Hi, can someone tell me the major applications of complex number? I just know several of them, evaluating integrals, summing series, proving trigonometric identities, theory on control systems, fluid flow analysis, etc. Can someone tell me if there is any other important application of it? Thank you very much!

Everywhere there are matrices or polynomial equations, complex numbers pop up. They even appear in number theory. Complex numbers are useful in geometry also. You can find several examples at my site: Napoleon theorem, Morley's theorem.