# Digit function properties

Let $f_n(x)$ be defined as the $n$th digit of the number $x$.

The result of $f_n(x)$ can be only ${0,1,2,3,4,5,6,7,8,9}$ for base 10.

For example, if $x=12.46$, then

$f_2(x)=0$;$f_1(x)=1$;$f_0(x)=2$;$f_{-1}(x)=4$; $f_{-2}(x)=6$ ; $f_{-3}(x)=0$.

If we have such function , we can write any real number easily as shown below:

$x=\sum \limits_{n=-\infty}^\infty f_n(x) 10^n$

I tried to find power series expression of the function. $f_n(x)=a_0(n)+a_1(n)x+a_2(n)x^2+\cdots$

\begin{align*} x&=\sum \limits_{n=-\infty}^\infty f_n(x) 10^n\\ &=\sum \limits_{n=-\infty}^\infty (a_0(n)+a_1(n)x+a_2(n)x^2+\cdots ) 10^n\\ \sum \limits_{n=-\infty}^\infty a_0(n) 10^n&=0\\ \sum \limits_{n=-\infty}^\infty a_1(n) 10^n&=1\\ \sum \limits_{n=-\infty}^\infty a_2(n) 10^n&=0 \end{align*}

But this do not give me so many thing to define $a_k(n)$

Is it possible to find $a_k(n)$ with some method that known?

I also wonder what the function properties of $f_n(x)$ are? (such as $f_n(x+y)$, $f_n(x.y)$ etc.) I wonder the literature about the function.

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Maclaurin series expansion (which is what you are doing) approximates the behavior of the function in a small interval around 0. But $f_n(x)$ is identically 0 in the interval $[0, 10^{-n}]$, so your Maclaurin expansion will have $a_i = 0$ for all $i$. Power series are not going to work here. – MJD May 6 '12 at 20:06

As @Mark Dominus said you won't be able to solve for your $a_k(n)$, but you can find a Fourier series for $f_n$.
First $f_n(x) = f_0(10^{-n}x)$ and $x = \sum 10^n f_0(10^{-n}x)$ so I will only deal with $f_0$.
$f_0(x+10) = f_0(x)$, so let us extend $f_0$ to the negative numbers by $f_0(x-10)=f_0(x)$ so $f_0$ is periodic over all $\mathbb{R}$. For cleanliness let's also define $$f_0(k)=\lim_{\epsilon\to 0}\frac{f_0(k-\epsilon)+f_0(k+\epsilon)}{2}$$ at the round integers $k$, so e.g. $f_0(2.9999\ldots) = f_0(3.0) = 2.5$ and $f_0(29.9999\ldots)=f_0(30.0)=4.5$. Also let $g_0(x) = f_0(x)-4.5$, then $g_0$ is an odd periodic function and has a Fourier sine series. In fact $g_0$ is the difference of two sawtooth waves. It's fairly straightforward to find $$g_0(x) = -\frac{10}{\pi}\sum_{k=1}^\infty b_k \sin\left(\frac{k\pi x}{5}\right)$$ where $$b_k = \begin{cases}0 & \mathrm{if}~10\mid k\\ 1/k & \mathrm{otherwise}\end{cases}$$
Really you shew very interesting results about the function.Thanks a lot. Can I find correctly nth digit of $\pi$ ,$e$ with your formula without calculating other digits. Can It be used for finding record digits of any irrational number? Thanks for advice. – Mathlover May 18 '12 at 10:31