mantissa function

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If we subtract from a real number $x$ the greatest integer not exceeding $x$, we obtain a number $y$ between 0 and 1, which can equal 0 if $x$ is an integer.\, In other \PMlinkescapetext{words},
$$y \;=\; x\!-\!\lfloor{x}\rfloor,$$
where $\lfloor{x}\rfloor$ is the floor of $x$.
Such a number $y$ is called the {\em mantissa} of $x$.\, So we have for example\\

$2.7-2 \;=\; 0.7$,\\
$1.7-1 \;=\; 0.7$,\\
$0.7-0 \;=\; 0.7$,\\
$-0.3\!-\!(-1) = 0.7$,\\
$-1.3\!-\!(-2) = 0.7,$\\

i.e. these numbers 2.7, 1.7, 0.7, $-0.3$, $-1.3$ at mutual distances an integer have the same mantissa (0.7).\, This is apparently always true --- thus the {\em mantissa function}
$$x \mapsto x\!-\!\lfloor{x}\rfloor$$

The mantissa is identic with the mantissa used in the Briggsian logarithm calculations.\\

When $x$ increases from an integer $n$ towards the next integer $n\!+\!1$, its mantissa $x\!-\!\lfloor{x}\rfloor$ increases with the same speed from 0 tending to 1, but at $n\!+\!1$ it falls back to 0.

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Being a periodic function, the \PMlinkname{Fourier expansion}{DeterminationOfFourierCoefficients} of the function is easy to form:
$$x\!-\!\lfloor{x}\rfloor \;=\; \frac{1}{2}-\sum_{n=1}^\infty\frac{\sin 2n\pi{x}}{n\pi}$$
This is valid for\, $x \not\in \mathbb{Z}$,\, since the series gives in the jump discontinuity points the arithmetic means ($= \frac{1}{2}$) of \PMlinkname{left and right limits}{OneSidedLimit}.
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