Question

Complex Analysis:

Residue theory

Verify the integral formula with aid of residue:
integral of (((x^2)+1)/((x^4)+1)) dx from 0 to infinity = PI / (Square-root of 2).

My thinking:
Let: a1 = square-root of i
a2 = square-root of -i

I can see the solution:
(1/2) (2PI * i) * res (a1) + res (a2)
Where: (1/2) is half of circle
(2PI*i) is the circle stuff
Problem: I can not seem to crack the res(a1)
and res(a2).
Where are you KingGeorge? THanks.

Regards
DL

Answer 1

\[
a_1=\sqrt{i}=\frac{1+i}{\sqrt{2}}
\]

Answer 2

Now find
\[

\lim{x->a1}
\left(\frac{\left(x-\frac{1+i}{\sqrt{2}}\right) \left(x^2+1\right)}{x^4+1}\right)=
-\frac{i}{2 \sqrt{2}}

\]
Do the steps necessary to prove that

Answer 3

Thanks so much EliasSaab for your help and time. It is appreciated.
As I am self learning, it is a little bit of difficulty to follow: sqrt(i) = (1+i)/sqrt(2).
Also, please let me know if I am err on the res(a2), also, the (1/2) part as well.
Basically, is my thinking generally correct or not? In the meantime, I will try your suggestions.
Again, thanks so much for your help and time. It is wonderful. Just another question,
do you know of anything about k-factor things since I am very interested in this
k-factor (loading analysis); but I could not find anything about it at all. Thanks again;

Answer 4

Thanks EliasSaab,
I can work out the srqt(i) you gave me. However, I wonder how in the world do you
cam up with that identity. Thanks.

Answer 5

Hi EliasSaab,
Thanks for your time. I worked out the limit:
((x^2)+1)/(x+sqrt(i)) * (x^2+i). as x->sqrt(i) = (1+i)/2.
(1/2)(2PI*i)*(-i/2*srqt(2)) = - (PI* (i^2))/sqrt(2) = PI/sqrt(2).
So things are fine now. However, I need to know if my thinking was err since
I did not do res (a2). Please let me know your comments, ideas or questions.
Thanks so much for your helps and time. It is appreciated.
Regards
DL

Answer 6

Write
\[
i = e^{ \frac {i \pi} 2 }\\
\sqrt i= \left( e^{ \frac {i \pi} 2 }\right)^{\frac 1 2}=
e^{ i\frac \pi 4} =\cos(\pi/4) + i \sin(\pi/4)= \frac 1 {\sqrt 2} + i \frac 1 {\sqrt 2}
\]

Answer 7

Thanks a lot. eliassaab.
Simply amazing @ the sqrt(i)=(1+i)/sqrt(2).