View source for Analytic expression

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''(formula)''

The totality of operations to be performed in a certain sequence on the value of an argument and on the constants in order to obtain the value of the function. Every function in one unknown $x$ with not more than a countable number of discontinuities has an analytic expression $A(x)$ involving only three operations (addition, multiplication, passing to the limit by rational numbers), performed not more than a countable number of times, starting from an argument $x$ and from the constants, e.g.
\[
\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}
\]
If there is at least one analytic expression describing a given function, there are infinitely many such expressions. Thus, the function which is identically equal to zero is expressed by the series
\[
  0 = \sum_{n=1}^\infty \frac{x^{n-1}(x-n)}{n!}+1
\]
and from any analytic expression $A(x)$ it is always possible to obtain another one which is identically equal to the first:
\[
  A(x) + B(x)\left(\sum_{n=1}^\infty \frac{x^{n-1}(x-n)}{n!}+1\right),
\]
where $B(x)$ is an arbitrary analytic expression.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Luzin,   "Theory of functions of a real variable" , Moscow  (1948)  (In Russian)</TD></TR></table>