The definition of an abscissa is the horizontal coordinate on a geometric plane. An example of an abscissa is the measurement along the y-axis parallel with the x-axis to point p. The first number in a pair; a number on an X-axis (= line) that shows the coordinate (= place along that line) of a point (Definition of abscissa from the Cambridge Academic Content Dictionary © Cambridge University Press). Definition of abscissa in the Definitions.net dictionary. Meaning of abscissa. What does abscissa mean? Information and translations of abscissa in the most comprehensive dictionary definitions resource on the web.

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Related to abscissa: Abscissa of convergence
abscissa
The coordinates for A are (4,3); the abscissa is 4 and the ordinate is 3.

ab·scis·sa

(ăb-sĭs′ə)n.pl.ab·scis·sas or ab·scis·sae(-sĭs′ē)
Symbol x The coordinate representing the position of a point along a line perpendicular to the y-axis in a plane Cartesian coordinate system.
[New Latin (līnea) abscissa, (line) cut off, from Latin abscissa, feminine past participle of abscindere, to abscise; see abscission.]
American Heritage® Dictionary of the English Language, Fifth Edition. Copyright © 2016 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.

abscissa

(æbˈsɪsə) n, pl-scissasor-scissae (-ˈsɪsiː)
(Mathematics) the horizontal or x-coordinate of a point in a two-dimensional system of Cartesian coordinates. It is the distance from the y-axis measured parallel to the x-axis. Also : abscissorabscisse Compare ordinate
[C17: New Latin, originally linea abscissa a cut-off line]
Collins English Dictionary – Complete and Unabridged, 12th Edition 2014 © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003, 2006, 2007, 2009, 2011, 2014

ab•scis•sa

(æbˈsɪs ə)
n., pl. -scis•sas, -scis•sae (-ˈsɪs i)
(in plane Cartesian coordinates) the x-coordinate of a point: its distance from the y-axis measured parallel to the x-axis. Compare ordinate.
[1690–1700; < Latin, feminine of abscissus, past participle of abscindere to cut off =ab-ab- + scindere to divide, tear]
Random House Kernerman Webster's College Dictionary, © 2010 K Dictionaries Ltd. Copyright 2005, 1997, 1991 by Random House, Inc. All rights reserved.
abscissa
The coordinates for A are (4,3); the abscissa is 4.

ab·scis·sa

(ăb-sĭs′ə)
The distance of a point from the y-axis on a graph in the Cartesian coordinate system. It is measured parallel to the x-axis. For example, a point having coordinates (2,3) has 2 as its abscissa. Compare ordinate.
The American Heritage® Student Science Dictionary, Second Edition. Copyright © 2014 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
ordinate, abscissa - The ordinate or Y axis is vertical; the abscissa or X axis is horizontal.
Farlex Trivia Dictionary. © 2012 Farlex, Inc. All rights reserved.
Noun1.abscissa - the value of a coordinate on the horizontal axis
Cartesian coordinate - one of the coordinates in a system of coordinates that locates a point on a plane or in space by its distance from two lines or three planes respectively; the two lines or the intersections of the three planes are the coordinate axes
Based on WordNet 3.0, Farlex clipart collection. © 2003-2012 Princeton University, Farlex Inc.

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d0329201.png$#A+1 = 131 n = 0$#C+1 = 131 : ~/encyclopedia/old_files/data/D032/D.0302920 Dirichlet series

(Automatically converted into $TeX$. Above some diagnostics.)

A series of the form

$$ tag{1 }sum _ { n=1 } ^ infty a _ {n} e ^ {- lambda _ {n} s } ,$$

where the $ a _ {n} $are complex coefficients, $ lambda _ {n} $, $ 0 < lambda _ {n} uparrow infty $, are the exponents of the series, and $ s = sigma + it $is a complex variable. If $ lambda _ {n} = mathop{rm ln} n $, one obtains the so-called ordinary Dirichlet series

$$ sum _ { n=1 } ^ infty frac{a _ {n} }{n ^ {s} } .$$

The series

$$ sum _ { n=1 } ^ infty frac{1}{n ^ {s} }$$

represents the Riemann zeta-function for $ sigma > 1 $. The series

$$ L (s) = sum _ { n=1 } ^ infty frac{chi (n) }{n ^ {s} } ,$$

where $ chi (n) $is a function, known as a Dirichlet character, were studied by P.G.L. Dirichlet (cf. Dirichlet $ L $-function). Series (1) with arbitrary exponents $ lambda _ {n} $are known as general Dirichlet series.

  • 2Dirichlet series with complex exponents.

General Dirichlet series with positive exponents.

