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11: 28.30 Expansions in Series of Eigenfunctions

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§28.30(i) Real Variable

►Let ${\widehat{\lambda }}_m$, $m=0,1,2,\mathrm{\dots }$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_m(x)$, $m=0,1,2,\mathrm{\dots }$, be the eigenfunctions, that is, an orthonormal set of $2\pi$-periodic solutions; thus ►

28.30.1 $$w_m^{\prime \prime }+({\widehat{\lambda }}_m+Q(x))w_m=0,$$

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Symbols:

$m$: integer, $x$: real variable, $w(z)$: Mathieu’s equation solution and ${\widehat{\lambda }}_m$: characteristic values

Permalink:

http://dlmf.nist.gov/28.30.E1

Encodings:

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See also:

Annotations for §28.30(i), §28.30 and Ch.28

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series ►

28.30.3 $$f(x)=\sum _{m=0}^{\mathrm{\infty }}{f}_m{w}_m(x),$$

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Symbols:

$m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E3

Encodings:

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See also:

Annotations for §28.30(i), §28.30 and Ch.28

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12: 1.8 Fourier Series

… ►Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function, …

13: 23.2 Definitions and Periodic Properties

… ►Hence $\mathrm{\wp }\left(z\right)$ is an elliptic function, that is, $\mathrm{\wp }\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\mathrm{\wp }\left(z\right)$ is meromorphic and has two periods whose ratio is not real. …

14: 28.29 Definitions and Basic Properties

… ► $\pi$ is the minimum period of $Q$. … ►where the function $P_{\nu }(z)$ is $\pi$-periodic. … ►The solutions of period $\pi$ or $2\pi$ are exceptional in the following sense. If (28.29.1) has a periodic solution with minimum period $n\pi$, $n=3,4,\mathrm{\dots }$, then all solutions are periodic with period $n\pi$. …

15: 4.2 Definitions

… ►The real and imaginary parts of $\ln z$ are given by … ►

§4.2(iii) The Exponential Function

… ►The function $\exp$ is an entire function of $z$, with no real or complex zeros. It has period $2\pi \mathrm{i}$: … ►When $a$ is real …

16: 25.16 Mathematical Applications

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25.16.6 $$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$

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Symbols:

$\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (3), p. 87)

Permalink:

http://dlmf.nist.gov/25.16.E6

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

►

25.16.7 $$H\left(s\right)=\frac{1}{2}\zeta \left(s+1\right)+\frac{\zeta \left(s\right)}{s-1}-\sum _{r=1}^k\left(\genfrac{}{}{0.0pt}{}{s+2r-2}{2r-1}\right)\zeta \left(1-2r\right)\zeta \left(s+2r\right)-\left(\genfrac{}{}{0.0pt}{}{s+2k}{2k+1}\right)\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{s+2k+1}}dx.$$

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Symbols:

$H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (6), p. 88)

Permalink:

http://dlmf.nist.gov/25.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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17: 22.3 Graphics

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§22.3(i) Real Variables: Line Graphs

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§22.3(ii) Real Variables: Surfaces

► $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, and $\mathrm{dn}(x,k)$ as functions of real arguments $x$ and $k$. The period diverges logarithmically as $k\to 1-$; see §19.12. … ►

§22.3(iii) Complex $z$; Real $k$

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18: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

… ►As a consequence of the factor $z$ on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as $z\to \pm \mathrm{\infty }$ on $ℝ$. …

19: 28.11 Expansions in Series of Mathieu Functions

… ►Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-infinite strip $S$ that contains the real axis, and $q$ be a normal value (§28.7). …

20: 32.10 Special Function Solutions

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32.10.34 $$w(z;0,0,0,\frac{1}{2})=\mathrm{\Lambda }(C_1{\varphi }_1+C_2{\varphi }_2,z),$$

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Symbols:

$z$: real, $\mathrm{\Lambda }(u,z)$: elliptic function, ${\varphi }_j$: periods and $C$: constant

Referenced by:

§32.10(vi)

Permalink:

http://dlmf.nist.gov/32.10.E34

Encodings:

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See also:

Annotations for §32.10(vi), §32.10 and Ch.32

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