Fourier transform formulation of the dirac delta

[ehrenfest]

I have seen two formulations of the dirac delta function with the Fourier transform. The one on wikipedia is
$$\int_{-\infty}^\infty 1 \cdot e^{-i 2\pi f t}\,dt = \delta(f)$$

and the one in my textbook (Robinett) is

$$1/2\pi \int_{-\infty}^\infty 1 \cdot e^{-i f t}\,dt = \delta(f)$$

I do not understand how they are equivalent? How can you just take the 2pi out of the integral?

 

[George Jones]

It's a simple change of variable: if $t'=2\pi t$, then $dt=dt'/2\pi$.

 

[denian]

The two formulations of the Dirac delta function using Fourier transform are indeed equivalent. The difference lies in the definition of the Fourier transform itself. In the first formulation, the Fourier transform is defined as:

F(f) = \int_{-\infty}^\infty f(t) e^{-i 2\pi ft}\,dt

whereas in the second formulation, the Fourier transform is defined as:

F(f) = \int_{-\infty}^\infty f(t) e^{-i ft}\,dt

The only difference between the two definitions is the factor of 2π in the exponent of the integrand. This factor is usually included in the definition of the Fourier transform to make it easier to work with in certain applications. However, mathematically, it is just a convention and does not affect the overall result.

To see how the two formulations are equivalent, we can rewrite the first formulation as:

F(f) = \int_{-\infty}^\infty f(t) e^{-i 2\pi ft}\,dt = \frac{1}{2\pi} \int_{-\infty}^\infty f(t) e^{-i ft}\,dt

This is equivalent to the second formulation, where we have taken the factor of 2π out of the integral. This is allowed because the factor is just a constant and can be pulled out of the integral. Therefore, both formulations give the same result for the Dirac delta function.

In summary, the two formulations of the Dirac delta function using Fourier transform are equivalent and the difference lies in the definition of the Fourier transform itself. The factor of 2π in the first formulation is just a convention and can be taken out of the integral without affecting the overall result.