 i \sin(\omega t)  \]`


(iii) The discrete Fourier transform

`\[z_n=\frac{1}{N}\sum_{m=0}^{N-1} Z_m e^{i2\pi nm/N},\quad\quad\quad  Z_m \equiv  \sum_{n=0}^{N-1}z_n e^{-i2\pi nm /N} \]`

(iv) The continuous Fourier transform

`\begin{equation}
  z(t) =\frac{1}{2\pi} \int_{-\infty}^{\infty} Z(\omega)\, e^{i\omega t} d\omega,\quad\quad Z(\omega)\equiv  \int_{-\infty}^{\infty} z(t)\, e^{-i\omega t}  d t
\end{equation}`

*Note (iii) becomes (iv) as a limit; (iv) gives the reflection formula*
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