Differential equations with complex functions?

[jonjacson]

Hi folks,

When you have a differential equation and the unknown function is complex, like in the Schrödinger equation, What methods should you use to solve it?

I mean, there is a theory of complex functions, Laurent series, Cauchy integrals and so on, I guess if it would be possible to integrate directly you could solve the equation using these methods but if you can't integrate directly, Do you just separate the equation in real and imaginary parts and then solve them separately?

If you know any book explaining how to solve differential equations that have complex unkown functions would be great!

 

[jasonRF]

The unknown function (the wave function) in the Schrödinger equation is a complex valued function of real variables. The derivatives are with respect to the real variables representing time and space; they are not with respect to complex variables so complex analysis (which deals with functions of complex variables) is not required.

You can solve these kinds of equations the same way you solve the differential equations you presumably already know, but just allow for complex constants. For example, solutions of
$$\frac{df}{dt} = i a f$$
for a $f$ a complex valued function of the real variable $t$ are of the form
$$f(t) = C \, e^{i a t}$$
where in general $C$ can be complex, depending on the initial conditions.

No special book required for this topic.

jason

 

[jonjacson]

I understand.

So you can consider i as just a constant like the mass?

 

[lurflurf]

My favorite such book is
Ordinary Differential Equations in the Complex Domain by Einar Hille
I wish there were a comparable book on partial differential equations.
Books on applied mathematics, differential equations, and and complex methods touch on this but not in great detail.
Obviously books on quantum mechanics spend a lot of time solving the Schrödinger equation and books on other topics spend time solving their equations as well.
As you point out there are obvious things to try extending real methods or splitting complex functions into parts.

 

[jasonRF]

I understand.

So you can consider i as just a constant like the mass?

Yep.

 

[jasonRF]

My favorite such book is
Ordinary Differential Equations in the Complex Domain by Einar Hille
I wish there were a comparable book on partial differential equations.
Books on applied mathematics, differential equations, and and complex methods touch on this but not in great detail.

That looks like an interesting, although not easy, book. I've only learned some of the basics, like contour integral approaches (almost like a generalized Laplace transform representation) that I first learned from Budden's "Propagation of Radio Waves". Very useful for getting representations for asymptotic expansions. Ince's classic book also covers this as well as other relevant topics int he second part of the book, but it is old-fashioned and difficult reading (at least for me).

I don't think the OP needs this if they are just starting out with the Schrödinger equation, though.

Jason

 

[lurflurf]

It is a shame that books like A treatise on differential equations by Andrew Russell Forsyth and the Hille and Ince books are out of fashion. They are full of gems. Funny how as we develop new methods we loose old ones. Kids today can use iphones but do they know how to skin a rabbit?

 

[jonjacson]

Thanks for all the answers folks.

lurflurf if you like classics you can read George Boole, I know it is more than 1 hundred years old but still a nice book!