What is the Fourier transform of this function ?

• hiyok
In summary, the speaker is having difficulty finding the Fourier transform of the function 1/sqrt(q^2 + m^2) and is seeking assistance. They have attempted to use contour integration but are unsure how to proceed due to the pole at q=im. They mention that the solution may involve Bessel functions and that they encountered this integral while trying to find the Green's function for a graphene ribbon.
hiyok
Hi, I have problems finding out the Fourier transform of the following function,

1/\sqrt{q^2 + m^2}, where m\neq 0 denotes a parameter.

It seems easy, but I don't know how. Could anybody show me how to do it ?

hiyok

Last edited:
$$f(q)=\frac{1}{\sqrt{q^2+m^2}}\\ \mathcal{F}(p)=\int_{-\infty}^\infty \frac{e^{-2\pi iqp}}{\sqrt{q^2+m^2}}\;\text{d}q$$ ... this correct?
i.e. you want the forward Fourier transform...

Last edited:
Yes, that is exactly what I meant.

I tried to make a contour and evaluate the residue around the pole q=im. But the order of this pole is not integer. I don't know how to proceed.

Thanks

I met this integral when trying to find out the Green's function for a graphene ribbon.

1. What is the Fourier transform of a sine function?

The Fourier transform of a sine function is a combination of two delta functions located at positive and negative frequencies, with the amplitude of the delta functions being half of the sine wave's amplitude.

2. How do I find the Fourier transform of a cosine function?

The Fourier transform of a cosine function is also a combination of two delta functions at positive and negative frequencies, but with the amplitude of the delta functions being half of the cosine wave's amplitude and with a phase difference of π/2.

3. Can any function have a Fourier transform?

Yes, any function that is square-integrable (meaning its integral over the entire real line is finite) can have a Fourier transform.

4. What is the relationship between a function and its Fourier transform?

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. This means that the Fourier transform gives us information about the frequency components of a function.

5. How does the Fourier transform help in signal processing?

The Fourier transform allows us to analyze signals in the frequency domain, which can be useful for filtering, compression, and other signal processing techniques. It also enables us to transform signals between the time and frequency domains, making it easier to work with certain types of signals.

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