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Difference between revisions of "Maximal term of a series"

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Applying this idea to the study of [[Power series|power series]]
 
Applying this idea to the study of [[Power series|power series]]
  
<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/m/m063/m063000/m0630001.png" /></td> </tr></table>
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$$ \sum_{k=0}^{\infty}c_{k}(z-a)^{k} $$
 
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in one complex variable $z$ with positive radius of convergence $R$, $0<R\leq\infty$, one has in mind the maximal term $\mu(r)$ of the series
in one complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063000/m0630002.png" /> with positive radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063000/m0630003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063000/m0630004.png" />, one has in mind the maximal term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063000/m0630005.png" /> of the series
 
 
 
<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/m/m063/m063000/m0630006.png" /></td> </tr></table>
 
  
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$$ \sum_{k=0}^{\infty}|c_{k}|r^{k},\quad 0<r=|z-a|<R. $$
 
Thus,
 
Thus,
  
<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/m/m063/m063000/m0630007.png" /></td> </tr></table>
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$$|c_{k}|r^{k}\leq\mu(r),\quad k=0,1,\dots$$
 
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The index $\nu(r)$ of the maximal term $\mu(r)$ is called the central index:
The index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063000/m0630008.png" /> of the maximal term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063000/m0630009.png" /> is called the central index:
 
 
 
<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/m/m063/m063000/m06300010.png" /></td> </tr></table>
 
 
 
If there are several terms in modulus equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063000/m06300011.png" />, then the central index is taken to be the largest of the indices of these terms. The function
 
 
 
<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/m/m063/m063000/m06300012.png" /></td> </tr></table>
 
 
 
is non-decreasing and convex; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063000/m06300013.png" /> is a step-function, increases at discontinuity points in natural numbers and is everywhere continuous from the right.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Valiron,  "Les fonctions analytiques" , '''Paris'''  (1954)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Wittich,  "Neuere Untersuchungen über eindeutige analytische Funktionen" , Springer  (1955)</TD></TR></table>
 
 
 
 
 
  
====Comments====
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$$\mu(r)=|c_{\nu(r)}|r^{\nu(r)} .$$
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If there are several terms in modulus equal to $\mu(r)$, then the central index is taken to be the largest of the indices of these terms. The function
  
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$$y=\ln\mu(e^x), \quad -\infty\leq x\leq\infty ,$$
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is non-decreasing and convex; the function $\nu(r)$ is a step-function, increases at discontinuity points in natural numbers and is everywhere continuous from the right.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Pólya,   G. Szegö,   "Problems and theorems in analysis" , '''2''' , Springer (1976) pp. Part IV, Chapt. 1 (Translated from German)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Valiron,  "Les fonctions analytiques" , '''Paris'''  (1954)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Wittich,  "Neuere Untersuchungen über eindeutige analytische Funktionen" , Springer  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G. Pólya, G. Szegö, "Problems and theorems in analysis" , '''2''' , Springer (1976) pp. Part IV, Chapt. 1 (Translated from German)</TD></TR></table>

Revision as of 21:16, 14 January 2012

The term of a convergent series of numbers or functions with positive terms the value of which is not less than the values of all other terms of this series.

Applying this idea to the study of power series

$$ \sum_{k=0}^{\infty}c_{k}(z-a)^{k} $$ in one complex variable $z$ with positive radius of convergence $R$, $0<R\leq\infty$, one has in mind the maximal term $\mu(r)$ of the series

$$ \sum_{k=0}^{\infty}|c_{k}|r^{k},\quad 0<r=|z-a|<R. $$ Thus,

$$|c_{k}|r^{k}\leq\mu(r),\quad k=0,1,\dots$$ The index $\nu(r)$ of the maximal term $\mu(r)$ is called the central index:

$$\mu(r)=|c_{\nu(r)}|r^{\nu(r)} .$$ If there are several terms in modulus equal to $\mu(r)$, then the central index is taken to be the largest of the indices of these terms. The function

$$y=\ln\mu(e^x), \quad -\infty\leq x\leq\infty ,$$ is non-decreasing and convex; the function $\nu(r)$ is a step-function, increases at discontinuity points in natural numbers and is everywhere continuous from the right.

References

[1] G. Valiron, "Les fonctions analytiques" , Paris (1954)
[2] H. Wittich, "Neuere Untersuchungen über eindeutige analytische Funktionen" , Springer (1955)
[3] G. Pólya, G. Szegö, "Problems and theorems in analysis" , 2 , Springer (1976) pp. Part IV, Chapt. 1 (Translated from German)
How to Cite This Entry:
Maximal term of a series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_term_of_a_series&oldid=18824
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article