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1: 31.3 Basic Solutions
31.3.2 a γ c 1 q c 0 = 0 ,
31.3.7 ( 1 z ) 1 δ H ( 1 a , ( ( 1 a ) γ + ϵ ) ( 1 δ ) + α β q ; α + 1 δ , β + 1 δ , 2 δ , γ ; 1 z ) .
2: 31.1 Special Notation
x , y real variables.
a complex parameter, | a | 1 , a 1 .
q , α , β , γ , δ , ϵ , ν complex parameters.
3: 15.7 Continued Fractions
15.7.1 𝐅 ( a , b ; c ; z ) 𝐅 ( a , b + 1 ; c + 1 ; z ) = t 0 u 1 z t 1 u 2 z t 2 u 3 z t 3 ,
4: 31.12 Confluent Forms of Heun’s Equation
31.12.1 d 2 w d z 2 + ( γ z + δ z 1 + ϵ ) d w d z + α z q z ( z 1 ) w = 0 .
31.12.2 d 2 w d z 2 + ( δ z 2 + γ z + 1 ) d w d z + α z q z 2 w = 0 .
31.12.3 d 2 w d z 2 ( γ z + δ + z ) d w d z + α z q z w = 0 .
31.12.4 d 2 w d z 2 + ( γ + z ) z d w d z + ( α z q ) w = 0 .
5: 31.14 General Fuchsian Equation
31.14.1 d 2 w d z 2 + ( j = 1 N γ j z a j ) d w d z + ( j = 1 N q j z a j ) w = 0 , j = 1 N q j = 0 .
31.14.3 w ( z ) = ( j = 1 N ( z a j ) γ j / 2 ) W ( z ) ,
31.14.4 d 2 W d z 2 = j = 1 N ( γ ~ j ( z a j ) 2 + q ~ j z a j ) W , j = 1 N q ~ j = 0 ,
6: 15.5 Derivatives and Contiguous Functions
15.5.5 ( z d d z z ) n ( z c a 1 ( 1 z ) a + b c F ( a , b ; c ; z ) ) = ( c a ) n z c a + n 1 ( 1 z ) a n + b c F ( a n , b ; c ; z ) .
7: 9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
8: 31.4 Solutions Analytic at Two Singularities: Heun Functions
31.4.2 q = a γ P 1 Q 1 + q R 1 P 2 Q 2 + q R 2 P 3 Q 3 + q ,
9: 31.6 Path-Multiplicative Solutions
10: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
31.5.1 [ 0 a γ 0 0 P 1 Q 1 R 1 0 0 P 2 Q 2 R n 1 0 0 P n Q n ] ,
31.5.2 𝐻𝑝 n , m ( a , q n , m ; n , β , γ , δ ; z ) = H ( a , q n , m ; n , β , γ , δ ; z )