休恩函数

Heun函数指HeunB、HeunC、HeunD、HeunG、HeunT等五个函数 HeunG函数是下列GeneralHeun方程的解 d 2 w d z 2 + [ γ z + δ z − 1 + ϵ z − a ] d w d z + α β z − q z ( z − 1 ) ( z − a ) w = 0. {\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)函数有6个参数，有四个奇点 H e u n G ( a , b , α , β , γ , δ , z ) = b ∗ z / ( γ ∗ a ) − ( 1 / 2 ) ∗ ( − b − b ∗ a ∗ δ − b ∗ γ ∗ a + b ∗ δ − b ∗ α − b ∗ β − b 2 + α ∗ β ∗ γ ∗ a ) ∗ z 2 / ( γ ∗ a 2 ∗ ( γ + 1 ) ) + O ( z 3 ) {\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})} H e u n G ( a , q , α , β , γ , δ , z ) = ( 1 − z ) ( 1 − δ ) ∗ H e u n G ( a , q − ( − 1 + δ ) ∗ γ ∗ a , β − δ + 1 , α − δ + 1 , γ , 2 − δ , z ) {\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)} [ H e u n G ( a , q , α , β , γ , δ , z ) = ( 1 − z / a ) ( − α − β + γ + δ ) ∗ H e u n G ( a , q − γ ∗ ( α + β − γ − δ ) , − β + γ + δ , − α + γ + δ , γ , δ , z ) , A n d ( a <> 0 ) ] {\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)]} [ H e u n G ( a , q , α , β , γ , δ , z ) = ( 1 − z ) ( 1 − δ ) ∗ ( 1 − z / a ) ( − α − β + γ + δ ) ∗ H e u n G ( a , q − γ ∗ ( ( − 1 + δ ) ∗ a + α + β − γ − δ ) , − β + γ + 1 , − α + γ + 1 , γ , 2 − δ , z ) , A n d ( 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)]} [ H e u n G ( a , q , α , β , γ , δ , z ) = H e u n G ( 1 / a , q / a , α , β , γ , α + β − γ − δ + 1 , z / a ) , A n d ( a <> 0 ) ] {\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)]} [ H e u n G ( a , q , α , β , γ , δ , z ) = ( 1 − z ) ( − α ) ∗ H e u n G ( a / ( − 1 + a ) , ( − q + γ ∗ α ∗ a ) / ( − 1 + a ) , α , α − δ + 1 , γ , α − β + 1 , z / ( z − 1 ) ) , A n d ( a <> 1 , z <> 1 ) ] {\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 } [ H e u n G ( a , q , α , β , γ , δ , z ) = h y p e r g e o m ( [ α , β ] , [ α + β − δ + 1 ] , z ) , A n d ( a = 0 , q = 0 ) ] {\displaystyle [HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\alpha +\beta -\delta +1],z),And(a=0,q=0)]} [ H e u n G ( a , q , α , β , γ , δ , z ) = h y p e r g e o m ( [ α , β ] , [ γ ] , z ) {\displaystyle [HeunG(a,q,\alpha ,\beta ,\gamma ,\delta ,z)=hypergeom([\alpha ,\beta ],[\gamma ],z)} ( A n d ( a = 1 , q = α ∗ β ) , A n d ( q = α ∗ β ∗ a , δ = α + β − γ + 1 ) ) ] {\displaystyle (And(a=1,q=\alpha *\beta ),And(q=\alpha *\beta *a,\delta =\alpha +\beta -\gamma +1))]} HeunB(a,b,c,d,z)是下列双合流方程的解 y z z − ( α + 2 ∗ z + z 2 ∗ α − 2 ∗ z 3 ) ( z + 1 ) 2 / ( z − 1 ) 2 ∗ y z + ( δ + ( 2 ∗ α + γ ) ∗ z + β ∗ z 2 ) ( z − 1 ) 3 / ( z + 1 ) 3 ∗ y ( z ) ; {\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(a,b,c,d,z)是下列微分方程的解 y z z − ( − z 2 ∗ a + ( − 2 − b − c + a ) ∗ z + b + 1 ) ∗ y z ) ( z ∗ ( z − 1 ) ) − ( 1 / 2 ) ∗ ( ( ( − b − c − 2 ) ∗ a − 2 ∗ d ) ∗ z + ( b + 1 ) ∗ a + ( − c − 1 ) ∗ b − c − 2 ∗ e ) ∗ y ( z ∗ ( z − 1 ) ) {\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))}}} [ H e u n C ( a , b , c , d , e , z ) = ( 1 − z ) ( − c ) ∗ H e u n C ( a , b , − c , d , e , z ) , A n d ( b :: ( N o t ( i n t e g e r ) ) , a b s ( z ) < 1 ) ] {\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)]} [ H e u n C ( a , b , c , d , e , z ) = e x p ( − z ∗ a ) ∗ H e u n C ( − a , b , c , d , e , z ) , A n d ( b :: ( N o t ( i n t e g e r ) ) , a b s ( 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(a,b,c,d,z)函数是下列微分方程的解 y z z − ( − 2 ∗ z 5 + 4 ∗ z 3 + z 4 ∗ a − 2 ∗ z − a ) ∗ y z ) ( ( z − 1 ) 3 ∗ ( z + 1 ) 3 ) − ( − z 2 ∗ b + ( − c − 2 ∗ a ) ∗ z − d ) ∗ y ( ( z − 1 ) 3 ∗ ( z + 1 ) 3 ) {\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})}}} [ H e u n D ( a , b , c , d , z ) = H e u n D ( − a , − d , − c , − b , 1 / z ) ] {\displaystyle [HeunD(a,b,c,d,z)=HeunD(-a,-d,-c,-b,1/z)]} HeunT(a,b,c,z)函数是下列微分方程的解 y z z − ( 3 ∗ z 2 + c ) ∗ y z − ( ( 3 − b ) ∗ z − a ) ∗ y {\displaystyle y_{z}z-(3*z^{2}+c)*y_{z}-((3-b)*z-a)*y} [ H e u n T ( a , b , c , z ) = H e u n T ( j ∗ a , b , j 2 ∗ c , j ∗ z ) , A n d ( j 3 = 1 ) ] {\displaystyle [HeunT(a,b,c,z)=HeunT(j*a,b,j^{2}*c,j*z),And(j^{3}=1)]} [ H e u n T ( a , b , c , z ) = H e u n T ( a , − b , c , − z ) ∗ e x p ( z 3 ) , A n d ( c = 0 ) ] {\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. (编), 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. (编), 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. 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