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?
 

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