Let, initially, the $ lambda _ {n} $be positive numbers. The analogue of the Abel theorem for power series is then valid: If the series (1) converges at a point $ s _ {0} = sigma _ {0} + it _ {0} $, it will converge in the half-plane $ sigma > sigma _ {0} $, and it will converge uniformly inside an arbitrary angle $ mathop{rm arg} ( s - s _ {0} ) < phi _ {0} < pi / 2 $. The open domain of convergence of the series is some half-plane $ sigma > c $. The number $ c $is said to be the abscissa of convergence of the Dirichlet series; the straight line $ sigma = c $is said to be the axis of convergence of the series, and the half-plane $ sigma > c $is said to be the half-plane of convergence of the series. As well as the half-plane of convergence one also considers the half-plane of absolute convergence of the Dirichlet series, $ sigma > a $: The open domain in which the series converges absolutely (here $ a $is the abscissa of absolute convergence). In general, the abscissas of convergence and of absolute convergence are different. But always:

$$ 0 leq a - c leq d , textrm{ where } d = overline{limlimits}; _ {nrightarrow infty } frac{ mathop{rm ln} n }{lambda _ {n} } ,$$

and there exist Dirichlet series for which $ a-c = d $. If $ d=0 $, the abscissa of convergence (abscissa of absolute convergence) is computed by the formula

Abscissa Define

$$ a = c = overline{limlimits}; _ {n rightarrow infty } frac{ mathop{rm ln} a _ {n} }{lambda _ {n} } ,$$

which is the analogue of the Cauchy–Hadamard formula. The case $ d>0 $is more complicated: If the magnitude

$$ beta = overline{limlimits}; _ {n rightarrow infty } frac{1}{lambda _ {n} } mathop{rm ln} left sum _ { i=1 } ^ { n } a _ {i} right $$

is positive, then $ c = beta $; if $ beta leq 0 $and the series (1) diverges at the point $ s = 0 $, then $ c=0 $; if $ beta leq 0 $and the series (1) converges at the point $ s = 0 $, then

$$ c = overline{limlimits}; _ {n rightarrow infty } frac{1}{lambda _ {n} } mathop{rm ln} left sum _ { i=1 } ^ infty a _ {i} right .$$

The sum of the series, $ F (s) $, is an analytic function in the half-plane of convergence. If $ sigma rightarrow + infty $, the function $ F ( sigma ) $asymptotically behaves as the first term of the series, $ a _ {1} e ^ {- lambda _ {1} sigma } $(if $ a _ {1} neq 0 $). If the sum of the series is zero, then all coefficients of the series are zero. The maximal half-plane $ sigma > h $in which $ F (s) $is an analytic function is said to be the half-plane of holomorphy of the function $ F (s) $, the straight line $ sigma = h $is known as the axis of holomorphy and the number $ h $is called the abscissa of holomorphy. The inequality $ hleq c $is true, and cases when $ h<c $are possible. Let $ q $be the greatest lower bound of the numbers $ beta $for which $ F (s) $is bounded in modulus in the half-plane $ sigma > beta $($ q leq a $). The formula

$$ a _ {n} = limlimits _ {T rightarrow infty } frac{1}{2T} intlimits _ { p-iT } ^ { p+iT } F (s) e ^ {lambda _ {n} s } ds, n=1, 2 dots p>q,$$

is valid, and entails the inequalities

$$ a _ {n} leq frac{M ( sigma ) }{e ^ {- lambda _ {n} sigma } } , M ( sigma ) = sup _ {- infty < t < infty } F (sigma + it ) ,$$

which are analogues of the Cauchy inequalities for the coefficients of a power series.

The sum of a Dirichlet series cannot be an arbitrary analytic function in some half-plane $ sigma > h $; it must, for example, tend to zero if $ sigma rightarrow + infty $. However, the following holds: Whatever the analytic function $ phi (s) $in the half-plane $ sigma > h $, it is possible to find a Dirichlet series (1) such that its sum $ F (s) $will differ from $ phi (s) $by an entire function.

If the sequence of exponents has a density

$$ tau = limlimits _ {n rightarrow infty } frac{n}{lambda _ {n} } < infty ,$$

the difference between the abscissa of convergence (the abscissas of convergence and of absolute convergence coincide) and the abscissa of holomorphy does not exceed

$$ delta = overline{limlimits}; _ {n rightarrow infty } frac{1}{lambda _ {n} } mathop{rm ln} left frac{1}{L ^ prime ( lambda _ {n} ) } right , L ( lambda ) = prod _ {n = 1 } ^ infty left ( 1 - frac{lambda ^ {2} }{lambda _ {n} ^ {2} } right ) ,$$

and there exist series for which this difference equals $ delta $. The value of $ delta $may be arbitrary in $ [ 0 , infty ] $; in particular, if $ lambda _ {n+1} - lambda _ {n} geq q > 0 $, $ n = 1 , 2 dots $then $ delta = 0 $. The axis of holomorphy has the following property: On any of its segments of length $ 2 pi tau $the sum of the series has at least one singular point.

If the Dirichlet series (1) converges in the entire plane, its sum $ F (s) $is an entire function. Let

$$ overline{limlimits}; _ {n rightarrow infty } frac{ mathop{rm ln} n }{lambda _ {n} } < infty ;$$

then the R-order of the entire function $ F (s) $(Ritt order) is the magnitude

$$ rho = overline{limlimits}; _ {sigma rightarrow - infty } frac{ { mathop{rm ln} mathop{rm ln} } M ( sigma ) }{- sigma } .$$

Its expression in terms of the coefficients of the series is

$$ - frac{1} rho = overline{limlimits}; _ {n rightarrow infty } frac{ mathop{rm ln} a _ {n} }{lambda _ {n} mathop{rm ln} lambda _ {n} } .$$

One can also introduce the concept of the R-type of a function $ F (s) $.

