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Fourier Transform

Tool to calculate the Fourier transform of an integrable function on R, the Fourier transform is denoted by ^f or F.

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Fourier Transform -

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# Fourier Transform

## Fourier Transform Calculator

 FT Definition used $\frac{1}{\sqrt{2\pi}} \int e^i$ (recomended for physics) $\int e^{-i}$ (appears in mathematics) $\int e^{-i2\pi}$ (alternative in physics)

### What is the Fourier Transform? (Definition)

The Fourier transformation of a function $f$ is denoted $\hat{f}$ (or sometimes $F$), its result (the transform) describes the frequency spectrum of $f$.

Several definitions of the Fourier transform coexist, they are identical except for a multiplicative coefficient (which can simplify the calculations)

For any function $f$ integrable on $\mathbb{R}$, the 3 most common Fourier transforms of $f$ are:

— $(1)$ most used definition in physics / mechanics / electronics, with time $t$ and frequency $\omega$ in rad/sec:

$$\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(t) \, \exp(i \omega t) \, \mathrm{d} t \tag{1}$$

The advantage of the factor $\frac{1}{\sqrt{2\pi}}$ is that it can be reused for the inverse Fourier transform.

— $(2)$ basic mathematical definition, without coefficient:

$$\hat{f}(\omega) = \int_{-\infty}^{+\infty} f(x) \, \exp(-i \omega x) \, \mathrm{d} x \tag{2}$$

— $(3)$ alternative definition in physics:

$$\hat{f}(\omega) = \int_{-\infty}^{+\infty} f(t) \, \exp(-i 2 \pi \omega t) \, \mathrm{d} t \tag{3}$$

### How to calculate the Fourier transform?

The calculation of the Fourier transform is an integral calculation (see definitions above).

On dCode, indicate the function, its variable, and the transformed variable (often $\omega$ or $w$ or even $\xi$).

Example: $f(x) = \delta(t)$ and $\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}}$ with the $\delta$ Dirac function.

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