e^{iθ} = cos(θ) + i sin(θ)

C = a + ib

C = p + iq

C^{∗} = a − ib

C^{∗} = p − iq

C + C^{∗} = (a + ib) + (a − ib) = 2a

C + C^{∗} = (p + iq) + (p − iq) = 2p

A_{0} + ∑_{n=1}^{∞}[A_{n} cos(nω_{1}t) + B_{n} sin(nω_{1}t)]

A_{0} + ∑_{n=1}^{∞}[A_{n} cos(nω_{1}t) + B_{n} sin(nω_{1}t)] = C_{0} + ∑_{n=1}^{∞}[C_{n}e^{inω_{1}t} + C_{n}^{∗}e^{−inω_{1}t}]

A_{0} = C_{0}

A_{1} cos(ω_{1}t) + B_{1} sin(ω_{1}t) = C_{1}e^{iω_{1}t} + C_{1}^{∗}e^{−iω_{1}t}

C_{1} = p + iq C_{1}^{∗} = p − iq

A_{1} cos(ω_{1}t) + B_{1} sin(ω_{1}t) = (p + iq)e^{iω_{1}t} + (p − iq)e^{−iω_{1}t}

A_{1} cos(ω_{1}t) + B_{1} sin(ω_{1}t) = (p + iq)[cos(ω_{1}t) + i sin(ω_{1}t)] +

(p − iq)[cos(ω_{1}t) − i sin(ω_{1}t)]

A_{1} cos(ω_{1}t) + B_{1} sin(ω_{1}t) =  p cos(ω_{1}t) + ip sin(ω_{1}t)

iq cos(ω_{1}t) + i^{2}q sin(ω_{1}t) p cos(ω_{1}t) − ip sin(ω_{1}t)

− iq cos(ω_{1}t) − i^{2}q sin(ω_{1}t)

A_{1} cos(ω_{1}t) + B_{1} sin(ω_{1}t) = 2p cos(ω_{1}t) − 2q sin(ω_{1}t)

p = \frac{1}{2}A_{1} q = −\frac{1}{2}B_{1}

\begin{aligned}C_{1} & = \frac{1}{2}A_{1} − i\frac{1}{2}B_{1} \\ C_{1}^{∗} & = \frac{1}{2}A_{1} + i\frac{1}{2}B_{1} \end{aligned}

Fred says y(t) = (30 cm) sin(ωt) Jane says y(t′) = (30 cm) cos(ωt′)