Last geometric statement of Jacobi: Difference between revisions

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{{context|date=February 2012}}
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{{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.  
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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}}

Revision as of 07:39, 5 March 2014

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Decide to try our secure and totally undetectable GT RACING 2 HACK TOOL. GT Racing 2: The Real automobile knowledge is a true-to-life automotive journey featuring the absolute most prestigious cars in the world! I suppose you wish to download GT Racing 2: The Real vehicle Experience Hack appliance and Cheats! Right here you can get cheats and hack for GT Racing 2: the actual automobile Experience at no cost!

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