Fractional/Integer Based integrals

Question

I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?

Answer 1

$y=\{x\}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.

In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.

They might appear when working with the Gauss transformation $T(x)=\frac{1}{x}-\lfloor\frac{1}{x}\rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $\mu(B)=\frac{1}{\ln(2)}\int_{B}\frac{dt}{1+t}$.

Answer 2

Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have $$\int_a^bf([x])\,\mathrm{d}x=\sum_{k=a}^{b-1}f(k).$$ Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-\infty$ and $\infty$, respectively.