Complex Representation.

Complex number provide are very useful in representing oscillations. The amplitude and phase of the oscillation can be combined into a single complex number which we shall refer to as the complex amplitude

$$\tilde{A}= A e^{i \phi} \,.$$ (1.6)

Note that we have introduced the symbol $\, \tilde{} \,$ (tilde) to denote complex numbers. The property that

$$e^{i \phi}= \cos \phi + i \sin \phi$$ (1.7)

allows us to represent any oscillating quantity $x(t)=A \cos(\omega_0 t + \phi)$ as the real part of the complex number $\tilde{x}(t)=\tilde{A} e^{i \omega_0 t}$,

$$\tilde{x}(t)= A e^{i (\omega_0t + \phi)}= A [ \cos (\omega_0t + \phi) + i \sin (\omega_0t + \phi)] \,.$$ (1.8)

We calculate the velocity $v$ in the complex representation $\tilde{v}=\dot{\tilde{x}}$. which gives us

Taking only the real part we calculate the particle's velocity

$$v(t)=-\omega_0 A \sin (\omega_0 t + \phi) \,.$$ (1.10)

The complex representation is a very powerful tool which, as we shall see later, allows us to deal with oscillating quantities in a very elegant fashion.
Problem 3: A SHO has position $x_0$ and velocity at the initial time $t=0$. Calculate the complex amplitude $\tilde{A}$ in terms of the initial conditions and use this to determine the particle's position $x(t)$ at a later time $t$.
Solution The initial conditions tell us that ${\rm Re}(\tilde{A})=x_0$ and ${\rm Re}(i \omega_0 \tilde{A})=v_0$. Hence $\tilde{A}=x_0 - i v_0/\omega_0$ which implies that $x(t)=x_0 \cos(\omega_0 t)+(v_0/\omega_0) \sin(\omega_0 t)$.