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

HeunG 函数
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HeunG function
HeunG函数**是下列GeneralHeun方程的解**

{\displaystyle {\frac {d^{2}w}{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.}
### 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

**解析失败 (SVG（MathML可通过浏览器插件启用）：从服务器"http://localhost:6011/zh.wikipedia.org/v1/"返回无效的响应（"Math extension cannot connect to Restbase."）：): {\displaystyle 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)}**

### 关系式

**解析失败 (SVG（MathML可通过浏览器插件启用）：从服务器"http://localhost:6011/zh.wikipedia.org/v1/"返回无效的响应（"Math extension cannot connect to Restbase."）：): {\displaystyle 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) }**

{\displaystyle [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)]}
{\displaystyle [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)]}
**解析失败 (SVG（MathML可通过浏览器插件启用）：从服务器"http://localhost:6011/zh.wikipedia.org/v1/"返回无效的响应（"Math extension cannot connect to Restbase."）：): {\displaystyle [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)] }**

**解析失败 (SVG（MathML可通过浏览器插件启用）：从服务器"http://localhost:6011/zh.wikipedia.org/v1/"返回无效的响应（"Math extension cannot connect to Restbase."）：): {\displaystyle [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)] }**

{\displaystyle }
### 超几何函数关系

{\displaystyle [HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\alpha +\beta -\delta +1],z),And(a=0,q=0)]}
{\displaystyle [HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\gamma ],z)}{\displaystyle (And(a=1,q=\alpha \*\beta ),And(q=\alpha \*\beta \*a,\delta =\alpha +\beta -\gamma +1))]}

HeunB函数
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HeunB
HeunB(a,b,c,d,z)是下列双合流方程的解
{\displaystyle y\_{z}z-{\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)}};}

HeunC函数
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HeunC
HeunC(a,b,c,d,z)是下列微分方程的解
{\displaystyle y\_{z}z-{\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))}}}

### 关系式

{\displaystyle [HeunC(a,b,c,d,e,z)=(1-z)^{(}-c)\*HeunC(a,b,-c,d,e,z),And(b::(Not(integer)),abs(z)<1)]}
{\displaystyle [HeunC(a,b,c,d,e,z)=exp(-z\*a)\*HeunC(-a,b,c,d,e,z),And(b::(Not(integer)),abs(z)<1)]}

HeunD函数
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HeunD
HeunD(a,b,c,d,z)函数是下列微分方程的解
{\displaystyle y\_{z}z-{\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})}}}

### 关系式

{\displaystyle [HeunD(a,b,c,d,z)=HeunD(-a,-d,-c,-b,1/z)]}

HeunT函数
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HeunT
HeunT(a,b,c,z)函数是下列微分方程的解

**解析失败 (SVG（MathML可通过浏览器插件启用）：从服务器"http://localhost:6011/zh.wikipedia.org/v1/"返回无效的响应（"Math extension cannot connect to Restbase."）：): {\displaystyle y\_zz-(3\*z^2+c)\*y\_z-((3-b)\*z-a)\*y}**

### 关系式

**解析失败 (SVG（MathML可通过浏览器插件启用）：从服务器"http://localhost:6011/zh.wikipedia.org/v1/"返回无效的响应（"Math extension cannot connect to Restbase."）：): {\displaystyle [HeunT(a, b, c, z) = HeunT(j\*a, b, j^2\*c, j\*z), And(j^3 = 1)]}**

{\displaystyle [HeunT(a,b,c,z)=HeunT(a,-b,c,-z)\*exp(z^{3}),And(c=0)]}

参考文献
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* Forsyth, Andrew Russell, New York: Dover Publications: 158, 1959 [1906], MR 0123757
* Heun, Karl, Mathematische Annalen, 1889, **33**: 161, doi:10.1007/bf01443849
* Maier, Robert S. Mathematics of Computation, 2007, **76** (258): 811–843, MR 2291838, arXiv:math/0408317 可免费查阅, doi:10.1090/S0025-5718-06-01939-9
* Ronveaux, A. (编), Oxford Science Publications, The Clarendon Press Oxford University Press, 1995, ISBN 978-0-19-859695-0, MR 1392976
* Sleeman, B. D.; Kuznetzov, V. B. Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
* Valent, Galliano, , World Sci. Publ. Hackensack, NJ: 664–686, 2007, MR 2451210, arXiv:math-ph/0512006 可免费查阅, doi:10.1142/9789812770752\_0057