Last geometric statement of Jacobi - Revision history

en>Rjwilmsi: /* References */Journal cites, Added 1 doi to a journal cite using AWB (10365)

2014-08-15T07:59:40Z

References: Journal cites, Added 1 doi to a journal cite using AWB (10365)

← Older revision		Revision as of 09:59, 15 August 2014
Line 1:		Line 1:
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en>David Eppstein: source

2014-03-05T05:39:29Z

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	~~{{context\|date=February 2012}}~~		Hi people GT Racing 2 the actual Car Exp! Un point positif de ce hack, c'est qu'il fonctionne avec toutes les versions d'android et ios. Gt race 2 hack - Gt race 2 triche illimité Cash et Credits a été créé afin de rendre votre jeu plus facilement l'expérience. We worked very hard with this hack device therefore we would love in the event that you could share also along with your pals!<br><br>
	~~{{Refimprove\|date=February 2012}}~~
	:''~~See~~ also ~~[[Nevanlinna theory]]~~

	~~In [[mathematics]]~~, in the ~~field~~ of ~~[[complex analysis]]~~, ~~a '''Nevanlinna function''' is a [[complex function]] which is an [[analytic function]] on~~ the ~~open [[upper half-plane]] ''H''~~ and ~~has non-negative [[imaginary part]]. Functions with~~ this ~~property are sometimes also known~~ as '''~~Herglotz''~~', ~~'''Pick'''~~ or ~~'''R''' functions~~.		If there is certainly a new upgrade, the file will immediately download and install it to ensure that the hack will assist newest version of the overall game. Click the begin switch and wait a couple of minutes when it comes to hack file to detect and apply necessary patch. 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	~~==Integral representation==~~

	~~Every Nevanlinna function ''N'~~' ~~admits~~ a ~~representation~~

	~~:<math> N(z) = C + Dz + \int_{\mathbb{R}} \left(\frac{1}{\lambda~~ - ~~z} - \frac{\lambda}{1~~+~~\lambda^~~2~~} \right) d\mu(\lambda), \quad z\~~in~~\mathbb{H}~~,~~</math>~~

	~~where ''C'~~' is a ~~real constant~~, ~~''D'' is a non~~-~~negative constant~~ and ~~μ is a~~ [~~[Borel measure]] on ''R'' satisfying the growth condition~~

	:~~<math> \int_{\mathbb{R}} \frac{d\mu(\lambda)}{1+\lambda^~~2~~} < \infty~~.~~</math>~~

	~~Conversely~~, ~~every function of~~ this ~~form turns out~~ to ~~be a Nevanlinna function~~.
	~~The constants~~ in ~~this representation~~ are ~~related to the function ''N'' via~~

	:<~~math~~> ~~C = \mathrm{Re}(N(i)) \qquad\text{and}\qquad D = \lim_{y\rightarrow\infty} \frac{N(iy)}{iy}~~ <~~/math~~>

	and the ~~Borel measure μ can be recovered from~~ '~~'N'' by employing~~ the ~~Stieltjes inversion formula~~

	~~:<math> \mu((\lambda_1~~,~~\lambda_2]) = \lim_{\delta\rightarrow0} \lim_{\varepsilon\rightarrow 0} \frac{1}{\pi} \int_{\lambda_1+\delta}^{\lambda_2+\delta} \mathrm{Im}(N(\lambda+i\varepsilon))d\lambda.</math>~~

	~~==Examples==~~

	* Some elementary examples of ~~Nevanlinna functions are (~~with ~~appropriately chosen branch cuts for~~ the ~~first two)~~

	~~::<math> \sqrt{z}, \quad \ln~~(z)~~, \quad \tanh(z)~~ <~~/math~~>

	* A [[linear fractional transformation]]

	::<~~math~~> ~~\frac{az+b}{cz+d} </math>~~

	~~: is a Nevanlinna function if~~

	~~::<math> a^\ast d - bc^\ast = 1 \qquad\text{and}\qquad \mathrm{Im}(b^\ast d) = \mathrm{Im}(a^\ast~~ c~~) = 0 </math>~~

	* If ''S'~~' is a~~ [~~[self~~-~~adjoint operator]] in a [[Hilbert space]]~~ and ~~f is an arbitrary vector, then the function~~

	::<~~math~~> ~~\langle(S-z)^{-1} f,f\rangle~~ <~~/math~~>

	~~: is a Nevanlinna~~ function.

