查看“︁休恩函数”︁的源代码

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'''Heun函数'''指HeunB、HeunC、HeunD、HeunG、HeunT等五个函数


==HeunG 函数==
[[File:HeunG.gif|thumb|300px|HeunG  function]]
HeunG函数'''是下列GeneralHeun方程的解

:<math>\frac {d^2w}{dz^2} + 
\left[\frac{\gamma}{z}+ \frac{\delta}{z-1} + \frac{\epsilon}{z-a} \right] 
\frac {dw}{dz} 
+ \frac {\alpha \beta z -q} {z(z-1)(z-a)} w = 0.</math>

===奇点===
HeunG(a, b, \alpha, \beta, \gamma, \delta, z)函数有6个参数，有四个奇点

===展开式===
<math>HeunG(a, b, \alpha, \beta, \gamma, \delta, z) =b*z/(\gamma*a)-(1/2)*(-b-b*a*\delta-b*\gamma*a+b*\delta-b*\alpha-b*\beta-b^2+\alpha*\beta*\gamma*a)*z^2/(\gamma*a^2*(\gamma+1))+O(z^3)</math>
===关系式===
:<math>HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = (1-z)^(1-\delta)*HeunG(a, q-(-1+\delta)*\gamma*a, \beta-\delta+1, \alpha-\delta+1, \gamma, 2-\delta, z)         </math>
:<math>  [HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = (1-z/a)^(-\alpha-\beta+\gamma+\delta)*HeunG(a, q-\gamma*(\alpha+\beta-\gamma-\delta), -\beta+\gamma+\delta, -\alpha+\gamma+\delta, \gamma, \delta, z), And(a <> 0)]       </math>
:<math>[HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = (1-z)^(1-\delta)*(1-z/a)^(-\alpha-\beta+\gamma+\delta)*HeunG(a, q-\gamma*((-1+\delta)*a+\alpha+\beta-\gamma-\delta), -\beta+\gamma+1, -\alpha+\gamma+1, \gamma, 2-\delta, z), And(a <> 0)]         </math>
:<math> [HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = HeunG(1/a, q/a, \alpha, \beta, \gamma, \alpha+\beta-\gamma-\delta+1, z/a), And(a <> 0)]        </math>
:<math>  [HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = (1-z)^(-\alpha)*HeunG(a/(-1+a), (-q+\gamma*\alpha*a)/(-1+a), \alpha, \alpha-\delta+1, \gamma, \alpha-\beta+1, z/(z-1)), And(a <> 1, z <> 1)]       </math>
:<math>         </math>
===超几何函数关系===
:<math>[HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = hypergeom([\alpha, \beta], [\alpha+\beta-\delta+1], z), And(a = 0, q = 0)]</math>
:<math>[HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = hypergeom([\alpha, \beta], [\gamma], z)</math><math>(And(a = 1, q = \alpha*\beta), And(q = \alpha*\beta*a, \delta = \alpha+\beta-\gamma+1))]</math>
==HeunB函数==
[[File:HeunB.gif|thumb|HeunB]]
HeunB(a,b,c,d,z)是下列双合流方程的解
<math>y_zz-\frac{(\alpha+2*z+z^2*\alpha-2*z^3)}{(z+1)^2/(z-1)^2}*y_z+\frac{(\delta+(2*\alpha+\gamma)*z+\beta*z^2)}{(z-1)^3/(z+1)^3*y(z)};</math>

==HeunC函数==
[[File:HeunC.gif|thumb|HeunC]]
HeunC(a,b,c,d,z)是下列微分方程的解
<math>y_zz-\frac{(-z^2*a+(-2-b-c+a)*z+b+1)*y_z)}{(z*(z-1))}-\frac{(1/2)*(((-b-c-2)*a-2*d)*z+(b+1)*a+(-c-1)*b-c-2*e)*y}{(z*(z-1))}</math>
===关系式===
:<math>[HeunC(a, b, c, d, e, z) = (1-z)^(-c)*HeunC(a, b, -c, d, e, z), And(b::(Not(integer)), abs(z) < 1)]</math>
:<math>[HeunC(a, b, c, d, e, z) = exp(-z*a)*HeunC(-a, b, c, d, e, z), And(b::(Not(integer)), abs(z) < 1)]</math>

==HeunD函数==
[[File:HeunD.gif|thumb|HeunD]]
HeunD(a,b,c,d,z)函数是下列微分方程的解
<math>y_zz-\frac{(-2*z^5+4*z^3+z^4*a-2*z-a)*y_z)}{((z-1)^3*(z+1)^3)}-\frac{(-z^2*b+(-c-2*a)*z-d)*y}{((z-1)^3*(z+1)^3)}</math>
===关系式===
<math>[HeunD(a, b, c, d, z) = HeunD(-a, -d, -c, -b, 1/z)]</math>

==HeunT函数==
[[File:HeunT.gif|thumb|HeunT]]
HeunT(a,b,c,z)函数是下列微分方程的解

<math>y_zz-(3*z^2+c)*y_z-((3-b)*z-a)*y</math>

===关系式===
<math>[HeunT(a, b, c, z) = HeunT(j*a, b, j^2*c, j*z), And(j^3 = 1)]</math>
<math>[HeunT(a, b, c, z) = HeunT(a, -b, c, -z)*exp(z^3), And(c = 0)]</math>

==参考文献==
<references/>
*{{Citation | last1=Forsyth | first1=Andrew Russell | title=Theory of differential equations.  4. Ordinary linear equations | origyear=1906 | publisher=[[Dover Publications]] | location=New York  | mr=0123757 | year=1959|url=http://www.archive.org/details/theorydiffeq04forsrich|pages=158}}
* {{citation|first=Karl |last=Heun |url=http://www.digizeitschriften.de/resolveppn/GDZPPN00225140X |title= Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten|journal= Mathematische Annalen |volume=33|page= 161 |year=1889 |doi=10.1007/bf01443849}}
*{{Citation | last1=Maier | first1=Robert S. | title=The 192 solutions of the Heun equation | arxiv=math/0408317 | doi=10.1090/S0025-5718-06-01939-9 | mr=2291838 | year=2007 | journal=[[Mathematics of Computation]]   | volume=76 | issue=258 | pages=811–843}}
*{{Citation | editor1-last=Ronveaux | editor1-first=A. | title=Heun's differential equations | publisher=The Clarendon Press Oxford University Press | series=Oxford Science Publications | isbn=978-0-19-859695-0 | mr=1392976 | year=1995}}
*{{dlmf|id=31|title=Heun functions|first=B. D.|last=Sleeman|first2=V. B. |last2=Kuznetzov}}
*{{Citation | last1=Valent | first1=Galliano | title=Difference equations, special functions and orthogonal polynomials | arxiv=math-ph/0512006 | publisher=World Sci. Publ., Hackensack, NJ | doi=10.1142/9789812770752_0057 | mr=2451210 | year=2007 | chapter=Heun functions versus elliptic functions | pages=664–686}}

{{DEFAULTSORT:Heun}}
[[Category:特殊函数]]
[[Category:常微分方程]]