Vibrations and Waves · Problem 1-5

The superposition of periodic motions — A.P. French

Physics — Waves · complex-exponential

Problem

To take successive derivatives of

e^{j\theta}

with respect to

\theta

, one merely multiplies by

j

\frac{d}{d\theta}(Ae^{j\theta}) = jAe^{j\theta}

Show that this prescription works if the sinusoidal representation

e^{j\theta} = \cos\theta + j\sin\theta

is used.