	* If ''M'' and ''N'' are ~~non-constant Nevanlinna functions~~, ~~then their composition is~~ a ~~Nevanlinna function as well~~.

	~~==References==~~
	* {{cite book \| title=Modern analysis and ~~applications \| page=27 \| editor=Vadim Adamyan \| isbn=3-7643-9918-X \| year=2009 \| page=27 }}~~
	* {{cite book \| title=Theory of ~~linear operators in Hilbert space \| author=[[Naum Ilyich Akhiezer]]~~ and I. ~~M. Glazman \| isbn=0~~-~~486~~-~~67748-6 \| year=1993 }}~~
	* {{cite book \| title=Topics in ~~Hardy Classes~~ and ~~Univalent Functions \| author=Marvin Rosenblum~~ and ~~James Rovnyak \| isbn=3-7643-5111-X \| year=1994 }}~~

	~~[[Category~~:~~Mathematical analysis]]~~

	~~{{mathanalysis-stub}}~~

en>Magioladitis: Fix category spacing + general fixes, added orphan tag using AWB (8232)

2012-08-07T14:15:11Z

Fix category spacing + general fixes, added orphan tag using AWB (8232)

New page

{{context|date=February 2012}}
{{Refimprove|date=February 2012}}
:''See also [[Nevanlinna theory]]

In [[mathematics]], in the field of [[complex analysis]], a '''Nevanlinna function''' is a [[complex function]] which is an [[analytic function]] on the open [[upper half-plane]] ''H'' and has non-negative [[imaginary part]]. Functions with this property are sometimes also known as '''Herglotz''', '''Pick''' or '''R''' functions.

==Integral representation==

Every Nevanlinna function ''N'' admits a representation

:<math> N(z) = C + Dz + \int_{\mathbb{R}} \left(\frac{1}{\lambda - z} - \frac{\lambda}{1+\lambda^2} \right) d\mu(\lambda), \quad z\in\mathbb{H},</math>

where ''C'' is a real constant, ''D'' is a non-negative constant and μ is a [[Borel measure]] on ''R'' satisfying the growth condition

:<math> \int_{\mathbb{R}} \frac{d\mu(\lambda)}{1+\lambda^2} < \infty.</math>

Conversely, every function of this form turns out to be a Nevanlinna function.
The constants in this representation are related to the function ''N'' via

:<math> C = \mathrm{Re}(N(i)) \qquad\text{and}\qquad D = \lim_{y\rightarrow\infty} \frac{N(iy)}{iy} </math>

and the Borel measure μ can be recovered from ''N'' by employing the Stieltjes inversion formula

:<math> \mu((\lambda_1,\lambda_2]) = \lim_{\delta\rightarrow0} \lim_{\varepsilon\rightarrow 0} \frac{1}{\pi} \int_{\lambda_1+\delta}^{\lambda_2+\delta} \mathrm{Im}(N(\lambda+i\varepsilon))d\lambda.</math>

==Examples==

* Some elementary examples of Nevanlinna functions are (with appropriately chosen branch cuts for the first two)

::<math> \sqrt{z}, \quad \ln(z), \quad \tanh(z) </math>

* A [[linear fractional transformation]]

::<math> \frac{az+b}{cz+d} </math>

: is a Nevanlinna function if

::<math> a^\ast d - bc^\ast = 1 \qquad\text{and}\qquad \mathrm{Im}(b^\ast d) = \mathrm{Im}(a^\ast c) = 0 </math>

* If ''S'' is a [[self-adjoint operator]] in a [[Hilbert space]] and f is an arbitrary vector, then the function

::<math> \langle(S-z)^{-1} f,f\rangle </math>

: is a Nevanlinna function.

* If ''M'' and ''N'' are non-constant Nevanlinna functions, then their composition is a Nevanlinna function as well.

==References==
* {{cite book | title=Modern analysis and applications | page=27 | editor=Vadim Adamyan | isbn=3-7643-9918-X | year=2009 | page=27 }}
* {{cite book | title=Theory of linear operators in Hilbert space | author=[[Naum Ilyich Akhiezer]] and I. M. Glazman | isbn=0-486-67748-6 | year=1993 }}
* {{cite book | title=Topics in Hardy Classes and Univalent Functions | author=Marvin Rosenblum and James Rovnyak | isbn=3-7643-5111-X | year=1994 }}

[[Category:Mathematical analysis]]

{{mathanalysis-stub}}