Classical mechanics

I helps to simplify the manipulation and solution of equations.

d'Alembert's solution to the wave equation would be,

u(x,t) = F(x-ct) + G(x+ct)

which can be re-written as

u(x,t) = f(x,t) + g(x,t)

and we can write

f(x,t) = A.exp[i(kx - wt)]
g(x,t) = B.exp[i(kx + wt)]

where

w = kc.

Although the deformation u(x,t) will clearly have to be a real number for all values of x and t, it turns out to be useful to consider complex soutions to the wave equation as well. The real and imaginary parts of a complex solution will individually satisfy the wave equation, so a complex solution encodes two real solutions.

Where a solution involves trigonometric functions, e.g.

u(x,t) = A.cos(x - ct) + B.cos(x + ct)

rather than a complex exponential function, then the solution and algebraic manipulation of the latter is often much easier than the former.