Legendrova funkcija hi: Razlika med redakcijama

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Legendrova funkcija <math>\chi_{\nu} (z)\, </math> se pojavlja v [[diskretna Fourierjeva transformacija|diskretni Fourierjevi transformaciji]] glede na red ν [[Hurwitzeva funkcija zeta|Hurwitzeve funkcije ζ(''s'', ''q'')]] in tudi kot [[Eulerjev polinom|Eulerjevi polinomi]] z eksplicitnimi zvezami podanimi v posameznih člankih.
 
Legendrova funkcija <math>\chi_{\nu} (z)\, </math> je posebni primer [[Lercheva funkcija zeta|Lerchevega transcendenta]] <math>\Phi(z, s, \alpha)\, </math> in je na ta način podana kot:
 
: <math> \chi_{\nu} (z) = 2^{-\nu}z\,\Phi (z^{2},\nu,1/2) \!\, . </math>
 
== Enakosti ==
 
: <math> \chi_{2} (x) + \chi_{2} (1/x)= \frac{\pi^{2}}{4}-\frac{i \pi}{2}|\ln x|, \qquad ( x > 0) \!\, , </math>
: <math> \frac{\mathrm{d}}{\mathrm{d} x}\chi_{2} (x) = \frac{{\operatorname{arc\, tanh} \,} x}{x} \!\, . </math>
 
== Integralske zveze ==
 
: <math> \int_{0}^{\pi/2} \operatorname{arc\, sin} (r \sin \theta) \mathrm{d} \theta
= \chi_{2} \left( r \right) \!\, , </math>
: <math> \int_{0}^{\pi/2} \operatorname{arc\, tg} (r \sin \theta) \mathrm{d} \theta
= -\frac{1}{2}\int_{0}^{\pi} \frac{ r \theta \cos \theta}{1+ r^{2} \sin^{2} \theta} \mathrm{d}\theta
= 2 \chi_{2} \left( \frac{\sqrt{1+r^{2}}- 1}{r} \right) \!\, , </math>
: <math> \int_{0}^{\pi/2} \operatorname{arc\, tg} (p \sin \theta) \operatorname{arc\, tg} (q \sin \theta) \mathrm{d} \theta = \pi \chi_{2} \left( \frac{\sqrt{1+p^{2}}- 1}{p}\cdot\frac{\sqrt{1+q^{2}}- 1}{q} \right) \!\, , </math>
: <math> \int_{0}^{\alpha}\int_{0}^{\beta} \frac{\mathrm{d} x \mathrm{d} y}{1-x^{2} y^{2}} = \chi_{2}(\alpha\beta), \qquad ( |\alpha\beta|\le 1 ) \!\, . </math>
 
== Viri ==
 
* {{citat|last1= Cvijović|first1= Djurdje|last2= Klinowski|first2= Jacek|url= http://www.ams.org/journal-getitem?pii=S0025-5718-99-01091-1|title= Values of the Legendre chi and Hurwitz zeta functions at rational arguments|journal= [[Mathematics of Computation]]|date= 1999|volume= 68|issue= |pages= 1623-1630|ref= harv}}
* {{navedi splet|last1= Cvijović|first1= Djurdje|year= 2006|url= http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK2-4MG1X3C-6&_user=1793225&_coverDate=11%2F30%2F2006&_alid=512412473&_rdoc=2&_fmt=summary&_orig=search&_cdi=6894&_sort=d&_docanchor=&view=c&_acct=C000053038&_version=1&_urlVersion=0&_userid=1793225&md5=d64e4c1e1d59beb223eefd865b64e422|title= Integral representations of the Legendre chi function|publisher= Elsevier|accessdate= 2006-12-15|language= en|ref= harv}}
 
== Zunanje povezave ==
 
* {{MathWorld|urlname=LegendresChi-Function|title=Legendre's Chi Function}}
* [http://math.stackexchange.com/questions/555882/integral-int-01-frac-arctan2x-sqrt1-x2dx Mathematics Stack Exchange] {{ikona en}}
 
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