Difference between revisions of "Analytic expression"
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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 | + | 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 | 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. | |
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− | where | ||
====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) |
Analytic expression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_expression&oldid=29016