Heun函数

**Heun 函數**指 HeunB、HeunC、HeunD、HeunG、HeunT 等五個函數

 

## HeunG 函數

HeunG 函數**是下列 GeneralHeun 方程的解**

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

### 奇點

HeunG (a, b, \alpha, \beta, \gamma, \delta, z) 函數有 6 個參數，有四個奇點

### 展開式

$$
HeunG(a,b,\alpha ,\beta ,\gamma ,\delta ,z)=b*z/(\gamma *a)-(1/2)*(-b-b*a*\delta -b*\gamma *a+b*\delta -b*\alpha -b*\beta -b^{2}+\alpha *\beta *\gamma *a)*z^{2}/(\gamma *a^{2}*(\gamma +1))+O(z^{3})
$$

### 關係式

　
: $HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=(1-z)^{(}1-\delta )*HeunG(a,q-(-1+\delta )*\gamma *a,\beta -\delta +1,\alpha -\delta +1,\gamma ,2-\delta ,z)$

　
: $[HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=(1-z/a)^{(}-\alpha -\beta +\gamma +\delta )*HeunG(a,q-\gamma *(\alpha +\beta -\gamma -\delta ),-\beta +\gamma +\delta ,-\alpha +\gamma +\delta ,\gamma ,\delta ,z),And(a<>0)]$

　
: $[HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=(1-z)^{(}1-\delta )*(1-z/a)^{(}-\alpha -\beta +\gamma +\delta )*HeunG(a,q-\gamma *((-1+\delta )*a+\alpha +\beta -\gamma -\delta ),-\beta +\gamma +1,-\alpha +\gamma +1,\gamma ,2-\delta ,z),And(a<>0)]$

　
: $[HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=HeunG(1/a,q/a,\alpha ,\beta ,\gamma ,\alpha +\beta -\gamma -\delta +1,z/a),And(a<>0)]$

　
: $[HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=(1-z)^{(}-\alpha )*HeunG(a/(-1+a),(-q+\gamma *\alpha *a)/(-1+a),\alpha ,\alpha -\delta +1,\gamma ,\alpha -\beta +1,z/(z-1)),And(a<>1,z<>1)]$

　
: $$

### 超幾何函數關係

　
: $[HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\alpha +\beta -\delta +1],z),And(a=0,q=0)]$

　
: $[HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\gamma ],z)$ $(And(a=1,q=\alpha *\beta ),And(q=\alpha *\beta *a,\delta =\alpha +\beta -\gamma +1))]$

## HeunB 函數

HeunB (a,b,c,d,z) 是下列雙合流方程的解 $y_{z}z-{\frac {(\alpha +2*z+z^{2}*\alpha -2*z^{3})}{(z+1)^{2}/(z-1)^{2}}}*y_{z}+{\frac {(\delta +(2*\alpha +\gamma )*z+\beta *z^{2})}{(z-1)^{3}/(z+1)^{3}*y(z)}};$

## HeunC 函數

HeunC (a,b,c,d,z) 是下列微分方程的解 $y_{z}z-{\frac {(-z^{2}*a+(-2-b-c+a)*z+b+1)*y_{z})}{(z*(z-1))}}-{\frac {(1/2)*(((-b-c-2)*a-2*d)*z+(b+1)*a+(-c-1)*b-c-2*e)*y}{(z*(z-1))}}$

### 關係式

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

　
: $[HeunC(a,b,c,d,e,z)=exp(-z*a)*HeunC(-a,b,c,d,e,z),And(b::(Not(integer)),abs(z)<1)]$

## HeunD 函數

HeunD (a,b,c,d,z) 函數是下列微分方程的解 $y_{z}z-{\frac {(-2*z^{5}+4*z^{3}+z^{4}*a-2*z-a)*y_{z})}{((z-1)^{3}*(z+1)^{3})}}-{\frac {(-z^{2}*b+(-c-2*a)*z-d)*y}{((z-1)^{3}*(z+1)^{3})}}$

### 關係式

$$
[HeunD(a,b,c,d,z)=HeunD(-a,-d,-c,-b,1/z)]
$$

## HeunT 函數

HeunT (a,b,c,z) 函數是下列微分方程的解

$$
y_{z}z-(3*z^{2}+c)*y_{z}-((3-b)*z-a)*y
$$

### 關係式

 $[HeunT(a,b,c,z)=HeunT(j*a,b,j^{2}*c,j*z),And(j^{3}=1)]$
 $[HeunT(a,b,c,z)=HeunT(a,-b,c,-z)*exp(z^{3}),And(c=0)]$

## 參考文獻

* Forsyth, Andrew Russell, Theory of differential equations. 4. Ordinary linear equations, New York: Dover Publications: 158, 1959 [1906], MR 0123757
* Heun, Karl, Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten, Mathematische Annalen, 1889, **33**: 161, doi:10.1007/bf01443849
* Maier, Robert S., The 192 solutions of the Heun equation, Mathematics of Computation, 2007, **76** (258): 811–843, MR 2291838, arXiv:math/0408317 , doi:10.1090/S0025-5718-06-01939-9
* Ronveaux, A. (編), Heun's differential equations, Oxford Science Publications, The Clarendon Press Oxford University Press, 1995, ISBN 978-0-19-859695-0, MR 1392976
* Sleeman, B. D.; Kuznetzov, V. B., Heun functions, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
* Valent, Galliano, Heun functions versus elliptic functions, Difference equations, special functions and orthogonal polynomials, World Sci. Publ., Hackensack, NJ: 664–686, 2007, MR 2451210, arXiv:math-ph/0512006 , doi:10.1142/9789812770752\_0057