查看“︁克劳森函数”︁的源代码

[[File:Clausen function.png|thumb|350px|Clausen function plot]]
'''克劳森函数'''是[[丹麦]][[数学家]]{{tsl|en|Thomas_Clausen_(mathematician)|托马斯·克劳森}}最先研究的[[特殊函数]]，定义如下：

:<math>\operatorname{Cl}_2(\varphi)=-\int_0^{\varphi} \log\Bigg|2\sin\frac{x}{2} \Bigg|\, dx:</math>

克劳森函数的[[傅立叶级数]]为

:<math>\operatorname{Cl}_2(\varphi)=\sum_{k=1}^{\infty}\frac{\sin k\varphi}{k^2} = \sin\varphi +\frac{\sin 2\varphi}{2^2}+\frac{\sin 3\varphi}{3^2}+\frac{\sin 4\varphi}{4^2}+ \, \cdots </math>
==基本性质==

:<math>\text{Cl}_2(m\pi) =0, \quad m= 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \cdots </math>

极大值点 :<math>\theta = \frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]</math>

:<math>\text{Cl}_2\left(\frac{\pi}{3}+2m\pi \right) =1.01494160 \cdots </math>

极小值点 :<math>\theta = -\frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]</math>

:<math>\text{Cl}_2\left(-\frac{\pi}{3}+2m\pi \right) =-1.01494160 \cdots </math>

:<math>\text{Cl}_2(\theta+2m\pi) = \text{Cl}_2(\theta) </math>

:<math>\text{Cl}_2(-\theta) = -\text{Cl}_2(\theta) </math>

<ref>Lu and Perez, 1992, </ref>.
==与伯努力多项式的关系==

:<math>B_{2n-1}(x)=\frac{2(-1)^n(2n-1)!}{(2\pi)^{2n-1}} \, \sum_{k=1}^{\infty}\frac{\sin 2\pi kx}{k^{2n-1}}</math>

:<math>B_{2n}(x)=\frac{2(-1)^{n-1}(2n)!}{(2\pi)^{2n}} \, \sum_{k=1}^{\infty}\frac{\cos 2\pi kx}{k^{2n}}</math>


:<math>\text{Sl}_{2m}(\theta) = \frac{(-1)^{m-1}(2\pi)^{2m}}{2(2m)!} B_{2m}\left(\frac{\theta}{2\pi}\right)</math>

:<math>\text{Sl}_{2m-1}(\theta) = \frac{(-1)^{m}(2\pi)^{2m-1}}{2(2m-1)!} B_{2m-1}\left(\frac{\theta}{2\pi}\right)</math>

其中：

:<math>B_n(x)=\sum_{j=0}^n\binom{n}{j} B_jx^{n-j}</math>

:<math> \text{Sl}_1(\theta)= \frac{\pi}{2}-\frac{\theta}{2} </math>

:<math> \text{Sl}_2(\theta)= \frac{\pi^2}{6}-\frac{\pi\theta}{2}+\frac{\theta^2}{4}  </math>

:<math> \text{Sl}_3(\theta)= \frac{\pi^2\theta}{6} -\frac{\pi\theta^2}{4}+\frac{\theta^3}{12}  </math>

:<math> \text{Sl}_4(\theta)= \frac{\pi^4}{90}-\frac{\pi^2\theta^2}{12}+\frac{\pi\theta^3}{12}-\frac{\theta^4}{48} </math>

==与多重对数函数的关系==
<math>Cl2 := -(1/2*I)*(polylog(2, exp(I*\phi))-polylog(2, exp(-I*\phi)))</math>

==参考文献==
<references/>
* {{cite arXiv| first1=Viktor. S. | last1=Adamchik | eprint=math/0308086v1 | title=Contributions to the Theory of the Barnes Function}}
*{{Cite journal | last1=Clausen | first1=Thomas | title=Über die Function sin φ + (1/2<sup>2</sup>) sin 2φ +  (1/3<sup>2</sup>) sin 3φ + etc. | url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0008 | year=1832 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=8 | pages=298–300 | ref=harv | access-date=2015-03-07 | archive-date=2013-10-04 | archive-url=https://web.archive.org/web/20131004215758/http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0008 | dead-url=no }}
* {{cite journal | first1=Van E. | last1=Wood | title=Efficient calculation of Clausen's integral
| url=https://archive.org/details/sim_mathematics-of-computation_1968-10_22_104/page/883 |journal=Math. Comp. | year=1968 | volume=22 | number=104 | pages=883–884 | mr=0239733
|doi = 10.1090/S0025-5718-1968-0239733-9}}
* Leonard Lewin, (Ed.). ''Structural Properties of Polylogarithms'' (1991)  American Mathematical Society, Providence, RI. ISBN 0-8218-4532-2
* {{cite journal| first1=Kurt Siegfried | last1=K&ouml;lbig | title=Chebyshev coefficients for the Clausen function Cl<sub>2</sub>(x)
| url=https://archive.org/details/sim_journal-of-computational-and-applied-mathematics_1995-12-20_64_3/page/n100 |journal=J. Comput. Appl. Math. |year=1995 |volume=64 | number=3 |pages=295–297
|mr=1365432 |doi=10.1016/0377-0427(95)00150-6}} 
* {{cite web |first1=Jonathan M. |last1=Borwein |first2=Armin |last2=Straub |url=http://www.thecarma.net/jon/nielsenrelations.pdf |title=Relations for Nielsen Polylogarithms |accessdate=2015-03-07 |archive-date=2013-12-12 |archive-url=https://web.archive.org/web/20131212084540/http://www.thecarma.net/jon/nielsenrelations.pdf |dead-url=yes }} 
* {{cite journal |first1=Jonathan M. |last1=Borwein |first2=David M. |last2=Bradley |first3=Richard E. |last3=Crandall |title=Computational Strategies for the Riemann Zeta Function |journal=J. Comp. App. Math. |year=2000 |volume=121 |mr=1780051 |pages=247–296 |ref=harv |url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf |doi=10.1016/s0377-0427(00)00336-8 |access-date=2015-03-07 |archive-date=2006-09-25 |archive-url=https://web.archive.org/web/20060925091659/http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf |dead-url=no }}
* {{cite journal|first1=Mikahil Yu. | last1=Kalmykov | first2=A. | last2=Sheplyakov
|title=LSJK - a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine integral
|journal=Comput. Phys. Comm. |year=2005 | volume=172 | pages=45–59
|doi=10.1016/j.cpc.2005.04.013 }} {{arxiv| archive=hep-ph | id=0411100}}
* {{cite arXiv| first1=R. J. | last1=Mathar | eprint=1309.7504 | title=A C99 implementation of the Clausen sums}}
* {{cite web | first1=Hung Jung | last1=Lu | first2=Christopher A. | last2=Perez | url=http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-5809.pdf | title=Massless one-loop scalar three-point integral and associated Clausen, Glaisher, and L-functions | year=1992 | accessdate=2015-03-07 | archive-date=2015-09-24 | archive-url=https://web.archive.org/web/20150924102600/http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-5809.pdf | dead-url=no }}

[[Category:Ζ函數與L函數]]