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Difference between revisions of "Analytic expression"

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''(formula)''
 
''(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012230/a0122301.png" /> with not more than a countable number of discontinuities has an analytic expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012230/a0122302.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012230/a0122303.png" /> and from the constants, e.g.
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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.
 
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\[
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012230/a0122304.png" /></td> </tr></table>
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\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}
 
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\]
 
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
 
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
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\[
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  0 = \sum_{n=1}^\infty \frac{x^{n-1}(x-n)}{n!}+1
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\]
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and from any analytic expression $A(x)$ it is always possible to obtain another one which is identically equal to the first:
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\[
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  A(x) + B(x)\left(\sum_{n=1}^\infty \frac{x^{n-1}(x-n)}{n!}+1\right),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012230/a0122305.png" /></td> </tr></table>
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where $B(x)$ is an arbitrary analytic expression.
 
 
and from any analytic expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012230/a0122306.png" /> it is always possible to obtain another one which is identically equal to the first:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012230/a0122307.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012230/a0122308.png" /> 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 14:03, 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=29016
This article was adapted from an original article by B.V. Kutuzov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article