Page 883 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 863

The integrand has two simple poles enclosed by the unit circle, and these are the square roots of (β − α)/(β + α).
Compute the residue of f (z) at each of these poles to obtain 1/(8αβ) and evaluate the integral.
$ −z 2 $ −z 2
15. Call the rectangular path . By Cauchy’s theorem, e dz = 0. Parametrize each side of , and write e dz as a

sum of three integrals
R R

e −x 2 dx − e β 2 e −x 2 cos(2βx)dx
−R −R
β

t 2
+ 2e −R 2 e sin(2Rt)dt = 0.
0
Let R →∞ to obtain

∞ √
e −x 2 cos(2βx)dx = πe −β 2 .
−∞
17. First show by a change of variable that
∞ x sin(αx) 1 ∞ x sin(αx)

dx = dx.
x + β 4 2 −∞ x + β 4
4
4
0
4
4
iαz
Show that ze /(z + β ) has simple poles in the upper half-plane at βe iπ/4 and βe 3πi/4 , and evaluate the residues there
to obtain the requested integral.
Section 22.4 Residues and the Inverse Laplace Transform
1. cos(3t)
−1 1 1

2t
3. + t e + e −4t
36 6 36
1
2 −5t
5. t e
2
√ √ √ √
√ % &
2 2 2 2 2
7. cosh t sin t − sinh t cos t
2 2 2 2 2
2
9. (1 + 4t + 2t )e 2t
CHAPTER TWENTY THREE CONFORMAL MAPPINGS AND APPLICATIONS
Section 23.1 Conformal Mappings
z
x
x
1. Under w = e = u + iv = e (cos(y) + i sin(y)), vertical lines x = x 0 map to circles |w|= e 0 , and horizontal lines
y = y 0 map onto half-lines (rays) arg(w) = y 0 .
3. w = sin(z) = 4sin(x)cosh(y) + 4i cos(x)sinh(y) maps vertical lines x = kπ (k any integer) to the vertical axis in the
w-plane. Vertical lines x = (2k + 1)π/2 map to the part of the u-axis |u|≥ 4. Other vertical lines map to hyperbolas
2 2
u v

− = 1.
4sin(x 0 ) 4cos(x 0 )
The horizontal line y = 0 maps onto |u|≤ 4, while other horizontal lines map to ellipses
2 2

u v
+ = 1.
4cosh(y 0 ) 4sinh(y 0 )
3 3iθ
3
iθ
5. If z =re then w = z =r e , yielding the second quadrant of the w-plane if π/6 ≤ θ ≤ π/3.
7. If θ = k, check that
1 1 1 1
u = r + cos(k) and v = r − sin(k),
2 r 2 r
so
u 2 v 2
− = 1
2
2
cos (k) sin (k)
if sin(k)
= 0andcos(k)
= 0.
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October 14, 2010 17:50 THM/NEIL Page-863 27410_25_Ans_p801-866