Problem:
first factor to get: f(x)
= x/ (x - 1
+ i)(x - 1 - i)

In
terms of the z variable:

f(z) =
z/ (z - 1 + i)(z
- 1 - i)

In
the

*upper half plane*we need to obtain Res f(z+) for z = 1+i
lim

_{ z }**[z/(z - 1 + i)**_{® 1+i}^{ }] = (1 +i)/ 2i
= ½ -
½ i

Then: Res f(z+) =
2 pi [½ -
½ i ] = pi
- pi

^{2}
= pi
+ p = p (i + 1)

In
the

*lower half plane*we need to obtain Res(f(z-) for z = 1- i
lim

_{ z }**[(z - 1 + i)**_{® 1-i}^{ }z/ (z - 1 + i)^{ }(z - 1 - i)]
= lim

_{ z }**[z/(z - 1 - i)**_{® 1-i}^{ }] = (1 -i)/ (– 2i)
=
½ +
½ i

Then: Res f(z-) =
- 2 pi [½ +
½ i ] = -pi
- pi

^{2}
=
-pi - p

^{ }= - p(i - 1)
∫

_{-}_{¥}^{¥}^{ }x dx**/**(x**- 2x + 2) = p (i + 1)**^{2}
For
the lower half plane:

∫

_{-}_{¥}^{¥}^{ }x dx**/**(x**- 2x + 2) = - p(i - 1)**^{2}Note: To modify the development for any case where the singularity is in the lower half plane, or m < 0 (i.e. m = -i) we have:

∫

_{-}_{¥}^{¥}^{ }exp(im x) f(x) dx =**- 2****p****i**å (Res)Question: What would you obtain if Res f(z+) and Res f(z-) are added together?

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