What is the application of the Residue Theorem in finance?

In summary, a Brazilian mathematician studying finance is struggling to understand an expression from a paper by Lewis (2001) that uses the Residue theorem. The expression involves an integral and they are looking for a reference to help them understand the relationship between a contour integral and the real part of another integral. They have asked for suggestions on how to solve the problem.
  • #1
Feanor
2
0
Hi everyone.
I'm a brazilian mathmatician that didn't studied complex analysis. I study finance and now I'm needing to study that.
In a paper of Lewis (2001) I found an expression that I couldn't understand.
Does anyone can help me with that? They say they use the Residue theorem but I couldn't make the calculations using the versions of this theorem that I found.
The equality is the following:


$ \int_{i Im(u)-\infty} ^{i Im(u)+ \infty} \left( \int_{0} ^{\infty} e^{iuA_t} \Phi^{\ast}(u)dx \right) du=
\pi + 2 \left( \int_{0} ^{\infty} Re \left[ \frac{e^{-iulnK} \Phi^{\ast}(-u) } {iu}\right] du \right) $
(jpg attached for non tex users)
Could you send me reference that I could read and understand the above?

Thanks!
 

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  • #2
Come on people...
Any suggestions on how to solve it?
I basically need a relation between a contour integral and the real part of another one.

Thanks!
 

What is the Residue Theorem and how is it applied in scientific research?

The Residue Theorem is a mathematical concept used to evaluate complex integrals. It states that the value of a contour integral around a closed path is equal to the sum of the residues of the function at the singular points inside the contour. In scientific research, it is commonly used in the fields of physics, engineering, and economics to solve difficult integrals that arise in various applications.

What are the key assumptions made when applying the Residue Theorem?

The Residue Theorem assumes that the function being integrated is analytic and has no singularities on the contour of integration, except for isolated poles. It also assumes that the contour of integration is a simple, closed curve.

What are some common applications of the Residue Theorem in scientific research?

The Residue Theorem has many applications in scientific research, such as in the evaluation of Fourier and Laplace transforms, solving differential equations, and calculating certain physical quantities in quantum mechanics. It is also used in the analysis of electrical circuits, signal processing, and fluid dynamics.

What are the advantages of using the Residue Theorem in scientific research?

The Residue Theorem allows for the evaluation of complex integrals that would otherwise be difficult or impossible to solve using traditional methods. It also provides a systematic approach to solving these integrals and can be applied to a wide range of mathematical problems in various fields of science and engineering.

What are some limitations of the Residue Theorem?

While the Residue Theorem is a powerful tool for solving complex integrals, it does have some limitations. It can only be applied to functions that are analytic, and it may not work for functions with multiple poles on the contour of integration. It also requires some knowledge of complex analysis and may not be suitable for beginners.

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