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7. (12 pts) The gamma function is defined by l(a) = Soy-le-Vdy for a > 0....
Exercise 8 The pdf of Gamma(α, λ) is f(x)-ra)r"-le-Az for x 0. a. Let X ~ Gamma (a, λ). Show that E( )--A for α > 1 b. Let Ux2. Show that E()for n > 2 n-2
n=7 Question 3 3 pts Find the Fourier Sine series for the function defined by f(x) = { 0, 2n, 0 <*n n<<2n and write down, 1. The period T and the frequency wo of the Fourier Sine series 2. The coefficients for r = 1,2,3,...
For s > 0 define the gamma function I (s) by T () = [co-dt. Show that I (8) extends to an analytic function in the half-plane 20 = {ZEC: Rez >0}, and that the above formula continues to hold there. Hint: Show that S T. (s) ds = 0 for every triangle T in C where I (8) = le-+48-1dt for S E C and 0 <€ < 1.
I-x, 0 < x < 1 1 defined within 0 < x < 2 (note: L = 2) 1. (20pts) Consider the function f(x)- a) Write down the full sine series of f(x), as well as the partial sum of first 3 non-zero terms. b) Plot the odd periodic extension of nx) and the first non-zero term in the sine series, for-6 < er the function /(x)-x,0 6 (recall that L=2)
4. Prove that the external measure is not "additive", in the sense that [0, 1]le < Vle+ |[0, 1] \Vle. 5. Let 7(x) := lim sup fx(x). +00 Prove that, for every a € R, {7 > a} = UN U{n>a + m}: MENJEN
Roots (20 points). Consider the loop-gain transfer function L(S) = TS-a)n-m where n and m are integers such that n > m and a € R. Also, consider the characteristic equation 1+ KL(S) = 0, with 0 <KER, which can be equivalently written as nam (s– an-m + K = TI (s – rj) = 0. Show that num ri=(n - m), for any 0 <KER.
n=2 Question 3 3 pts Find the Fourier Sine series for the function defined by 0<c<n f() = { 0, 2n, n<3 < 2n and write down, 1. The period T and the frequency wo of the Fourier Sine series 2. The coefficients bn for n = 1,2,3,...
Type or paste question here 3. (20 pts.) Consider the function f defined on (0, 2) by 2+1 f(x) = = { 0<x< 1 1<x< 2 (a) Denote by fs the sum of the sine Fourier series of f (on (0,2]). Plot the graph of the function fs for x € (-2, 4), indicating the values at each point in that interval. Compute fs(0) and fs(2). [You do not have to compute the coefficients of the Fourier series.] (b) Denote...
1) (2 pts) Write a script to plot the function 15V4r +10 r29 10 r<0 for-5ss 30
The Ackermann function is usually defined as follows: In+1 A(m, n) = {Am - 1,1) ( Alm – 1, A(m, n - 1)) if m =0 if m >0 and n=0 if m >0 and n > 0. Use the definition of the Ackermann function to find Ack(3,2). Please show your work step by step.