Heun函數 指HeunB、HeunC、HeunD、HeunG、HeunT等五個函數
HeunG function HeunG函數是下列GeneralHeun方程的解
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{\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.}
HeunG(a, b, \alpha, \beta, \gamma, \delta, z)函數有6個參數，有四個奇點
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{\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})}
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{\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)}
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{\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)]}
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{\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)]}
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{\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)]}
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{\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)]}
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{\displaystyle [HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\alpha +\beta -\delta +1],z),And(a=0,q=0)]}
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{\displaystyle [HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\gamma ],z)}
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{\displaystyle (And(a=1,q=\alpha *\beta ),And(q=\alpha *\beta *a,\delta =\alpha +\beta -\gamma +1))]}
HeunB HeunB(a,b,c,d,z)是下列雙合流方程的解
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{\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 HeunC(a,b,c,d,z)是下列微分方程的解
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{\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))}}}
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{\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)]}
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{\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 HeunD(a,b,c,d,z)函數是下列微分方程的解
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{\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})}}}
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{\displaystyle [HeunD(a,b,c,d,z)=HeunD(-a,-d,-c,-b,1/z)]}
HeunT HeunT(a,b,c,z)函數是下列微分方程的解
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{\displaystyle y_{z}z-(3*z^{2}+c)*y_{z}-((3-b)*z-a)*y}
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{\displaystyle [HeunT(a,b,c,z)=HeunT(j*a,b,j^{2}*c,j*z),And(j^{3}=1)]}
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{\displaystyle [HeunT(a,b,c,z)=HeunT(a,-b,c,-z)*exp(z^{3}),And(c=0)]}
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 978-0-19-859695-0MR 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 978-0521192255MR 2723248 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 
![{\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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67a0b10a0c9676e965486e9615ad58c9be85f22e)
![{\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)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1dc7d1865b9b608d6480fa55affcb1ae495c9b4)
![{\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)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c79c04b6d36c697bc7fbddda0a38d1cc3e98cec)
![{\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)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc51a60dd379e9d99ea04853e01fb6ca1abfd9b)
![{\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)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24cd4eed452c1ed5ff3b3e10958d687fff00b4c0)
![{\displaystyle [HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\alpha +\beta -\delta +1],z),And(a=0,q=0)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be2390b71ea4ca39f819e36df775b12302a51ef5)
![{\displaystyle [HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\gamma ],z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f125f498e045eaedd0b02e704f726cab3310088a)
![{\displaystyle (And(a=1,q=\alpha *\beta ),And(q=\alpha *\beta *a,\delta =\alpha +\beta -\gamma +1))]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efa9df4215fa77e376d3c4bfd8edd2f9ae54e4e1)
![{\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)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d7cac550278b8fa7210df802aefccc7282f353)
![{\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)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/776c9b8f4ad58f6e93166ac112441c36deb0bc6c)
![{\displaystyle [HeunD(a,b,c,d,z)=HeunD(-a,-d,-c,-b,1/z)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbde009fb92e9225022e0d5e611d95e17d25970)
![{\displaystyle [HeunT(a,b,c,z)=HeunT(j*a,b,j^{2}*c,j*z),And(j^{3}=1)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abf1c8b346c923bd1929bcb3776fe9cea9ad6b5e)
![{\displaystyle [HeunT(a,b,c,z)=HeunT(a,-b,c,-z)*exp(z^{3}),And(c=0)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1149467c6fbb3ee5b73984bd0fb7ffa742e4e9)
, doi:10.1090/S0025-5718-06-01939-9