{n}(x) defined by

f_{n}(x) = \{ \begin{matrix}n & if 0 < x < \frac{1}{n} \\ 0 & otherwise \end{matrix}

defines the generalized function δ(x) (Dirac delta function)

(b) Let f(x) be a good function, prove that δ(x) f(x) = f(0) δ(x),

(c) Let a ≠ 0 then prove that δ(ax) = \frac{1}{|a|}δ(x)