Residue theorem to solve integration

In summary, the speaker is seeking advice for solving an integral equation involving complex functions and exponential terms. They have attempted to use the residue theorem and L'Hopital's rule, but their solution did not match the expected plot. They are asking for advice on how to approach the problem again.
  • #1
Elsasw
10
0
Hi all, i have an integral equation of

∫1/[(z+ia)^2 *(z-ia)]*exp(-p^2[(A+iBz)/(C(z^2+a^2))])*exp(-ikbz)dz

from the limit of 0 t0 l

I tried perform residue theorem but due to the 1/[(z+ia)(z-ia)] factor in the exponential term complicated it...I also tried incorporate L'Hopita rules in the exp term. I did manage to solve it but did not seems to be right because when i plotted in with Matlab, the plot is not correct. Perhaps my approach is wrong.

Need advise to solve this again.
Thank you
 
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  • #2
What contour were you using?
 
  • #3
HallsofIvy said:
What contour were you using?

Contour? Does this referred to residues at z=ia and z=-ia?
 

1. What is the residue theorem?

The residue theorem is a mathematical tool used in complex analysis to evaluate certain types of integrals. It states that if f(z) is a function that is analytic except for a finite number of isolated singularities, then the integral of f(z) over a closed contour C is equal to 2πi times the sum of the residues of f(z) at the singular points inside C.

2. How is the residue theorem used to solve integrals?

The residue theorem is used to evaluate integrals by first identifying the singular points of the function being integrated and then calculating the residues at those points. Once the residues are found, the integral can be evaluated using the residue theorem formula.

3. What types of integrals can the residue theorem be used for?

The residue theorem can be used to evaluate integrals of functions that have singularities, such as poles or branch points, located inside the contour of integration. It is most commonly used for integrals involving rational functions or trigonometric functions.

4. Are there any limitations to using the residue theorem to solve integrals?

Yes, there are some limitations to using the residue theorem. It can only be used for integrals over closed contours, and the function being integrated must be analytic except for a finite number of isolated singularities. In addition, the contour of integration must be simple, meaning it does not intersect itself or any singularities of the function.

5. Can the residue theorem be extended to higher dimensions?

Yes, the residue theorem can be extended to higher dimensions through the use of multidimensional residues. This allows for the evaluation of integrals over surfaces or volumes in higher dimensional space. However, the concept of residues becomes more complex in higher dimensions and is not as commonly used as in the one-dimensional case.

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