Evaluate the following residues. A. (10 points) Res ( rez B. (10 points) Res (2.44 +3:1)
Use residues to evaluate the following: 271 a) S." 3 sin 0+5 1 - dᎾ b) S* sin’ x sin 7x dx Note: it is not necessary to show every step of the calculations in detail here. There's one step in particular where you will want to draw a conclusion “by inspection”.
3.) a.) Evaluate the following integral. (15 points) in(1 +5x?) dx b.) Evaluate the following integral. (10 points) tanº (6x) sec 10 (6x) dx
1) Evaluate the following real integrals using Residues, L***2+1&P +27 +2) (1) + drva
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)
6. Evaluate le.lda A. ( 10 points) C is the unit circle 11-1 B. (10 points) C is the upper unit semicirele from 1 to -1
use the contour integration and theory of residues to evaluate too COS Xdx -O [1+ x41
Evaluate the following integral using residues: I = { cos(bx)-cos(ax) dx. x2 Let a and b: real constants such that a > b>0. Note: cos(bz)-cos(az) is well-behaved along the real axis (singularity at z = 0 is removable), ejbz-ejaz has a pole at the origin. Make sure to handle this point correctly 22
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
Use the theory of residues to evaluate the following real, definite integrals: 210 do Hint: integrate on the unit circle, z = elo; dz = izdo. Hint: find the residue in the upper half plane. ſº I dx 13+5 sin e dx b. So y2 + 4x +5 J-ox? +dx
I sinta fosinta 3. (40 points) Evaluate the following integrals: (a) (10 points) sin(2 + 7)dz, where C is the square with vertices at 2i, 3i, 1+ 3i and 1+2i, in this order. (b) (10 points) sin(22) $c 2+1 where C is the positively oriented (counter-clockwise) triangle with vertices (0,0), (2,0) and (0,5). (c) (10 points) cosh(22) -dz, (3-2) where is the negatively oriented (clockwise) circle centered at (1,1) of radius 2. (d) (10 points) dz, 2-1 where C consist...