Digital Library of Mathematical Functions

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28 Mathieu Functions and Hill’s Equation Mathieu Functions of Noninteger Order 28.12 Definitions and Basic Properties 28.14 Fourier Series

§28.13 Graphics

Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.13

Contents

§28.13(i) Eigenvalues $\mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right)$ for General $\nu$
§28.13(ii) Solutions $\mathop{\mathrm{ce}_{{\nu}}\/}\nolimits\!\left(x,q\right)$ , $\mathop{\mathrm{se}_{{\nu}}\/}\nolimits\!\left(x,q\right)$ , and $\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(x,q\right)$ for General $\nu$

§28.13(i) Eigenvalues $\mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right)$ for General $\nu$

Notes:: Figures 28.13.1 and 28.13.2 were produced at NIST.
Keywords:: Mathieu’s equation
Permalink:: http://dlmf.nist.gov/28.13.i

See accompanying text

Figure 28.13.1: $\mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right)$ as a function of

for $\nu=0.5(1)3.5$ and $\mathop{a_{{n}}\/}\nolimits\!\left(q\right),\mathop{b_{{n}}\/}\nolimits\!\left% (q\right)$ for

(

’s),

(

’s). (Compare Figure 28.2.1.)

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Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter, : integer, $\nu$ : complex parameter and : parameter
Referenced by:: §28.12(i), §28.13(i), §28.16
Permalink:: http://dlmf.nist.gov/28.13.F1
Encodings:: pdf, png

See accompanying text

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Figure 28.13.2: $\mathop{\lambda_{{\nu}}\/}\nolimits\!\left(q\right)$ for $-2<\nu<2$ , $0\leq q\leq 10$ .

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Symbols:: $\mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and $\nu$ : complex parameter
Referenced by:: §28.12(i), §28.13(i), §28.16
Permalink:: http://dlmf.nist.gov/28.13.F2
Encodings:: VRML, X3D, pdf, png

§28.13(ii) Solutions $\mathop{\mathrm{ce}_{{\nu}}\/}\nolimits\!\left(x,q\right)$ , $\mathop{\mathrm{se}_{{\nu}}\/}\nolimits\!\left(x,q\right)$ , and $\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(x,q\right)$ for General $\nu$

Notes:: Figures 28.13.3–28.13.5 were produced at NIST.
Permalink:: http://dlmf.nist.gov/28.13.ii

See accompanying text

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Figure 28.13.3: $\mathop{\mathrm{ce}_{{\nu}}\/}\nolimits\!\left(x,1\right)$ for $-1<\nu<1$ , $0\leq x\leq 2\pi$ .

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Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : real variable and $\nu$ : complex parameter
Referenced by:: §28.12(iii), §28.13(ii)
Permalink:: http://dlmf.nist.gov/28.13.F3
Encodings:: VRML, X3D, pdf, png

See accompanying text

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Figure 28.13.4: $\mathop{\mathrm{se}_{{\nu}}\/}\nolimits\!\left(x,1\right)$ for $0<\nu<1$ , $0\leq x\leq 2\pi$ .

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Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.13.F4
Encodings:: VRML, X3D, pdf, png

See accompanying text

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Figure 28.13.5: $\mathop{\mathrm{me}_{{i\mu}}\/}\nolimits\!\left(x,1\right)$ for $0.1\leq\mu\leq 0.4$ , $-\pi\leq x\leq\pi.$

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Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function and : real variable
Referenced by:: §28.13(ii), §28.17
Permalink:: http://dlmf.nist.gov/28.13.F5
Encodings:: VRML, X3D, pdf, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 28.12 Definitions and Basic Properties 28.14 Fourier Series

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