I am evaluating the fourier transform of a function + a constant: $\frac{1}{2\pi}\int_{-\infty}^{+\infty}(f(x)+c)e^{-ikx}dx$ equals what?

I am evaluating the fourier transform of a function plus a constant $$c$$:

$$\frac{1}{2\pi}\int_{-\infty}^{+\infty}(f(x)+c)e^{-ikx}dx.$$

As a result, I should get the fourier transform of the function $$f(x)$$ plus a dirac delta: $$\frac{1}{2\pi}\int_{-\infty}^{+\infty}(f(x)+c)e^{-ikx}dx=\hat{f}(k)+c\delta(k).$$

However, I cannot understand whether I am making some mistake or not and the meaning of such a result.

• the integral $\int_{-\infty}^\infty ce^{-ikt}\, dt$ doesnt exists. And the Dirac delta is not a function, so Im not sure what you mean by $\delta(k)$ – Masacroso Jan 8 at 12:11
• Your result is almost correct, the Fourier transform of $1$ is $\delta$ with your normalization, so you should have $c \delta(k)$. – Botond Jan 8 at 12:15
• @Botond I cannot understand the physical meaning. The Fourier transform can be associated to an energy spectrum. In this case do I have an infinite energy at zero frequency? – ARF Jan 8 at 13:13
• I can't really say anything about it without context. But why don't you ask it on PSE? – Botond Jan 8 at 13:35
• I can only assume you are talking about the physical meaning as I do not see a question otherwise. In that case superimposing $\cos(\omega_0 t)$ with low $\omega_0$ to a signal $f(t)$ will result in a disturbed signal on top. In the frequency domain you will see spikes at $\pm \omega_0$. Think of a low constant frequency bass sound on top of a melody. Now lot $\omega_0 \rightarrow 0$. – Diger Jan 8 at 14:32