Analytic expression

(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

[1]	N.N. Luzin, "Theory of functions of a real variable" , Moscow (1948) (In Russian)