查看“︁隆梅尔函数”︁的源代码

[[File:LommelS1.png|thumb|right|200px|第一类隆梅尔函数]]
[[File:LommelS2.png|thumb|right|200px|第二类隆梅尔函数]]

'''隆梅尔函数'''是下列隆梅尔方程的两类解：

:<math>z^2 \frac{d^2y}{dz^2} + z \frac{dy}{dz} + (z^2 - \nu^2)y =  z^{\mu+1}.</math>

1880年数学家{{Link-en|隆梅尔|Eugen von Lommel}}首先给出隆梅尔方程的两个解，称为隆梅尔函数：

: <math>s_{\mu,\nu}(z) = \frac{1}{2} \pi  \left[ Y_\nu (z) \int_0^z z^\mu J_\nu (z)\, dz - J_\nu (z) \int_0^z z^\mu Y_\nu (z)\, dz\right]</math>
: <math>\displaystyle S_{\mu,\nu}(z) = s_{\mu,\nu}(z)  -\frac{2^{\mu-1}\Gamma(\frac{1+\mu+\nu}{2})}{\pi\Gamma(\frac{\nu-\mu}{2})}
\left(J_\nu(z)-\cos(\pi(\mu-\nu)/2)Y_\nu(z)\right)</math>

其中 ''J''<sub>ν</sub>(''z'') 是第一类[[贝塞尔函数]]，  ''Y''<sub>ν</sub>(''z'') 是第二类贝塞尔函数。

==参考文献==
*{{Citation | last1=Erdélyi | first1=Arthur | last2=Magnus | first2=Wilhelm | author2-link=Wilhelm Magnus | last3=Oberhettinger | first3=Fritz | last4=Tricomi | first4=Francesco G. | title=Higher transcendental functions. Vol II | publisher=McGraw-Hill Book Company, Inc., New York-Toronto-London | mr=0058756 | year=1953 | url=http://apps.nrbook.com/bateman/Vol2.pdf | accessdate=2014-05-31 | archive-date=2011-07-14 | archive-url=https://web.archive.org/web/20110714210423/http://apps.nrbook.com/bateman/Vol2.pdf | dead-url=no }}
*{{citation|first=E.|last= Lommel|title=Ueber eine mit den Bessel'schen Functionen verwandte Function|journal=  Math. Ann. |volume= 9  |year=1875|pages= 425–444|doi=10.1007/BF01443342|issue=3}}
*{{citation|first=E.|last= Lommel|title=Zur Theorie der Bessel'schen Funktionen IV|journal=  Math. Ann. |volume= 16  |year=1880|pages= 183–208|doi=10.1007/BF01446386|issue=2}}
*{{dlmf|id=11.9|first=R. B. |last=Paris}}
*{{springer|id=l/l060800|first=E.D. |last=Solomentsev}}

==外部链接==
* Weisstein, Eric W. [http://mathworld.wolfram.com/LommelDifferentialEquation.html "Lommel Differential Equation."] {{Wayback|url=http://mathworld.wolfram.com/LommelDifferentialEquation.html |date=20120831043808 }} From MathWorld—A Wolfram Web Resource.
* Weisstein, Eric W. [http://mathworld.wolfram.com/LommelFunction.html "Lommel Function."] {{Wayback|url=http://mathworld.wolfram.com/LommelFunction.html |date=20120831042945 }} From MathWorld—A Wolfram Web Resource.

[[Category:特殊函数]]