Find all complex variables z E C with Izl < 10 that satisfy e: +2e-:-3. 23
3i)16 in polar form: z r(cos 0isin 0) where (1 Write the complex number z and e= The angle should satisfy 0 0 < 2«.
Help please e Find the Laplace transform of f(t) = e-.if Ost<3 O, if t 23
Problem 10.7. Show that azn-e* has n roots with Izl < 1 if lal > e. Problem 10.8. Suppose that f is holomorphic inside and on a toy contour γ. Suppose f has no zeros on the contour γ and that zi, ,«n are the zeros of f inside γ, the order of being kj. Show that j-1 for any function g which is holomorphic inside and on γ. Problem 10.9. Let u(z,y)-zy(x2-уг). Find the maximum and minimum values of...
2. Shade the region of the complex plane defined by <z +4 + 3i : 3 < 3 < 5,2 EC}. Include the appropriate axis labels and any significant points.
Find the Peano range of the Cauchy problem. Z=38 {r' = (2 = (Z -t)y,-3<t< 3; y(1) = 2
For a standard normal distribution, find: P(-1.95<z<0.09)
in each case: (e) Compute y = sin(z)cos(r) for 0 < z < π/2
2. Let the random variables X and Y have the joint PDF given below: 2e -y 0 xyo0 fxy (x, y) otherwise 0 (a) Find P(X Y < 2) (b) Find the marginal PDFs of X and Y (c) Find the conditional PDF of Y X x (d) Find P(Y< 3|X = 1)
find the inverse z transform X(z) = 1-2-3 with [2]<1
Find all integers x, y, 0 < x, y < n, that satisfy each of the following pairs of congruences. If no solutions exist, explain why. (a) x + 5y = 3(mod n), and 4x + y = 1(mod n), for n = 8. (b) 7x + 2y = 3(mod n), and 9x + 4y = 6(mod n), for n=5.