_{}

^{}

1)
∫

_{-}_{¥}^{¥}^{ }sin x dx**/**(x**+ 2x + 4)**^{2}
In
terms of f(z) we can write:

f(z)= exp(iz)
dz / (z

^{2}- 2z + 4)
so
the integral becomes:

∫

_{-}_{¥}^{¥}^{ }exp(iz) dz / (z^{2}- 2z + 4)
Factor
f(z) to get:

exp(iz) dz / (z

^{2}- 2z + 4) = exp(iz)/ (z + 1 - iÖ3) (z + 1 + iÖ3)
Then
singularities (poles ) occur at:

z=
-1 + -iÖ3 and z = -1 -iÖ3

Therefore:

Res
f(z) = lim

_{ z }_{® 1+i}**[exp(iz)/ (z + 1 + iÖ3)]**_{Ö3}
= exp i(- 1+ -iÖ3)/ 2iÖ3
= exp(-i) exp(-Ö3) / 2i Ö3 exp (-Ö3)

But
recall exp (-i) = sin(-1) = -sin
(1) so that:

Res
f(z) = - sin(1)/ 2i Ö3 exp (-Ö3)

Then:

∫

_{-}

_{¥}

^{¥}

^{ }exp(iz) dz / (z

^{2}- 2z + 4) =

2
pi [- sin(1)/ 2i Ö3 exp (-Ö3)]
= -pi [- sin(1)

**/**Ö3 e^{-}^{Ö}^{3}]
= ∫

_{-}_{¥}^{¥}^{ }sin x dx**/**(x**+ 2x + 4)**^{2}
Note:
this soln. is for the upper half plane.
To modify the development for the case where the singularity is in the lower
half plane, or m < 0 (i.e. m = -i) then we have:

∫

_{-}_{¥}^{¥}^{ }exp(im x) f(x) dx = - 2 pi å (Res)
(See e.g. the accompanying graphic. Note that in the lower graphic, that should be

**- i Ö****3**along the Im -axis)
2) ∫

_{-}_{¥}^{¥}^{ }cos x dx**/**(x**+ 1)**^{4}
To
obtain singularities:

x

**+ 1 = 0 or x**^{4}**= -1**^{4}
Then:
x = (-1)

^{1/4}= (-1 + 0i)^{ 1/4}
=

^{4}Ö1 [cos (p + 2kp/ 4) + isin((p + 2kp/ 4)]
For
k= 0:

(-1)

^{1/4}= cis(p/ 4) = exp (ip/ 4)
For
k=1:

(-1)

^{1/4}= cis(3p/ 4) = exp (i3p/ 4)
Both
poles are in the upper half plane (see e,g, diagram for Jan. 27 post)

The
above are associated with

*two simple poles*, for which:
Res
f(z)

_{z = exp }**= lim**_{pi/4}_{ z }_{® exp }**exp(iz)**_{pi/4}**/**4 z^{3}=
exp(- p/4) / 4 exp(3pi/4) = exp(- p/4) /
2Ö2 – 2iÖ2 =

exp(-
p/4) / 2Ö2 (1– i)

And:

Res
f(z)

_{z = exp 3}**= lim**_{pi/4}_{ z }_{® exp 3}**exp(iz)**_{pi/4}**/**4 z^{3}=
exp(- 3p/4) / 4 [exp(3pi/4)]

^{3}= exp(- 3p/4) / 4 exp(7pi/4)
= exp(- 3p/4) / 4[ cos(9p/4) + i sin(9p/4)]

= exp(- 3p/4) /
2Ö2 + 2iÖ2 =
exp(- 3p/4)
/ 2Ö2 (1 +i)

Finally:

∫

_{-}_{¥}^{¥}^{ }exp(iz) dz**/**(z**+ 1) =**^{4}
2 pi [exp(- p/4) / 2Ö2 (1– i)
+ exp(- 3p/4) / 2Ö2 (1 +i) ]

=
∫

_{-}_{¥}^{¥}^{ }cos x dx**/**(x**+ 1)**^{4}
## No comments:

Post a Comment