2. It turns out that for s in the interior of that interval (away from the endpoints), the function t →f(t)est will be integrable on the whole real line (you do not need to prove this). Therefore, thebilateral Laplace transform,∞(Lf)(s) = T [ f(t)e-st dl,and then for,∞is well defined for such s. Taking for granted that you can exchange derivatives in s and integrals in t(a version of Leibniz' rule for differentiation under the integral sign), give an expression ford" (Lf)(s)dsnd" (Lf)(s)dsnmn :=(−1)ns=0where n is an arbitrary natural number. Note that this question is about derivatives of the Laplacetransform, not the Laplace transform of derivatives discussed in class in the unilateral case.9

We are given the following information. For s in the interior of that interval (away from the…

Answered: It turns out that for s in the interior…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Similar questions

(Continuation of Exercise 31.) If ƒ is continuously differentiable on [0, a] for a > 0, and f(a) = f(0) = b, prove that r) dx = f(x) dx – 2ab + 12
Show that the function f (t) 5 1yt2 does not possess a Laplace transform. [Hint: Write +{1yt2} as two improper integrals.
Use the definition of Laplace Transform, F(s)=∫∞0e-sx*f(s)dx to show that L{f'(x)}=s*F(s)-f(0).
Prove the following continuous time Fourier transform relationship. sin(at) 1 X(N) = Continuous Time x(1) = = sinc(t) nt otherwise Fourie Transf.
Compute the gradients of each of the following functions: o = xyz & = ln [(x² - y²)-¹/2] r = || = √√√x² + y² + x² 1/r Note: Remember that your tutorial has Hints in it, at the end of each chapter. If you're stuck, they are often quite useful! • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, shift and - for subscripts, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx= x², Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {A₂, Ay, A₂}. Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. Vo= V = Vr = V= (2A, 2A, 2A₂).
Explain why the Laplace transform of f(t) = 1 / ₁ t > 0 is not defined for any s, while the Laplace transform of 1 √t' g(t) = t> 0 is actually well defined for any s > 0. You do not need to calculate the explicit expression of the Laplace transform for either of the functions above.
2) For a random variable X, its kth-order moment is defined as E(X*). Given the p.d.f. fx(x) of X, its Fourier transform Fx(u) = S fx(x)e¬j2muz dx is also called moment generation function. a) Determine an expression of E(X*) in terms of Fx(u). Hint: consider the kth-order derivative of Fx (u) with respect to u and examine its relationship to the definition of E(X*). b) If X is N(0, o²), we know that Fx (u) = exp (-}(2u)°o²). Apply the above idea to determine E(X3) and E(Xª) in terms of ơ².
2. Let f(t) be a continuous function on the interval [0, 27]. Suppose the 1st order Fourier approximation to f(t) is 1 – cos(t). Prove that " f(t)²dt > 3n.

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