\magnification=\magstep1


$$
\eqalign{
\cos(\theta) \quad &= \quad \cos(0) \ + \ {{{\theta}^1}\over{1!}} (-\sin(0)) 
                                   \ + \ {{{\theta}^2}\over{2!}} (-\cos(0))
                                   \ + \ {{{\theta}^3}\over{3!}} (\sin(0))
                                   \ + \ {{{\theta}^4}\over{4!}} (\cos(0))
                                   \ + \ ... \cr
                   &= \quad 1      \quad + \quad 0  \quad 
                                   \ - \quad {{{\theta}^2}\over{2!}}  \quad 
                                   \ - \quad 0 \quad 
                                   \ + \quad {{{\theta}^4}\over{4!}}  \quad 
                                   \ + \quad ... \cr 
}
$$

$$
\eqalign{
\sin(\theta) \quad &= \quad \sin(0) \ + \ {{{\theta}^1}\over{1!}} (\cos(0)) 
                                   \ + \ {{{\theta}^2}\over{2!}} (-\sin(0))
                                   \ + \ {{{\theta}^3}\over{3!}} (-\cos(0))
                                   \ + \ {{{\theta}^4}\over{4!}} (\sin(0))
                                   \ + ... \cr
                   &= \quad 0      \quad + \quad {{{\theta}^1}\over{1!}}  \quad 
                                   \ - \quad 0 \quad 
                                   \ - \quad {{{\theta}^3}\over{3!}}  \quad 
                                   \ + \quad 0 \quad 
                                   \ + \quad {{{\theta}^5}\over{5!}} \quad 
                                   \ + \quad ... \cr 
}
$$


$$
\eqalign{
e^{i\theta} \quad &= \quad e^{0}  \ + \ {{{i\theta}^1}\over{1!}} (e^{0}) 
                                   \ + \ {{{i^2\theta}^2}\over{2!}} (e^{0})
                                   \ + \ {{{i^3\theta}^3}\over{3!}} (e^{0})
                                   \ + \ {{{i^4\theta}^4}\over{4!}} (e^{0})
                                   \ + ... \cr
                   &= \quad 1      \quad + \quad {{{i\theta}^1}\over{1!}}  \quad 
                                   \ - \quad {{{\theta}^2}\over{2!}} \quad 
                                   \ - \quad {{{i\theta}^3}\over{3!}}  \quad 
                                   \ + \quad {{{\theta}^4}\over{4!}} \quad
                                   \ + \quad {{{i\theta}^5}\over{5!}} \quad 
                                   \ + \quad ... \cr 
}
$$

$$
{ {d^2 y}\over{dt^2} } \quad = \quad 
            - \thinspace \left( {{k}\over{m}} \right) \thinspace y
                  \quad = \quad - \ \omega^2 y
$$

$$
\omega \quad \equiv  \quad \sqrt{ {{k}\over{m}} }
$$

$$
y(t) \quad = \quad A \cos(\omega t + \phi) 
$$

$$
y(t) \quad = \quad B \cos(\omega t) \ + \ C \sin(\omega t) 
$$

$$
y(0) \quad = \quad 0 {\rm \ m}
$$

$$
{{dy}\over{dt}}|_{t=0} \ = \ v_y(0) \quad = \quad 0.12 {\rm \ m/s} 
$$

$$
y(t) \quad = \quad (-0.04 {\rm \ m}) \cos(\omega t + {{\pi}\over{2}} ) 
$$

$$
y(t) \quad = \quad (0) \ \cos(\omega t) \ + \ (0.04 {\rm \ m})\  \sin(\omega t) 
$$

$$
y(t) \quad = \quad A e^{i(\omega t + \phi)}
$$

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

$$
y(t) \quad = \quad A e^{i(\omega t + \phi)} \quad = 
                    \quad A \cos(\omega t + \phi) \ + \  
                    i\thinspace A \sin(\omega t + \phi)
$$

$$
{{dy}\over{dt}} \quad = \quad v_y(t) \quad = \quad i \omega A e^{i(\omega t + \phi)}
                        \quad = \quad i \omega y(t)
$$

