Meixner, Josef; Schäfke, Friedrich Wilhelm Mathieusche Funktionen und Sphäroid-Funktionen mit Anwendungen auf physikalische und technische Probleme. (German) Zbl 0058.29503 (Die Grundlehren der mathematischen Wissenschaften. Bd. LXXI.) Berlin- Göttingen-Heidelberg: Springer-Verlag. 414 S. mit 29 Abb. (1954). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 2 ReviewsCited in 99 Documents Keywords:special functions × Cite Format Result Cite Review PDF Digital Library of Mathematical Functions: §28.10(i) Equations with Elementary Kernels ‣ §28.10 Integral Equations ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.11 Expansions in Series of Mathieu Functions ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.12(iii) Functions ce_𝜈(𝑧,𝑞), se_𝜈(𝑧,𝑞), when 𝜈∉ℤ ‣ §28.12 Definitions and Basic Properties ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.12(ii) Eigenfunctions me_𝜈(𝑧,𝑞) ‣ §28.12 Definitions and Basic Properties ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.12(i) Eigenvalues 𝜆_{𝜈+2𝑛}(𝑞) ‣ §28.12 Definitions and Basic Properties ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.14 Fourier Series ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.15(ii) Solutions me_𝜈(𝑧,𝑞) ‣ §28.15 Expansions for Small 𝑞 ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.15(i) Eigenvalues 𝜆_𝜈(𝑞) ‣ §28.15 Expansions for Small 𝑞 ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.16 Asymptotic Expansions for Large 𝑞 ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.17 Stability as 𝑥→±∞ ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.19 Expansions in Series of me_{𝜈+2𝑛} Functions ‣ Mathieu Functions of Noninteger Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation In §28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.20(iii) Solutions M^(𝑗)_𝜈 ‣ §28.20 Definitions and Basic Properties ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.20(ii) Solutions Ce_𝜈, Se_𝜈, Me_𝜈, Fe_𝑛, Ge_𝑛 ‣ §28.20 Definitions and Basic Properties ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.20(vii) Shift of Variable ‣ §28.20 Definitions and Basic Properties ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.20(vi) Wronskians ‣ §28.20 Definitions and Basic Properties ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.22(ii) Noninteger 𝜈 ‣ §28.22 Connection Formulas ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.22(i) Integer 𝜈 ‣ §28.22 Connection Formulas ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.23 Expansions in Series of Bessel Functions ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.25 Asymptotic Expansions for Large ℜ𝑧 ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.28(i) Equations with Elementary Kernels ‣ §28.28 Integrals, Integral Representations, and Integral Equations ‣ Modified Mathieu Functions ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.2(iii) Floquet’s Theorem and the Characteristic Exponents ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.2(ii) Basic Solutions 𝑤_”I”, 𝑤_”II” ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.2(iv) Floquet Solutions ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.2(vi) Eigenfunctions ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.2(v) Eigenvalues 𝑎_𝑛, 𝑏_𝑛 ‣ §28.2 Definitions and Basic Properties ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation In §28.33(ii) Boundary-Value Problems ‣ §28.33 Physical Applications ‣ Applications ‣ Chapter 28 Mathieu Functions and Hill’s Equation In §28.33(iii) Stability and Initial-Value Problems ‣ §28.33 Physical Applications ‣ Applications ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.33(ii) Boundary-Value Problems ‣ §28.33 Physical Applications ‣ Applications ‣ Chapter 28 Mathieu Functions and Hill’s Equation In §28.34(ii) Eigenvalues ‣ §28.34 Methods of Computation ‣ Computation ‣ Chapter 28 Mathieu Functions and Hill’s Equation In §28.34(iii) Floquet Solutions ‣ §28.34 Methods of Computation ‣ Computation ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.4(iii) Normalization ‣ §28.4 Fourier Series ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.4(ii) Recurrence Relations ‣ §28.4 Fourier Series ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.4(i) Definitions ‣ §28.4 Fourier Series ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.4(iv) Case 𝑞=0 ‣ §28.4 Fourier Series ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.4(vi) Behavior for Small 𝑞 ‣ §28.4 Fourier Series ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.4(v) Change of Sign of 𝑞 ‣ §28.4 Fourier Series ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.4(vi) Behavior for Small 𝑞 ‣ §28.4 Fourier Series ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.5(i) Definitions ‣ §28.5 Second Solutions fe_𝑛, ge_𝑛 ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation Wronskians ‣ §28.5(i) Definitions ‣ §28.5 Second Solutions fe_𝑛, ge_𝑛 ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.6(ii) Functions ce_𝑛 and se_𝑛 ‣ §28.6 Expansions for Small 𝑞 ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.6(i) Eigenvalues ‣ §28.6 Expansions for Small 𝑞 ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.7 Analytic Continuation of Eigenvalues ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.7 Analytic Continuation of Eigenvalues ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.8(i) Eigenvalues ‣ §28.8 Asymptotic Expansions for Large 𝑞 ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.8(ii) Sips’ Expansions ‣ §28.8 Asymptotic Expansions for Large 𝑞 ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.9 Zeros ‣ Mathieu Functions of Integer Order ‣ Chapter 28 Mathieu Functions and Hill’s Equation Chapter 28 Mathieu Functions and Hill’s Equation §30.10 Series and Integrals ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.11(iii) Asymptotic Behavior ‣ §30.11 Radial Spheroidal Wave Functions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.11(iii) Asymptotic Behavior ‣ §30.11 Radial Spheroidal Wave Functions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.11(iv) Wronskian ‣ §30.11 Radial Spheroidal Wave Functions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.11(vi) Integral Representations ‣ §30.11 Radial Spheroidal Wave Functions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.11(v) Connection with the 𝑃𝑠 and 𝑄𝑠 Functions ‣ §30.11 Radial Spheroidal Wave Functions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.11(vi) Integral Representations ‣ §30.11 Radial Spheroidal Wave Functions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.13(iii) Laplacian ‣ §30.13 Wave Equation in Prolate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.13(ii) Metric Coefficients ‣ §30.13 Wave Equation in Prolate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.13(i) Prolate Spheroidal Coordinates ‣ §30.13 Wave Equation in Prolate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.13(iv) Separation of Variables ‣ §30.13 Wave Equation in Prolate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids ‣ §30.13 Wave Equation in Prolate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids ‣ §30.13 Wave Equation in Prolate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.14(iii) Laplacian ‣ §30.14 Wave Equation in Oblate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.14(ii) Metric Coefficients ‣ §30.14 Wave Equation in Oblate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.14(i) Oblate Spheroidal Coordinates ‣ §30.14 Wave Equation in Oblate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.14(iv) Separation of Variables ‣ §30.14 Wave Equation in Oblate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids ‣ §30.14 Wave Equation in Oblate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids ‣ §30.14 Wave Equation in Oblate Spheroidal Coordinates ‣ Applications ‣ Chapter 30 Spheroidal Wave Functions §30.16(i) Eigenvalues ‣ §30.16 Methods of Computation ‣ Computation ‣ Chapter 30 Spheroidal Wave Functions §30.16(i) Eigenvalues ‣ §30.16 Methods of Computation ‣ Computation ‣ Chapter 30 Spheroidal Wave Functions §30.1 Special Notation ‣ Notation ‣ Chapter 30 Spheroidal Wave Functions §30.2(ii) Other Forms ‣ §30.2 Differential Equations ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.2(i) Spheroidal Differential Equation ‣ §30.2 Differential Equations ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.3(iii) Transcendental Equation ‣ §30.3 Eigenvalues ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.3(ii) Properties ‣ §30.3 Eigenvalues ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.3(i) Definition ‣ §30.3 Eigenvalues ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.3(iv) Power-Series Expansion ‣ §30.3 Eigenvalues ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.4(ii) Elementary Properties ‣ §30.4 Functions of the First Kind ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.4(i) Definitions ‣ §30.4 Functions of the First Kind ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.4(iv) Orthogonality ‣ §30.4 Functions of the First Kind ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.5 Functions of the Second Kind ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.5 Functions of the Second Kind ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.6 Functions of Complex Argument ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions Values on (-1,1) ‣ §30.6 Functions of Complex Argument ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.8(ii) Functions of the Second Kind ‣ §30.8 Expansions in Series of Ferrers Functions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.8(i) Functions of the First Kind ‣ §30.8 Expansions in Series of Ferrers Functions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.9(ii) Oblate Spheroidal Wave Functions ‣ §30.9 Asymptotic Approximations and Expansions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.9(i) Prolate Spheroidal Wave Functions ‣ §30.9 Asymptotic Approximations and Expansions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.9(ii) Oblate Spheroidal Wave Functions ‣ §30.9 Asymptotic Approximations and Expansions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions §30.9(i) Prolate Spheroidal Wave Functions ‣ §30.9 Asymptotic Approximations and Expansions ‣ Properties ‣ Chapter 30 Spheroidal Wave Functions Chapter 30 Spheroidal Wave Functions Notations P ‣ Notations Notations P ‣ Notations Notations Q ‣ Notations Notations Q ‣ Notations