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biconfluent Heun equation

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1: 31.2 Differential Equations
§31.2(i) Heun’s Equation
§31.2(ii) Normal Form of Heun’s Equation
§31.2(v) Heun’s Equation Automorphisms
Composite Transformations
2: 31.1 Special Notation
x , y real variables.
The main functions treated in this chapter are H ( a , q ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) Hf m ( a , q m ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) Hf m ν ( a , q m ; α , β , γ , δ ; z ) , and the polynomial Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) . …Sometimes the parameters are suppressed.
3: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
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 )
These solutions are the Heun polynomials. …
4: 29.2 Differential Equations
§29.2 Differential Equations
§29.2(i) Lamé’s Equation
For the Weierstrass function see §23.2(ii). Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
5: 30.2 Differential Equations
§30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
§30.2(iii) Special Cases
6: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
§15.10(i) Fundamental Solutions
15.10.1 z ( 1 - z ) d 2 w d z 2 + ( c - ( a + b + 1 ) z ) d w d z - a b w = 0 .
This is the hypergeometric differential equation. …
7: 32.2 Differential Equations
§32.2 Differential Equations
§32.2(i) Introduction
The six Painlevé equations P I P VI  are as follows: …
§32.2(ii) Renormalizations
8: 28.2 Definitions and Basic Properties
§28.2(i) Mathieu’s Equation
This is the characteristic equation of Mathieu’s equation (28.2.1). …
§28.2(iv) Floquet Solutions
9: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
When z is replaced by ± i z , (28.2.1) becomes the modified Mathieu’s equation:
28.20.1 w ′′ - ( a - 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 - 1 ) w ′′ + ζ w + ( 4 q ζ 2 - 2 q - a ) w = 0 , ζ = cosh z .
Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ζ 1 / 2 e ± 2 i h ζ as ζ in the respective sectors | ph ( i ζ ) | 3 2 π - δ , δ being an arbitrary small positive constant. …
10: 31.12 Confluent Forms of Heun’s Equation
Confluent Heun Equation
Doubly-Confluent Heun Equation
Biconfluent Heun Equation
Triconfluent Heun Equation