A short lecture on evaluation of Complex integrations for Geophysics
Avradip Ghosh, University of Houston-University Park
Initial Publication Date: February 10, 2021
Summary
In Geophysics, we heavily use Fourier/Laplace, or inverse Fourier/Laplace transforms in all branches. Anyone who has tried to evaluate them using pen and paper (especially an inverse Fourier/Laplace transform) knows that the integrations do not follow the general rules we use to evaluate integrals. The nature of this behavior is due to the involvement of complex numbers, and such integrals can only be evaluated using complex analysis. Sometimes it is automatically done by software, and we use it without knowing its underlining details. This short lecture will provide an introduction to the world of Complex analysis from a Geophysicst's perspective without going into too many mathematical details. The short course is a guide for the motivated reader to learn to evaluate such integrals and serves as a stepping stone for the readers interested in learning more about the subject of Complex analysis.
Context
Audience
The short lecture is designed mostly keeping in mind the needs of students studying Geophysics (especially in the theoretical fields). Nevertheless, any reader who is interested in knowing Complex analysis in a quick introduction is welcome.
Needless to say a preliminary knowledge of algebra and calculus is necessary.
Skills and concepts that students must have mastered
Preliminary knowledge of algebra and calculus is necessary
How the activity is situated in the course
Any reader who is interested in knowing Complex analysis in a quick introduction is welcome
Goals
Content/concepts goals for this activity
The short lecture is aimed at providing a quick guide for the evaluation of Complex integrals. These processes will be useful for finding the inverse Fourier/Laplace transforms of complex functions.
Higher order thinking skills goals for this activity
Other skills goals for this activity
Description and Teaching Materials
As a student of Geophysics(theoretical) myself, I was left confused and troubled when faced with the task of inverse Fourier/Laplace transforming complex functions. I had to go and study the underlying math from scratch which does have a high learning curve.
Thus, keeping in mind my own struggles I have tried providing a quick guide for students seeking to learn to evaluate Complex integrals without going into too many mathematical details. The interested reader can of course use this as a stepping stone to learn more about complex analysis.
Complex integration guide (Acrobat (PDF) 1.3MB Feb10 21)
Teaching Notes and Tips
To successfully learn the reader must solve the problems in the lectures by themselves although the solution has been provided.
The same philosophy applies to understanding the algebra too.
Assessment
There are no assessments. Interested students can find several problems to be solved in the references of the books provided or on the internet for practice or for further knowledge.
References and Resources
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