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Inverse Fourier Transform

Tool to calculate the inverse Fourier transform of a function having undergone a Fourier transform, denoted by ^f or F.

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Inverse Fourier Transform -

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# Inverse Fourier Transform

## Inverse Fourier Transform Calculator

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

## Answers to Questions (FAQ)

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

The inverse Fourier transform (IFT) is the reciprocal operation of a Fourier transform.

Several variants of the Fourier transform exist and differ only by a multiplicative coefficient.

For any transformed function $\hat{f}$, the 3 usual definitions of inverse Fourier transforms are:

— $(1)$ widespread definition for physics / mechanics / electronics calculations, with $t$ the time and $\omega$ in radians per second:

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

— $(2)$ mathematical definition:

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

— $(3)$ alternative definition in physics:

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

### How to calculate the inverse Fourier transform?

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

On dCode, indicate the function, its transformed variable (often $\omega$ or $w$ or even $\xi$) and it's initial variable (often $x$ or $t$).

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

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Inverse Fourier Transform on dCode.fr [online website], retrieved on 2024-06-25, https://www.dcode.fr/inverse-fourier-transform

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