\magnification=\magstep1

$$
e^{i\theta} \quad = \quad \cos(\theta) \ + \ i \sin(\theta)
$$

$$
C \quad = \quad a \ + \ i b
$$

$$
C \quad = \quad p \ + \ i q
$$


$$
C^* \quad = \quad a \ - \ i b
$$


$$
C^* \quad = \quad p \ - \ i q
$$


$$
C + C^* \quad = \quad (a + i b) \ + \ (a - i b) \quad  = \quad 2 a
$$

$$
C + C^* \quad = \quad (p + i q) \ + \ (p - i q) \quad  = \quad 2 p
$$


$$
A_0 \ + \ \sum_{n=1}^{\infty} \bigl[ A_n \cos(n \omega_1 t) \ + \ B_n \sin(n \omega_1 t)  \bigr] 
$$

$$
A_0 \ + \ \sum_{n=1}^{\infty} \bigl[ A_n \cos(n \omega_1 t) \ + \ B_n \sin(n \omega_1 t)  \bigr]  \quad = \quad
               C_0  \ + \ 
             \sum_{n=1}^{\infty} \bigl[ C_n e^{i n \omega_1 t} \ + \ 
                                        C_n^* e^{-i n \omega_1 t} \bigr]
$$


$$
A_0 \quad = \quad C_0
$$


$$
 A_1 \cos(\omega_1 t) \ + \ B_1 \sin(\omega_1 t)  \quad = \quad
              C_1 e^{i \omega_1 t} \ + \ 
                                        C_1^* e^{-i \omega_1 t} 
$$

$$
C_1 \quad = \quad p + iq \qquad\qquad C_1^* \quad = \quad p - iq
$$


$$
 A_1 \cos(\omega_1 t) \ + \ B_1 \sin(\omega_1 t)  \quad = \quad
               (p + iq) e^{i \omega_1 t} \ + \ 
                                        (p - iq)  e^{-i \omega_1 t} 
$$


$$
\eqalign{
 A_1 \cos(\omega_1 t) \ + \ B_1 \sin(\omega_1 t)  \quad = \quad
      &(p + iq) \bigl[ \cos(\omega_1 t) \ + \  i \sin(\omega_1 t) \bigr]  \ + \  \cr
      &(p - iq) \bigl[ \cos(\omega_1 t) \ - \ i \sin(\omega_1 t) \bigr]  \cr
}
$$

$$
\eqalign{
 A_1 \cos(\omega_1 t) \ + \ B_1 \sin(\omega_1 t)  \quad = \quad
      &\phantom{-\ }p \cos(\omega_1 t) \ + \  i p \sin(\omega_1 t)  \cr
      &\phantom{-\ }iq \cos(\omega_1 t) \ + \  i^2 q \sin(\omega_1 t)  \cr
      &\phantom{-\ }p \cos(\omega_1 t) \ - \  i p \sin(\omega_1 t)  \cr
      &-iq \cos(\omega_1 t) \ - \  i^2 q \sin(\omega_1 t)  \cr
}
$$

$$
 A_1 \cos(\omega_1 t) \ + \ B_1 \sin(\omega_1 t)  \quad = \quad
      2p \cos(\omega_1 t) \ - \ 2q  \sin(\omega_1 t)  
$$

$$
p \quad = \quad {{1}\over{2}} A_1  \qquad \qquad 
  q \quad = \quad -{{1}\over{2}} B_1
$$

$$
\eqalign{
C_1 \quad &= \quad {{1}\over{2}} A_1 \ - \ i {{1}\over{2}} B_1  \cr
 \phantom{.} \cr
C_1^* \quad &= \quad {{1}\over{2}} A_1 \ + \ i {{1}\over{2}} B_1  \cr
}
$$

$$
{\rm Fred\ says\ } \qquad\qquad  y(t) \quad = \quad (30\thinspace {\rm cm}) \ \sin(\omega t)
$$

$$
{\rm Jane\ says\ } \qquad\qquad  y(t') \quad = \quad (30\thinspace {\rm cm}) \ \cos (\omega t')
$$





\bye
