|
|
| Line 1: |
Line 1: |
| − | Two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a0136601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a0136602.png" /> are called asymptotically equal as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a0136603.png" /> if in some neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a0136604.png" /> (except possibly at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a0136605.png" /> itself) | + | {{TEX|done}} |
| | + | Two functions $f(x)$ and $g(x)$ are called asymptotically equal as $x\to x_0$ if in some neighbourhood of the point $x_0$ (except possibly at $x_0$ itself) |
| | | | |
| − | <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/a013/a013660/a0136606.png" /></td> </tr></table>
| + | $$f(x)=\epsilon(x)g(x),$$ |
| | | | |
| | where | | where |
| | | | |
| − | <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/a013/a013660/a0136607.png" /></td> </tr></table>
| + | $$\lim_{x\to x_0}\epsilon(x)=1,$$ |
| | | | |
| | i.e. | | i.e. |
| | | | |
| − | <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/a013/a013660/a0136608.png" /></td> </tr></table>
| + | $$f(x)=g(x)[1+o(1)],$$ |
| | | | |
| − | as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a0136609.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366010.png" /> is a finite or an infinite point of the set on which the functions under consideration are defined). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366011.png" /> does not vanish in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366012.png" />, this condition is equivalent to the requirement | + | as $x\to x_0$ ($x_0$ is a finite or an infinite point of the set on which the functions under consideration are defined). If $g(x)$ does not vanish in some neighbourhood of $x_0$, this condition is equivalent to the requirement |
| | | | |
| − | <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/a013/a013660/a01366013.png" /></td> </tr></table>
| + | $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1.$$ |
| | | | |
| − | In other words, asymptotic equality of two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366015.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366016.png" /> means, in this case, that the relative error of the approximate equality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366018.png" />, i.e. the magnitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366020.png" />, is infinitely small as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366021.png" />. Asymptotic equality of functions is meaningful for infinitely-small and infinitely-large functions. Asymptotic equality of two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366023.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366024.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366025.png" />, and is reflexive, symmetric and transitive. Accordingly, the set of infinitely-small (infinitely-large) functions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366026.png" /> is decomposed into equivalence classes of such functions. An example of asymptotically-equal functions (which are also called equivalent functions) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366027.png" /> are the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366032.png" />. | + | In other words, asymptotic equality of two functions $f(x)$ and $g(x)$ as $x\to x_0$ means, in this case, that the relative error of the approximate equality of $f(x)$ and $g(x)$, i.e. the magnitude $[f(x)-g(x)]/g(x)$, $g(x)\neq0$, is infinitely small as $x\to x_0$. Asymptotic equality of functions is meaningful for infinitely-small and infinitely-large functions. Asymptotic equality of two functions $f(x)$ and $g(x)$ is denoted by $f(x)\sim g(x)$ as $x\to x_0$, and is reflexive, symmetric and transitive. Accordingly, the set of infinitely-small (infinitely-large) functions as $x\to x_0$ is decomposed into equivalence classes of such functions. An example of asymptotically-equal functions (which are also called equivalent functions) as $x\to x_0$ are the functions $u(x)$, $\sin u(x)$, $\ln[1+u(x)]$, $e^{u(x)}-1$, where $\lim_{x\to x_0}u(x)=0$. |
| | | | |
| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366034.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366035.png" />, then | + | If $f\sim f_1$ and $g\sim g_1$ as $x\to x_0$, then |
| | | | |
| − | <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/a013/a013660/a01366036.png" /></td> </tr></table>
| + | $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=\lim_{x\to x_0}\frac{f_1(x)}{g_1(x)},$$ |
| | | | |
| | where the existence of any one of the limits follows from the existence of the other one. See also [[Asymptotic expansion|Asymptotic expansion]] of a function; [[Asymptotic formula|Asymptotic formula]]. | | where the existence of any one of the limits follows from the existence of the other one. See also [[Asymptotic expansion|Asymptotic expansion]] of a function; [[Asymptotic formula|Asymptotic formula]]. |
| Line 26: |
Line 27: |
| | | | |
| | ====Comments==== | | ====Comments==== |
| − | One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366038.png" /> are of the same order of magnitude at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013660/a01366039.png" /> instead of asymptotically equal. | + | One also says that $f(x)$ and $g(x)$ are of the same order of magnitude at $x_0$ instead of asymptotically equal. |
| | | | |
| | ====References==== | | ====References==== |
| | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Courant, "Differential and integral calculus" , '''1''' , Blackie (1948) pp. Chapt. 3, Sect. 9 (Translated from German)</TD></TR></table> | | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Courant, "Differential and integral calculus" , '''1''' , Blackie (1948) pp. Chapt. 3, Sect. 9 (Translated from German)</TD></TR></table> |
Two functions $f(x)$ and $g(x)$ are called asymptotically equal as $x\to x_0$ if in some neighbourhood of the point $x_0$ (except possibly at $x_0$ itself)
$$f(x)=\epsilon(x)g(x),$$
where
$$\lim_{x\to x_0}\epsilon(x)=1,$$
i.e.
$$f(x)=g(x)[1+o(1)],$$
as $x\to x_0$ ($x_0$ is a finite or an infinite point of the set on which the functions under consideration are defined). If $g(x)$ does not vanish in some neighbourhood of $x_0$, this condition is equivalent to the requirement
$$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1.$$
In other words, asymptotic equality of two functions $f(x)$ and $g(x)$ as $x\to x_0$ means, in this case, that the relative error of the approximate equality of $f(x)$ and $g(x)$, i.e. the magnitude $[f(x)-g(x)]/g(x)$, $g(x)\neq0$, is infinitely small as $x\to x_0$. Asymptotic equality of functions is meaningful for infinitely-small and infinitely-large functions. Asymptotic equality of two functions $f(x)$ and $g(x)$ is denoted by $f(x)\sim g(x)$ as $x\to x_0$, and is reflexive, symmetric and transitive. Accordingly, the set of infinitely-small (infinitely-large) functions as $x\to x_0$ is decomposed into equivalence classes of such functions. An example of asymptotically-equal functions (which are also called equivalent functions) as $x\to x_0$ are the functions $u(x)$, $\sin u(x)$, $\ln[1+u(x)]$, $e^{u(x)}-1$, where $\lim_{x\to x_0}u(x)=0$.
If $f\sim f_1$ and $g\sim g_1$ as $x\to x_0$, then
$$\lim_{x\to x_0}\frac{f(x)}{g(x)}=\lim_{x\to x_0}\frac{f_1(x)}{g_1(x)},$$
where the existence of any one of the limits follows from the existence of the other one. See also Asymptotic expansion of a function; Asymptotic formula.
One also says that $f(x)$ and $g(x)$ are of the same order of magnitude at $x_0$ instead of asymptotically equal.
References
| [a1] | R. Courant, "Differential and integral calculus" , 1 , Blackie (1948) pp. Chapt. 3, Sect. 9 (Translated from German) |