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paraxial wave equation

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1: 30.1 Special Notation
β–ΊThe main functions treated in this chapter are the eigenvalues Ξ» n m ⁑ ( Ξ³ 2 ) and the spheroidal wave functions π–―π—Œ n m ⁑ ( x , Ξ³ 2 ) , π–°π—Œ n m ⁑ ( x , Ξ³ 2 ) , 𝑃𝑠 n m ⁑ ( z , Ξ³ 2 ) , 𝑄𝑠 n m ⁑ ( z , Ξ³ 2 ) , and S n m ⁒ ( j ) ⁑ ( z , Ξ³ ) , j = 1 , 2 , 3 , 4 . …Meixner and Schäfke (1954) use ps , qs , Ps , Qs for π–―π—Œ , π–°π—Œ , 𝑃𝑠 , 𝑄𝑠 , respectively. β–Ί
Other Notations
β–ΊFlammer (1957) and Abramowitz and Stegun (1964) use Ξ» m ⁒ n ⁒ ( Ξ³ ) for Ξ» n m ⁑ ( Ξ³ 2 ) + Ξ³ 2 , R m ⁒ n ( j ) ⁑ ( Ξ³ , z ) for S n m ⁒ ( j ) ⁑ ( z , Ξ³ ) , and …
2: 30.11 Radial Spheroidal Wave Functions
§30.11 Radial Spheroidal Wave Functions
β–Ί
§30.11(i) Definitions
β–Ί
Connection Formulas
β–Ί
§30.11(ii) Graphics
β–Ί
§30.11(iv) Wronskian
3: 30.2 Differential Equations
§30.2 Differential Equations
β–Ί
§30.2(i) Spheroidal Differential Equation
β–Ίβ–ΊThe Liouville normal form of equation (30.2.1) is … β–Ί
§30.2(iii) Special Cases
4: 31.2 Differential Equations
§31.2 Differential Equations
β–Ί
§31.2(i) Heun’s Equation
β–Ί β–Ί
§31.2(v) Heun’s Equation Automorphisms
β–Ί
Composite Transformations
5: 29.2 Differential Equations
§29.2 Differential Equations
β–Ί
§29.2(i) Lamé’s Equation
β–Ί
§29.2(ii) Other Forms
β–ΊEquation (29.2.10) is a special case of Heun’s equation (31.2.1).
6: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
β–Ί
§15.10(i) Fundamental Solutions
β–Ί
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 .
β–ΊThis is the hypergeometric differential equation. … β–Ί
7: 32.2 Differential Equations
§32.2 Differential Equations
β–Ί
§32.2(i) Introduction
β–ΊThe six Painlevé equations P I P VI  are as follows: … β–Ί
§32.2(ii) Renormalizations
β–Ί
8: 28.2 Definitions and Basic Properties
β–Ί
§28.2(i) Mathieu’s Equation
β–Ί
28.2.1 w ′′ + ( a 2 ⁒ q ⁒ cos ⁑ ( 2 ⁒ z ) ) ⁒ w = 0 .
β–ΊThis is the characteristic equation of Mathieu’s equation (28.2.1). … β–Ί
§28.2(iv) Floquet Solutions
β–Ί
9: 28.20 Definitions and Basic Properties
β–Ί
§28.20(i) Modified Mathieu’s Equation
β–ΊWhen z is replaced by ± i ⁒ z , (28.2.1) becomes the modified Mathieu’s equation: β–Ί
28.20.1 w ′′ ( a 2 ⁒ q ⁒ cosh ⁑ ( 2 ⁒ z ) ) ⁒ w = 0 ,
β–Ί
28.20.2 ( ΞΆ 2 1 ) ⁒ w ′′ + ΞΆ ⁒ w + ( 4 ⁒ q ⁒ ΞΆ 2 2 ⁒ q a ) ⁒ w = 0 , ΞΆ = cosh ⁑ z .
β–ΊThen from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ΞΆ 1 / 2 ⁒ e ± 2 ⁒ i ⁒ h ⁒ ΞΆ as ΞΆ in the respective sectors | ph ⁑ ( βˆ“ i ⁒ ΞΆ ) | 3 2 ⁒ Ο€ Ξ΄ , Ξ΄ being an arbitrary small positive constant. …
10: 36.10 Differential Equations
§36.10 Differential Equations
β–Ί K = 1 , fold: (36.10.1) becomes Airy’s equation9.2(i)) … β–Ί
§36.10(iii) Operator Equations
β–ΊEquation (36.10.17) is the paraxial wave equation.