Problem 6 (8 points) For each of the following functions, find and classify all singulari- ties,...
i want all this answer in the complex number
ili-Let f be entire and If (2)l s Izl2 for all sufficiently large values of Izl>To.Prove that f must be a polynomial of degree at most2. ii-Classify the zeros of f(z)cos ( iii-Find Residue of g at points of singularity,g(z) = cotrz. -Find the radius of convergence of Σ-o oo (z-2i)n 1 Tl f(z)sinz
ili-Let f be entire and If (2)l s Izl2 for all sufficiently large values of Izl>To.Prove that...
Problem 7.1 (10 points) Express the unilateral z-transforms of the following functions as rational functions. Find also the ROC. You may use tables. (a) xl[n]-1-0.2)" (b) x2[n] (0.3)" +2(-5) -0.2n Problem 7.2 (10 points) Express the unilateral z-transforms of the following functions as rational functions. Find also the ROC. You may use tables. (a) xl[n] = 3e-j02" (b) x2[n]- 5cos(5n) (c) x3[n] = e-0.gn sin(0.7n) Problem 7.3 (10 points) The signals given are sampled every 0.3 s, beginning att-0. Find...
please, be explicit
4. Find all fixed points for each of the following maps and classify them as cobweb for the typical trajectories. a) f(x) 2r(1-); b) f(z) -; c) f(z) ; d) f() 4+ attracting, repelling, or neutral. Draw a
4. Find all fixed points for each of the following maps and classify them as cobweb for the typical trajectories. a) f(x) 2r(1-); b) f(z) -; c) f(z) ; d) f() 4+ attracting, repelling, or neutral. Draw a
6. (16 points) Sections 6.1-6.3 For each of the singularities of the following functions: i. find the residues ii. find the principal parts iii. classify the singularities as one of the types: removable, pole or essential sin 22 a. cosh b, C. 2 + 2 2² - 32 d. ze
Find and classify the critical points of these functions (that
is, are they local maxima, minima, saddle points, or points where
the function is not differentiable)
(a) h(x, y) = (12-2) (b) k(x,y) = sin(I) cos(y), with the domain {(1,y) |+ y2 < 4}.
1. Find all critical points for the given function and classify each as a local maximum, local minimum, or saddle point. a) f(x,y)= 2 +2y2-2xy + 4x-6y-5 b) f(z, y) = 813 + 6xy2-24r2-6y2 + 4 d) f(x,y) = cosx cos y,-r<1<T,-π < y < π
Exercise 12: Residues and real integrals (a) [6+4 points) Compute the residues for all isolated singularities of the following functions (i) f(2)== (2-) tan(2), (i) 9(2):= z2 sin () (b) (4+6+5 points) Compute (using the Residue theorem) (1) cos(72) ( d, A3 := {z € C:<3), 243 := {Z EC: | = 3}, 34, (2-1)(2 + 2)2(2-4) : 43 € C:21 <3}, po 12 To (x2 + 4)2 da, 24 2 + 4 cosat. J 5 + 4 sin(t)
2. For the transfer functions in problem 1 (a)(d)(e), find the corresponding impulse response functions h(t) using partial fraction expansion and determine the value of lim h(t) if the limit exists. Verify that lim- n(t)-0 for stable systems. (optional) After performing the partial fraction expansion by hand (required), yoiu are encouraged to use MATLAB to verify your results. MATLAB has a function called 'residue' that can calculate poles (pi) and residues (ci). For example, the following line will calculate the...
Problem #3: Find the residues of the following functions at z = 0 a) f(3) = 2* cos () b) f(3) = 1-cosa; c) f(3) = CS2 f(3) = 25(1 – 22) COS 2 COS 2 e) f(3) = 15e *e*1 f) f(3) = cosz - 1 9) f(3) = (sin 2)23 W f(z) = (eš – 1)2
2. (8 points) Find the Laplace transform of each of the following functions. 1. 2 f(t) = 14 + cos 3t + 3e-2t 2. 2 h(t) = (1 - 3t)? (Hint: expand...) 3. 2 g(t) = t sin’t (Hint: use half angle formula first...) 4. 2 h(t) = e-2 cos(v3t) - tet