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Difference between revisions of "Analytic expression"

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\[
 
\[
 
   A(x) + B(x)\left(\sum_{n=1}^\infty \frac{x^{n-1}(x-n)}{n!}+1\right),
 
   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.
 
where $B(x)$ is an arbitrary analytic expression.
  
 
====References====
 
====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>
 
<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>

Revision as of 14:04, 1 December 2012

(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)
How to Cite This Entry:
Analytic expression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_expression&oldid=29017
This article was adapted from an original article by B.V. Kutuzov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article