Show that the sequence (e2ni(mx-+ ry)y is an orthonormal set in し2 ([0, 1] × [0, 1]). Here the L...
b) 16 marks Assume that each set Vi, j = 1, 2, ...k, is a compact set in a metric space X. Prove that the (finite union) set V = V1 U V2 U... U Vk is a compact set. c) [7 marks] Let H be a Hilbert space with inner product < x, y > and the induced norm ||2|= << x, x >. (i) Show that ||* + y|l2 + ||* – y|l2 = 2(1|x1|2 + ||4||2) for...
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...
mx 'e'dVR Evaluate is bounded by 2 - xy, 2-0, y = x, y = 0. and x=1. mzdV R is bounded by z - Vy? -98°, z = 0, y = 3x, x = 0, and y=9.
2. Let {xn}nEN be a sequence in R converging to x 0. Show that the sequence R. Assume that x 0 and for each n є N, xn converges to 1. 3. Let A C R". Say that x E Rn is a limit point of A if every open ball around x contains a point y x such that y E A. Let K c Rn be a set such that every infinite subset of K has a limit...
(20 pts.) The Laguerre differential equation is ry" + (1 - )y' + Ay = 0. (a) Show that x = 0 is a regular singular point. (b) Determine the indicial equation, its roots, and the recurrence relation. (c) Find one solution (x > 0). Show that if = m, a positive integer, this solution reduces to a polynomial. When properly normalized, this polynomial is known as the Laguerre polynomial, L. (2).
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
Show that, if an ≥ 0 for all n ∈ N and (an) is a Cauchy sequence, then (√ an) is also a Cauchy sequence. Hint: x − y = (√ x − √y)(√ x + √y) Show that, if an > 0 for all n є N and (an) is a Cauchy sequence, then (Van) is also a Cauchy sequence. Hint: r -y- (V1-vu) (Va + vⓙ Show that, if an > 0 for all n є N and...
2 + COS- 2.ry dy d 1+y2 = y(y + sin x), 7(0) = 1. 3. [2cy cos(x+y) - sin x) dx + x2 cos (+²y) dy = 0. 4. Determine the values of the constants r and s such that (x,y) = x'y is an Integrating Factor for the following DE. (2y + 4x^y)dr + (4.6y +32)dy = 0. 2. C = -1 You need to find the solution in implicit form. 3. y = arcsin (C-cos) 4. r=...
y(n) w(n) 2 L1 -1 L2 し4 tions for the network sho figure relating x(n), w(n), and y(n). b. Find H() by transforming the equations in (a) and elimi- nating W). c. Find the state matrices A, b, s', d. d. Find H() from part (c) and check it with that from (b) y(n) w(n) 2 L1 -1 L2 し4 tions for the network sho figure relating x(n), w(n), and y(n). b. Find H() by transforming the equations in (a)...
Question 2 2 pts Consider the solution to the IVP y - ry=2; y(0) = 2 Find y' (0) Question 3 2 pts Consider the solution to the IVP y - ry=r; y(0) = 2 Find y" (0) Question 4 4 pts Consider the solution to the IVP w"-() = 0; y(0) = 1; 7 (0) = 2 Find the coefficient of in its Taylor expansion centered ato.