ex4.13 Ex. 4.13. Calculate the residues at the poles indicated 1 COS 2 2 (a) 2...
2. Caleulate the residues at the indicated poles coS 2 a)- z=0; il , 2 2 sin z (c) 따가, 1+22)22j.
2. Caleulate the residues at the indicated poles coS 2 a)- z=0; il , 2 2 sin z (c) 따가, 1+22)22j.
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)
Evaluate the following integral using residues: cos(bx)-cos(ax) I = dx. x2 Let a and b: real constants such that a > b >0. Note: cos(bz)-cos(az) has a singularity at z = 0 is removable, z2 ejbz-ejaz has a pole at the origin. Make sure to handle this point correctly 22
a. 1. Answer each of the following: Compute the Wrornskian of the set {ex cos 3x, ex sin 3x} b. Show that {e* cos 3x, ex sin 3x} form a fundamental solution set for y" – 2y' + 10y = 0 on the interval (-00,00) and write the general solution
Will give thumbs up!
limx40 sin(4.c) cos(3.c) tan 2x = ? limo–0 2–1 eX – 1 = ? lim.c10+ x ln x : ? S-|(10x4 + 1) dx = ?
Re -3 -2 -1 0 1 2 3 4 Note that C is not a simple curve, so Cauchy's integral formula does not directly apply. By breaking up C as needed, evaluate T z2+9 Jc (2+2-i)(2+1-i)z dz. Syntax notes: • When entering lists in the questions below, use commas to separate elements of the list. Order does not matter. • The complex number i is entered as I (capital i). z2+9 (a) The poles of (z+2-i)(2+1-i)z that lie inside Care...
Simplify the following trigonometric expression by following the indicated direction sin 0 1 + cos 0 Multiply 7- cos o by 1 + cos 0 sin e 1 + cos 0 1 - cos ' 1+ cos = *cos (Simplify your answer.)
Problem 1 Use residues to verify the formula poo cos(3x) – cos(x) dx = a Jo x² Hint: use the indented contour with two semicircles from the April 14 lecture. Problem 2 Use residues to verify the formula [ * nedz =
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
Supposez1 =4 cos 3 +isin 3 andz2 =2 cos 6 +isin 6 .
Computez1z2.
(a) 8(cos?π?+isin?π?) 22
(b) 4(cos?4π?+isin?4π?) 66
(c) 2(cos?π?+isin?π?) 66
(d) cos(π)+isin(π)
(e) 8(cos?π?+isin?π?)
66
17. Suppose z1 = 4 (cos (1) + i sin (5)) and z2 = 2 (cos () + i sin (7)). Compute z122. (a) 8(cos (7) + i sin (7)) (b) 4(cos (4) + i sin (*)) (c) 2(cos (7) + i sin ()) (d) cos(T) + i sin(TT) (e) 8(cos (7)...