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1: 31.3 Basic Solutions
31.3.2 a γ c 1 - q c 0 = 0 ,
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
15.5.11 ( c - a ) F ( a - 1 , b ; c ; z ) + ( 2 a - c + ( b - a ) z ) F ( a , b ; c ; z ) + a ( z - 1 ) F ( a + 1 , b ; c ; z ) = 0 ,
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: 15.1 Special Notation