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PHY.K02UF Molecular and Solid State Physics | ||||
The Fourier transform of a function $f(t)$ using the [1,-1] notation is,
$$\mathcal{F}(\omega)= \int\limits_{-\infty}^{\infty}f(t)e^{-i\omega t}dt.$$Using Euler's formula, you can think of this as the projecting $f(t)$ onto its cosine components and its sine components.
$$\mathcal{F}(\omega)= \int\limits_{-\infty}^{\infty}f(t)\cos(\omega t)dt-i\int\limits_{-\infty}^{\infty}f(t)\sin(\omega t)dt.$$Consider a function that is nonzero only in the interval $t_1 < t < t_2$. The Fourier transform in this case is,
$$\mathcal{F}(\omega)= \int\limits_{t_1}^{t_2}f(t)e^{-i\omega t}dt.$$The following form can be used to define $f(t)$ between $t_1$ and $t_2$. The function $f(t)$ as well as its Fourier transform $\mathcal{F}(\omega)$ are tabulated and plotted.