惠泰克函數,惠泰克1904推導合流超幾何函數,是下列惠泰克方程的解[1]
此方程在 0 有用正則奇點,在 ∞ 有非正則奇點.
惠泰克方程有兩個解[2]
M 與 U :
惠塔克M函數
W h i t t a k e r M = ∑ k = 0 ∞ ( 1 / 2 − a + b ) k ∗ z b + 1 / 2 + k e z / 2 ∗ k ! ∗ ( 1 + 2 b ) k {\displaystyle WhittakerM=\sum _{k=0}^{\infty }{\frac {(1/2-a+b)_{k}*z^{b+1/2+k}}{e^{z/2}*k!*(1+2b)_{k}}}}
[ W h i t t a k e r W ( a , b , z ) = ∑ | k 1 = 0 ∞ ( − P i ∗ ( z ( b + 1 / 2 + k 1 ) ∗ Γ ( 1 / 2 − a + b + k 1 ) ∗ Γ ( 1 − 2 ∗ b + k 1 ) − Γ ( 1 / 2 − a − b + k 1 ) ∗ z ( − b + 1 / 2 + k 1 ) ∗ Γ ( k 1 + 1 + 2 ∗ b ) ) / ( G A M M A ( k 1 + 1 ) ∗ G A M M A ( 1 / 2 − a + b ) ∗ G A M M A ( 1 / 2 − a − b ) ∗ s i n ( 2 ∗ P i ∗ b ) ∗ G A M M A ( k 1 + 1 + 2 ∗ b ) ∗ e x p ( ( 1 / 2 ) ∗ z ) ∗ G A M M A ( 1 − 2 ∗ b + k 1 ) ) , k 1 = 0.. i n f i n i t y ) , A n d ( b :: ( N o t ( n o n p o s i n t ) ) , ( 1 / 2 − a + b ) :: ( N o t ( n o n p o s i n t ) ) , ( 1 / 2 − a − b ) :: ( N o t ( n o n p o s i n t ) ) , a b s ( z ) < 1 ) ] {\displaystyle [WhittakerW(a,b,z)=\sum |_{k1=0}^{\infty }(-Pi*(z^{(}b+1/2+_{k}1)*\Gamma (1/2-a+b+_{k}1)*\Gamma (1-2*b+_{k}1)-\Gamma (1/2-a-b+_{k}1)*z^{(}-b+1/2+_{k}1)*\Gamma (_{k}1+1+2*b))/(GAMMA(_{k}1+1)*GAMMA(1/2-a+b)*GAMMA(1/2-a-b)*sin(2*Pi*b)*GAMMA(_{k}1+1+2*b)*exp((1/2)*z)*GAMMA(1-2*b+_{k}1)),_{k}1=0..infinity),And(b::(Not(nonposint)),(1/2-a+b)::(Not(nonposint)),(1/2-a-b)::(Not(nonposint)),abs(z)<1)]}