- #1
lurflurf said:You have a bigger problem then that, the residue at x=0 is zero (removeable singularity). This is a standard example and is often done as follows
consider
To fix the residue=0 problem add in an odd function the usual one is to use
exp(x i)/x=cos(x)/x+i sin(x)/x
the cos(x)/x diverges as an improper integral, but as lqg states cauch principle sense may be used
now consider ther contour formed by theese pieces
(-R,r) straight line
(-r,r) semicircle arc about z=0
(r, R) straight line
(R,r) semicircle arc about x=0
asi123 said:I didn't quite understand the contour you described there.
What's R and What's r?
Thanks a lot.
lurflurf said:Take limits R-> +infinity,r->+0
If you draw a pictuis an upper semicircle that in the limit is large with a tiny semicircle at the origin then you get for the various integrals
(-R,r) straight line
-infinity+(Integral you want)/2
(-r,r) semicircle arc about z=0
{+,-}[+ if it was upper - if it was lower] pi*i (residue theorem)
(r, R) straight line
-infinity+(integral you want)/2
(R,r) semicircle arc about x=0
0
asi123 said:Is it something like that?
Thanks again.
lurflurf said:That is it. I guess that is one reasone why some people do not like old math books with no pictures. Were you able to finish?
The Residue Theorem is a powerful tool in complex analysis that allows for the calculation of certain types of integrals. It states that the integral of a function around a closed curve is equal to the sum of the residues of the function at its singular points inside the curve.
The Residue Theorem simplifies the calculation of certain types of integrals by reducing them to a sum of residues, which are much easier to compute. This can save a significant amount of time and effort in solving complex integrals.
No, the Residue Theorem is specifically useful for integrals that involve rational functions or functions with simple poles. It cannot be used for integrals with essential singularities or branch points.
Yes, the Residue Theorem can only be used for integrals that are taken along closed curves. It also requires knowledge of complex analysis and the ability to identify singular points and their corresponding residues.
The Residue Theorem has many applications in physics, engineering, and mathematics. It can be used to solve problems involving contour integrals, real integrals, and infinite series. It is also used in the study of complex functions, such as in the calculation of residues and poles.