Exercise 8.3. Let c(n) denote Ramanujan's sum and let M(r)=En
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- Change Ax = b to x = (I -A)x + b. What are Sand T for this splitting? What matrix s-1y controls the convergence of Xk+l = (I -A)xk + b?Prove whether a function f(x)=x satisfies the Dirchlet conditions.Find the second order Taylor expansion of [15] f(x,y) = (1 +4x2 +y2)1/2 about the point (1,2) and use it to compute approximately (1.2,2.5).
- If we assume that all of the functions have continuous second order partialderivatives, show that any function of the formz=f(a+at)+g(a-at)is a solution to the wave equation02where a is some constantThe impulse response of a DT system is h[n] = 3^(n-1)u[n] . Calculate the input x[n] which generates an output y[n] = 2^(n-1)u[n-1] using z-transform method.Prove the following property of the compound Poisson process:1. E(xt) = λ t E(Y).
- Consider the geometric Brownian motion with σ = 1: dS = μSdt + SdX, and consider the function F(S) = A + BSα. Find any necessary conditions on A, B, and α such that the function F(S) follows a stochastic process with no drift.Determine the inverse laplace using partial fraction, show complete solutions and provide the checking. 1. F(s)= S+4/S²+2SProve that the trial solution of 2nd order homogeneous linear ( ODE ) with constant coefficients in the case of real double roots (λ= -a/2 ) is equal to : y(x)= (c1+ c2 * x ) e^((-a/2)*x) subject :- sconde order
- Compute the Taylor polynomial T5 (x) and use the Error Bound to maximize possible size of the error. f(x) = cos(x), a=0, x = 0.15In the study of a nonlinear spring with periodic forcing, the equation y''+ky+ry3= Acos(ωt) arises. Let k = 4, r = 8, A = 1, and ω = 6. Find the first three nonzero terms in the Taylor polynomial approximation to the solution with initial values y(0) = 0, y'(0)=1. The Taylor approximation to three nonzero terms is y(t) =?