What are applications of the theory of differential equations in complex domain in physical sciences or engineering? In which physical science differentiation wrt complex variable makes sense?
The theory of diferential equations in complex domain is very well developed and is one the most beautiful areas of mathematics. A well known old book on this subject is by E. Hill. There are many contemporary results in this area but all of them I found are in mathematics. I would like to know a physical science or engineering application.
Hi
I advice you to see these documents. You will find what you need.
Best regards
http://www.sciencedirect.com/science/article/pii/0022039687901410
https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=0CDEQFjACahUKEwjOqIHiu8TIAhUJ2hoKHc3eAso&url=http%3A%2F%2Fwww.math.utah.edu%2F~milicic%2FEprints%2Fde.pdf&usg=AFQjCNF7J93I5ztatDP0x8yIRPsRcWKelQ&sig2=nJ4tYtTnkl-lD2n_ZBwe4Q&bvm=bv.105039540,d.d2s
http://www.amazon.com/Ordinary-Differential-Equations-Complex-Mathematics/dp/0486696200
Hi Ahmed
Thanks for your interest in answering my query. But kindly see my question carefully. I am looking for applications of the theory in physical and engineering sciences. What you have pointed out are references on mathematics of differential equations in complex domain. Kindly tell me if you know an application to real world problem.
Virendra
both maxwell theory of electromagnetism and wave mechanics are full of examples with the use of complex functions describing waves, and the extended use of usually linear differential equations, both ordinary and partial.
Hi Juan
Thanks for your answer. I agree that differential equations in which independent variables are purely space (not time, which gives a causal structure) the differentiation can be extended to complex. In fact all classical orthogonal functions are eigenfunctions of Sturm-Liouville operators in complex Hilbert spaces. Here the independent variable can be treated as complex since no reference to time is usually necessary. It is truly the application to dynamical systems arising in Physical science or engineering is what should be intended in my question. So again in wave mechanics the time variable cannot be extended to a complex variable. Do you have suggestions?
In the quantum theory, the independent variables wheather spatial or time,
on which the wave function depends, area almost always real, as far as i know.
The wave function itself can be a Complex function. If there is a magnetic field
it must be complex.
For travelling waves you use something like
exp i(kx-omega t), k wavector and omega frequency. This is a complex function.
The quantum hall effect wheather integer or fractional makes heavy use of complex functions.
General references;
Jackson for electromagnetic theory is a good reference and full of math.
Landau and Lifschitz for the quantum theory is a good older reference.
Goldstein for Classical Mechanics is the standard reference
For a two dimensional problem, in principle , you can use complex variables as an independent variable.(spatial part) Maybe in hydrodynamics, or in Laughlins theory of the integer quantum hall effect, you can find this use.
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