Solution to Complex Integral Problem (Correction)

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

 Then:   Res f(z+) =

 

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)

 
Therefore (for upper half plane):

∫ _-¥  ^¥    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?