Problem 8 (15). Let f(2) 2+1)(22 + 2) (a) Find the residue of f(z) at the...
6: Problem 1 Previous Problem Problem List Next Problem (2 points) Let f(x) = z* In(t)dt (a) Evaluate f'(10) = (b) Evaluate (8-1)'(0) = 6: Problem 27 Problem List (1 point) Evaluate the integral p T/3 -9 In(tan(x)), 57/4 sin(x) cos(x) 6: Problem 29 Previous Problem Problem List Next Problem (1 point) Find the area of the region enclosed between f(x) = x2 – 3x + 8 and g(x) = 2x2 – x. Area = (Note: The graph above represents...
9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which lie in the upper half-plane fwEC: Rew> 0}. of =
9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which lie in the upper half-plane fwEC: Rew> 0}. of =
Problem 3. Evaluate the integral co sinx dx. Hint: Apply residue theorem to the function f(z) = and the contour y of the following shape:
12. (a) Show that 1y dt By letting R o, deduce that the residue of f ) at t 0. at zoo by the equation f (z) dz is given by 2πί times (b) When zoe is an isolated singular point, define the residue of f () Show that (e)d2miRes () Coo (c) Use the above result to evaluate the integral Ca2 + 22 z where C is any positive contour enclosing the points z 0, tia, and check the...
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the surface S be the part of the sphere x2 + y2 + x2 = 4 that lies above the plane 2=1 and be oriented downwards. (a) Find the divergence of F. (b) Compute the flux integral SS. F . ñ ds.
= and z= 8. Let A be the part of the cylinder x2 + y2 1 between the planes z = 2, where n points away from the z-axis. Let C be the counterclockwise boundary of A. Let F(x, y, z) = (2xz + 2yz, –2xz, x2 + y²). Verify Stokes' Theorem: (a) Evaluate the line integral in Stokes' Theorem. (Hint: C has two separate parts.] (b) Evaluate the surface integral in Stokes' Theorem. Hint: curl (F) = (2x +...
(2) Let F-1 + rj + yk and consider the integral- , ▽ × F. т. dS where s is the surface of the paraboloid z = 1-12-y2 corresponding to z 0, and n is a unit normal vector to S in the positive z-direction (a) Apply Stokes' theorem to evaluate the integral. (b) Evaluate the integral directly over the surface S rectlv over the new surface
(2) Let F-1 + rj + yk and consider the integral- , ▽...
Question one (9 marks total, 3 marks each) Let f(2)= Z 22-32+2 a. Find a Maclaurin series for f(z) in the region [z] < 1. b. Find a Laurent series for $(2) in the region 1 < lz[< 2. c. Find a Laurent series for S(z) in the region [2] > 2. Om01
16: Problem 8 Previous Problem ListNext 1 point) 1) Suppose that f(x) is a function that is positive and decreasing. Recall that by the integral test: f(z) dz < Σ f(n) Recall that e-Σ. ,,Suppose that tor each positive integer k f(k)- Find an upper bound Bor f(z) dz 2) A function is given by ts values may be found in tables. Make the change of variables y In(4) to express 1-4 d as a constant C times h(3). Find...
3. Find the residue at 2-0 of the function f(z) 2142