In what domain of physics are complex numbers used?

I rarely, or never seen complex numbers used to make physic discoveries.





can you give an example?In what domain of physics are complex numbers used?
A lot of times, complex numbers are used where real numbers can be. Many times, working with complex numbers is easier to manipulate than with their real number counterparts. For instance, in trigonometry, sin(x), cos(x)..etc can all be replace by complex exponents e^(ix) due to Euler's formula e^(ix)=cos(x)+i*sin(x) .. when you are doing lots of algebra with trig expressions... using complex numbers avoids having to know tons of trigonometric identities... it makes life simpler. Electrical engineers (and physicists) use this application a lot. Physicists use this complex number notation a lot when dealing with electromagnetic waves (among other things). As I have said before, for this stuff, one does not HAVE to use complex numbers.. it just makes things a little easier.





Complex numbers can also be used to evaluate some real number integrals that are generally 'impossible' to integrate, or at least impossible using standard methods (You can look up Cauchy Residue Theorem if you are interested in this aspect of complex numbers)





In general, quantum mechanics MUST use complex numbers. The theory does not work with just real numbers. So, absolutely complex numbers must be used in quantum mechanics. Depending on where you are at in your education, the quantum mechanics book by Griffiths is a good advanced undergraduate text.In what domain of physics are complex numbers used?
If you are in New York the position of Chicago could be specified by a distance (r) and a direction (u); but not just by a distance. The distance and direction is called a vector. The position could also be expressed as a complex number.


[r*sin(u) + i*r*cos(u)]


Both representation describe the same position for Chicago. Sometimes one is easier to use than the other. Sometimes using one over the other clarifies the problem. Choice is your's as long as you use them correctly.
I believe electronics uses them a lot.