Some of the derivations so far without the use of complex numbers have been algebraically quite cumbersome since, repeatedly, it was necessary to break up a harmonic function cos ( ?t ? ) in terms of a sum of cos( ?t)- and sin( ?t) terms and to use elementary trigonometric identities to arrive at a final answer. In problems that go much beyond the simple harmonic oscillator, this can become a considerable burden.

Euler s formula makes it possible to express a harmonic function of time in terms of an exponential function and to define a complex amplitude which contains the characteristics of the harmonic function under consideration.

In a similar manner, time derivatives of the harmonic function can also be expressed in terms of the complex amplitude and differential equations of motion which describe the problems that will be converted into algebraic equations for the complex variable. Once the solution for the complex variable has been obtained, the time dependent functions expressing the real solution can readily be retrieved. The use of complex amplitudes thus avoids many of the algebraic difficulties mentioned above and has been illustrated in examples in the text.