modified Mathieu equation
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1: 28.20 Definitions and Basic Properties
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§28.20(i) Modified Mathieu’s Equation
►When is replaced by , (28.2.1) becomes the modified Mathieu’s equation: ►
28.20.1
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28.20.2
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28.20.6
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2: 28.8 Asymptotic Expansions for Large
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►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1).
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3: 28.32 Mathematical Applications
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►The separated solutions can be obtained from the modified Mathieu’s equation (28.20.1) for and from Mathieu’s equation (28.2.1) for , where is the separation constant and .
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4: 28.35 Tables
§28.35 Tables
… ►Blanch and Clemm (1969) includes eigenvalues , for , , , ; 4D. Also and for , , and , respectively; 8D. Double points for ; 8D. Graphs are included.
5: 28.1 Special Notation
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►The main functions treated in this chapter are the Mathieu functions
…and the modified Mathieu functions
…The functions and are also known as the radial Mathieu functions.
►The eigenvalues of Mathieu’s equation are denoted by
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►The radial functions and are denoted by and , respectively.
6: 28.33 Physical Applications
§28.33 Physical Applications
… ►McLachlan (1947, Chapters XVI–XIX) for applications of the wave equation to vibrational systems, electrical and thermal diffusion, electromagnetic wave guides, elliptical cylinders in viscous fluids, and diffraction of sound and electromagnetic waves.
§28.33(iii) Stability and Initial-Value Problems
… ►Hence from §28.17 the corresponding Mathieu equation is stable or unstable according as is in the intersection of with the colored or the uncolored open regions depicted in Figure 28.17.1. … ►Torres-Vega et al. (1998) for Mathieu functions in phase space.
7: 28.28 Integrals, Integral Representations, and Integral Equations
8: 28.34 Methods of Computation
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§28.34(i) Characteristic Exponents
… ►§28.34(ii) Eigenvalues
… ►§28.34(iii) Floquet Solutions
… ►§28.34(iv) Modified Mathieu Functions
…9: 28.23 Expansions in Series of Bessel Functions
§28.23 Expansions in Series of Bessel Functions
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28.23.2
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28.23.6
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►When the series in the even-numbered equations converge for and , and the series in the odd-numbered equations converge for and .
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