Heun Functions and their uses in Physics
Heun Functions and their uses in Physics
复制标题
Heun 函数及其在物理学中的应用
DOI：
10.1142/9789814417532_0002
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
M. Hortaçsu
中科院分区：
文献类型：
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作者：
M. Hortaçsu
今天已知的大多数理论物理学都是用少量的微分方程来描述的。对于线性系统，不同形式的超几何方程或合流超几何方程往往足以描述所研究的系统。这些方程具有幂级数解，具有连续系数之间的简单关系，并且/或者可以用简单的积分变换表示。如果问题是非线性的，通常使用Painlev方程的一种形式。然而，在一些重要的例子中，我们必须使用高阶方程。Heun方程就是其中的一个例子，最近在广义相对论和天体物理学的问题中经常遇到它。它的特殊的和汇合的形式被命名为Mathieu， Lam和Coulomb球面方程。对于这些方程，每当写出幂级数的解时，我们找到的不是级数中系数之间的双向递推关系，而是三个或四个不同系数之间的递推关系。使用更简单的函数的积分变换解也无法得到。在后来的几年里，这个方程在物理和数学文献中的应用爆炸式增长，与1889年引入这个方程到2010年的时间相比，在过去的七年里，使用这些解的论文增加了一倍多。我们用SCI数据得出了这个结论，虽然不精确，但大致正确。使用更简单的函数的积分变换解也无法得到。这里将介绍这个方程，并举例说明它的应用，特别是在广义相对论文献中。
Most of the theoretical physics known today is described using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlev\'{e} equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lam\'{e} and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable.The use of this equation in physics and mathematical literature exploded in the later years, more than doubling papers with these solutions in the last seven years, compared to time period since this equation was introduced in 1889 up to 2010. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. An integral transform solution using simpler functions also is not obtainable. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given.