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 ® 1+i
[z/(z
- 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 ® 1-i
[(z - 1 +
i) z/ (z - 1
+ i) (z - 1 - i)]
= lim z ® 1-i
[z/(z
- 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 /
(x2 - 2x +
2) = p (i
+ 1)
For
the lower half plane:
∫ -¥ ¥ x dx /
(x2 - 2x +
2) = - p(i - 1)
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 pi å (Res)
Question: What would you obtain if Res f(z+) and Res f(z-) are added together?
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