Results of Mathieu Functions and Hill’s Equation

Notation

28.1 Special Notation

Mathieu Functions of Integer Order

28.2 Definitions and Basic Properties
28.3 Graphics
28.4 Fourier Series
28.5 Second Solutions ${\displaystyle{\displaystyle\mathrm{fe}_{n}}}$ , ${\displaystyle{\displaystyle\mathrm{ge}_{n}}}$
28.6 Expansions for Small ${\displaystyle{\displaystyle q}}$
28.7 Analytic Continuation of Eigenvalues
28.8 Asymptotic Expansions for Large ${\displaystyle{\displaystyle q}}$
28.9 Zeros
28.10 Integral Equations
28.11 Expansions in Series of Mathieu Functions

Mathieu Functions of Noninteger Order

28.12 Definitions and Basic Properties
28.13 Graphics
28.14 Fourier Series
28.15 Expansions for Small ${\displaystyle{\displaystyle q}}$
28.16 Asymptotic Expansions for Large ${\displaystyle{\displaystyle q}}$
28.17 Stability as ${\displaystyle{\displaystyle x\to\pm\infty}}$
28.18 Integrals and Integral Equations
28.19 Expansions in Series of ${\displaystyle{\displaystyle\mathrm{me}_{\nu+2n}}}$ Functions

Modified Mathieu Functions

28.20 Definitions and Basic Properties
28.21 Graphics
28.22 Connection Formulas
28.23 Expansions in Series of Bessel Functions
28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
28.25 Asymptotic Expansions for Large ${\displaystyle{\displaystyle\Re z}}$
28.26 Asymptotic Approximations for Large ${\displaystyle{\displaystyle q}}$
28.27 Addition Theorems
28.28 Integrals, Integral Representations, and Integral Equations

Hill’s Equation

28.29 Definitions and Basic Properties
28.30 Expansions in Series of Eigenfunctions
28.31 Equations of Whittaker–Hill and Ince

Applications

28.32 Mathematical Applications
28.33 Physical Applications

Computation

28.34 Methods of Computation
28.35 Tables
28.36 Software