antiperiodicity

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1: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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28.5.3

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Defines:

$f_n(z,q)$: functions (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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28.5.4

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Defines:

$g_n(z,q)$: functions (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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2: 28.3 Graphics

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Even $\pi$-Antiperiodic Solutions

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Odd $\pi$-Antiperiodic Solutions

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3: 28.29 Definitions and Basic Properties

… ►The $\pi$-periodic or $\pi$-antiperiodic solutions are multiples of $w_{\text{I}}(z,\lambda ),w_{\text{II}}(z,\lambda )$, respectively. …

4: 28.2 Definitions and Basic Properties

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§28.2(vi) Eigenfunctions

… ►Period $\pi$ means that the eigenfunction has the property $w(z+\pi )=w(z)$, whereas antiperiod $\pi$ means that $w(z+\pi )=-w(z)$. … ►

Table 28.2.2: Eigenfunctions of Mathieu’s equation.

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Eigenvalues	Eigenfunctions	Periodicity	Parity
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$a_{2n+1}\left(q\right)$	${\mathrm{ce}}_{2n+1}(z,q)$	Antiperiod $\pi$	Even
$b_{2n+1}\left(q\right)$	${\mathrm{se}}_{2n+1}(z,q)$	Antiperiod $\pi$	Odd
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