## Friday, January 31, 2014

### 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?