even or odd

SearchAdvancedHelp

(0.000 seconds)

21—30 of 105 matching pages

21: 24.17 Mathematical Applications

… ►

24.17.5

ⓘ

Symbols:

$B_n$: Bernoulli numbers, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $n$: integer, $x$: real or complex and $M_n(x)$: function

Permalink:

http://dlmf.nist.gov/24.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

…

22: 28.2 Definitions and Basic Properties

… ► $w_{\text{I}}(z;a,q)$ is even and $w_{\text{II}}(z;a,q)$ is odd. … ►Even parity means $w(-z)=w(z)$, and odd parity means $w(-z)=-w(z)$. … ►

Table 28.2.2: Eigenfunctions of Mathieu’s equation.

► ►►►►►

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

►

…

23: 29.5 Special Cases and Limiting Forms

… ►

29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

►

29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

…

24: 30.9 Asymptotic Approximations and Expansions

… ►As ${\gamma }^2\to -\mathrm{\infty }$, with $q=n+1$ if $n-m$ is even, or $q=n$ if $n-m$ is odd, we have …

25: 31.7 Relations to Other Functions

… ►The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

26: 34.7 Basic Properties: $\mathit{9}j$ Symbol

… ►Even (cyclic) permutations of either columns or rows, as well as transpositions, leave the $\mathit{9}j$ symbol unchanged. Odd permutations of columns or rows introduce a phase factor ${(-1)}^R$, where $R$ is the sum of all arguments of the $\mathit{9}j$ symbol. …

27: 36.8 Convergent Series Expansions

… ►

${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\exp \left(\mathrm{i}{\displaystyle \frac{\pi (2n+1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{2n+1}{K+2}}\right)a_{2n}(𝐱),$ $K$ even,

►

${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\mathrm{i}}^n\cos \left({\displaystyle \frac{\pi (n(K+1)-1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{n+1}{K+2}}\right)a_n(𝐱),$ $K$ odd,

…

28: 32.8 Rational Solutions

… ►

$\alpha =\frac{1}{2}{(m+\epsilon \gamma )}^2$ and $\beta =-\frac{1}{2}{n}^2$, where $n>0$, $m+n$ is odd, and $\alpha \ne 0$ when .

►

$\alpha =\frac{1}{2}{n}^2$ and $\beta =-\frac{1}{2}{(m+\epsilon \gamma )}^2$, where $n>0$, $m+n$ is odd, and $\beta \ne 0$ when .

►

$\alpha =\frac{1}{2}{a}^2$, $\beta =-\frac{1}{2}{(a+n)}^2$, and $\gamma =m$, with $m+n$ even.

►

$\alpha =\frac{1}{2}{(b+n)}^2$, $\beta =-\frac{1}{2}{b}^2$, and $\gamma =m$, with $m+n$ even.

…

29: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►

34.3.5

ⓘ

Symbols:

$!$: factorial (as in $n!$), : $\mathit{3}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(i), §34.3 and Ch.34

… ►Even permutations of columns of a $\mathit{3}j$ symbol leave it unchanged; odd permutations of columns produce a phase factor ${(-1)}^{j_1+j_2+j_3}$, for example, …

30: 23.2 Definitions and Periodic Properties

… ► $\mathrm{\wp }\left(z\right)$ is even and $\zeta \left(z\right)$ is odd. …The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. …