If

$$ overline{limlimits}; _ {n rightarrow infty } frac{n}{lambda _ {n} } = tau < infty$$

and if the function $ F (s) $is bounded in modulus in a horizontal strip wider than $ 2 pi tau $, then $ F (s) equiv 0 $(the analogue of one of the Liouville theorems).

Dirichlet series with complex exponents.

For a Dirichlet series

$$ tag{2 }F (s) = sum _ {n = 1 } ^ infty a _ {n} e ^ {- lambda _ {n} s }$$

with complex exponents $ 0 < lambda _ {1} leq lambda _ {2} leq dots $, the open domain of absolute convergence is convex. If

$$ limlimits _ {n rightarrow infty } frac{ mathop{rm ln} n }{lambda _ {n} } = 0 ,$$

the open domains of convergence and absolute convergence coincide. The sum $ F (s) $of the series (2) is an analytic function in the domain of convergence. The domain of holomorphy of $ F (s) $is, generally speaking, wider than the domain of convergence of the Dirichlet series (2). If

$$ limlimits _ {n rightarrow infty } frac{n}{lambda _ {n} } = 0,$$

then the domain of holomorphy is convex.

Let

$$ overline{limlimits}; _ {n rightarrow infty } frac{n}{ lambda _ {n} } = tau < infty ;$$

let $ L ( lambda ) $be an entire function of exponential type which has simple zeros at the points $ lambda _ {n} $, $ n geq 1 $; let $ gamma (t) $be the Borel-associated function to $ L ( lambda ) $(cf. Borel transform); let $ overline{D}; $be the smallest closed convex set containing all the singular points of $ gamma (t) $, and let

$$ psi _ {n} (t) = frac{1}{L ^ prime ( lambda _ {n} ) }intlimits _ { 0 } ^ infty frac{L ( lambda ) }{lambda - lambda _ {n} }e ^ {- lambda t } d lambda , n = 1 , 2 , . . . .$$

Then the functions $ psi _ {n} (t) $are regular outside $ overline{D}; $, $ psi _ {n} ( infty ) = 0 $, and they are bi-orthogonal to the system $ { e ^ {lambda _ {n} s } } $:

$$ frac{1}{2 pi i } intlimits _ { C } e ^ {lambda _ {m} t } psi _ {n} (t) d t = left { begin{array}{ll}0 , & m neq n , 1, & m =n , end{array} right .$$

where $ C $is a closed contour encircling $ overline{D}; $. If the functions $ psi _ {n} (t) $are continuous up to the boundary of $ overline{D}; $, the boundary $ partial overline{D}; $may be taken as $ C $. To an arbitrary analytic function $ F (s) $in $ D $(the interior of the domain $ overline{D}; $) which is continuous in $ overline{D}; $one assigns a series:

$$ tag{3 }F (s) sim sum _ {n = 1 } ^ infty a _ {n} e ^ {lambda _ {n} s } ,$$

$$ a _ {n} = frac{1}{2 pi i } intlimits _ {partial overline{D}; } F (t) psi _ {n} (t) d t , n geq 1 .$$

For a given bounded convex domain $ overline{D}; $it is possible to construct an entire function $ L ( lambda ) $with simple zeros $ lambda _ {1} , lambda _ {2} dots $such that for any function $ F (s) $analytic in $ D $and continuous in $ overline{D}; $the series (3) converges uniformly inside $ D $to $ F (s) $. For an analytic function $ phi (s) $in $ D $(not necessarily continuous in $ overline{D}; $) it is possible to find an entire function of exponential type zero,

$$ M ( lambda ) = sum _ {n = 0 } ^ infty c _ {n} lambda ^ {n} ,$$

and a function $ F (s) $analytic in $ D $and continuous in $ overline{D}; $, such that

$$ phi (s) = M ( D ) F (s) = sum _ {n=0 } ^ infty c _ {n} F ^ { (n) } (s) .$$

Then

$$ phi (s) = sum _ {n = 0 } ^ infty a _ {n} M ( lambda _ {n} )e ^ {lambda _ {n} s } , s in D .$$

The representation of arbitrary analytic functions by Dirichlet series in a domain $ D $was also established in cases when $ D $is the entire plane or an infinite convex polygonal domain (bounded by a finite number of rectilinear segments).

References

[1] A.F. Leont'ev, 'Exponential series' , Moscow (1976) (In Russian)
[2] S. Mandelbrojt, 'Dirichlet series, principles and methods' , Reidel (1972)

Abscissa Axis

Abscissa

Comments

References

Abscissa Pronunciation

[a1] G.H. Hardy, M. Riesz, 'The general theory of Dirichlet series' , Cambridge Univ. Press (1915) Zbl 45.0387.03
How to Cite This Entry:
Dirichlet series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_series&oldid=51116
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

Abscissa E Ordenada

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