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1: 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
2: 31.2 Differential Equations
§31.2 Differential Equations
§31.2(i) Heun’s Equation
§31.2(v) Heun’s Equation Automorphisms
Composite Transformations
3: 29.2 Differential Equations
§29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
4: 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. …
5: 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
6: 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
7: 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. …
8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Equation (1.18.19) is often called the completeness relation. … Applying equations (1.18.29) and (1.18.30), the complete set of normalized eigenfunctions being … For example, replacing 2 q cos ( 2 z ) of (28.2.1) by λ cos ( 2 π α n + θ ) , n gives an almost Mathieu equation which for appropriate α has such properties. … See §18.39 for discussion of Schrödinger equations and operators. …
9: Frank W. J. Olver
He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. … In a review of that volume, Jet Wimp of Drexel University said that the papers “exemplify a redoubtable mathematical talent, the work of a man who has done more than almost anyone else in the 20th century to bestow on the discipline of applied mathematics the elegance and rigor that its earliest practitioners, such as Gauss and Laplace, would have wished for it. …
10: Browsers
Thus, we are now using the new (almost-)standard for HTML, HTML5, as the primary format for our web content. …