It is convenient to represent waves using exponential functions. For example a stationary wave can be described by the function: ψ(x) = Aeikx, where the constant A is a real number. The function ψ(x) can be resolved into its real and imaginary parts using Euler's equation,
- ei θ = cosθ + isinθ,
- ψ(x) = Acos(kx) + iAsin(kx).
A travelling stationary wave can be described by exponential function
- ψ(x),(t) = Aei(kx - ωt)
The exponential function has mathematical properties, which makes it more convenient to use than trigonometric functions. For instance the product of an exponential function eA and a second exponential function eB can be evaluated by simply adding the exponents:
- eA + eB = eA + B