Confusion about the contour green's function

[wdlang]

i find that most books on green's function are burdened with too much formalism

i am now reading the book by Rammer, which deals with non-equilibrium physics.

The formalism is so lengthy and so confusing. You have to strive hard to remenber the various green's functions, and only to find that there is little concrete applications.

I am haunted with the puzzle: the idea of a closed contour seems very naive, and how this trick can solve non-equilibrium physics?

I still cannot biuld a picture of the green's function

 

[pam]

Study some complex variable theory to learn why Cauchy's theorem requires a closed contour.

 

[blechman]

I am by no means an expert on non-equilibrium physics, and therefore probably can't answer your q's very well. But I can tell you about greens fcns in general:

A Green's fcn is basically the solution of a differential equation with a point-source. For example: the Green's function of electrostatics is just the Coulomb potential. The great thing about Green's functions is that as long as your equation is linear, there is a superposition principle (if f(x) and g(x) are solutions, so is f(x)+g(x)). Therefore if you can break up your source as a sum of "point particles", you can effectively compute your solution to the diffEQ as a sum of Green's functions. Of course, for continuum mechanics, the sums become integrals, but that's the basic idea.

As to the closed-contour technique: that's just Cauchy's theorem from complex analysis (see Pam's response). I see nothing "naive" about relying on a mathematical theorem! The idea is simple enough: if the integrand vanishes fast enough at infinity, if you add a term that is the contour-integral over the semicircle at infinity, you are just adding zero and that is perfectly allowed. But now that you have added that term, you have closed the contour and can use Cauchy's theorem to evaluate the integral. That's all there is to it.

As to "solving non-equilibrium physics": you got me! I don't know what you mean by this!

Green's functions are dealt with pretty hard-core in Jackson's famous E&M textbook. Also, most math method textbooks (Arken & Weber, for example) go into it a bit.

Hope this helps!