$$
{{d^2y}\over{dt^2}} \quad = \quad a_y(t) \quad = \quad
                        - \omega^2 A e^{i(\omega t + \phi)}  \quad = \quad
                        - \omega^2 y(t)
$$


$$
y(t) \quad = \quad A e^{-i(\omega t + \phi)} \quad = 
                    \quad A \cos(\omega t + \phi) \ - \  
                    i\thinspace A \sin(\omega t + \phi)
$$

$$
y(t) \quad = \quad A e^{i(\omega t)} \ + \ B e^{-i(\omega t)} 
$$

$$
\vec{F}  \quad = \quad 
            - \thinspace k \vec{x} 
$$

$$
\vec{F}  \quad = \quad 
            - \thinspace k \vec{x} \ - \thinspace b \vec{v}
$$


$$
m \thinspace {{d^2 x}\over{dt^2} } \quad = \quad 
            - \thinspace k x \
                  - \thinspace b v
$$


$$
m \thinspace { {d^2 x}\over{dt^2} } \  + 
                  \ b  {{dx}\over{dt}} \ + 
                  \ k x
                  \quad = \quad 0
$$


$$
x(t) \quad = \quad A \cos(\omega t + \phi) \thinspace e^{-t/\tau}
$$

$$
x(t) \quad = \quad A e^{ i (\omega t + \phi)  - (t/\tau)}
$$

$$
v(t) \quad = \quad {{dx}\over{dt}} \quad = \quad 
                 \left[ i \omega - {{1}\over{\tau}} \right] 
                   A e^{ i (\omega t + \phi)  - (t/\tau)}  
               \quad = \quad
                 \left[ i \omega - {{1}\over{\tau}} \right] x(t) 
$$

$$
a(t) \quad = \quad {{d^2x}\over{dt^2}} \quad = \quad 
                 {\left[ i \omega - {{1}\over{\tau}} \right]}^2
                   A e^{ i (\omega t + \phi)  - (t/\tau)}  
               \quad = \quad
                 \left( [{{1}\over{\tau^2}} - \omega^2] 
                      \thinspace - \thinspace i ({{2 \omega}\over{\tau}}) \right) x(t) 
$$

$$
\left( [ {{m}\over{\tau^2}} - m \omega^2 ] \thinspace - 
          \thinspace i [ {{2 m \omega}\over{\tau}} ] \right) x(t)
   \ + \ 
     \left( -{{b}\over{\tau}} \thinspace + i
          \thinspace b \omega \right) x(t) 
   \ + \ 
     \left( k \right)  x(t) \quad = \quad 0
$$


$$
 -{{2 m \omega}\over{\tau}}  \ + \ b \omega \quad = \quad 0
$$

$$
\tau \quad = \quad {{2 m}\over{b}}
$$

$$
{{m}\over{\tau^2}} \ - \ m \omega^2 \ - \ {{b}\over{\tau}} \ + \ k \quad = \quad 0
$$

$$
\omega \quad = \quad \sqrt{ {{k}\over{m}} - {{b^2}\over{4 m^2}} }
$$


$$
\omega \quad = \quad \sqrt{ {\omega_0}^2 - {{b^2}\over{4 m^2}} }
$$


$$
x(t) \quad = \quad A e^{ i (\omega t + \phi)  - (t/\infty)} \quad 
            = \quad A e^{ i (\omega t + \phi) }  
$$

$$
x(t) \quad = \quad A e^{ i (0 + \phi)  - (t/\tau)} \quad 
            = \quad A C e^{ -t/{\tau} }  
$$

$$
x(t) \quad = \quad B t e^{- t/\tau} 
$$

$$
x(t) \quad = \quad A e^{- t/\tau} \ + \ B t e^{- t/\tau} 
$$


$$
\gamma \quad = \quad \sqrt{ {{b^2}\over{4 m^2}}  - {{k}\over{m}} }
$$

$$
\omega \quad = \quad \pm i \gamma
$$

$$
x(t) \quad = \quad A e^{-(\gamma + {{1}\over{\tau}}) t} \ + 
                   \ B e^{-({{1}\over{\tau}} - \gamma) t} 
$$
















\bye 
