even or odd

SearchAdvancedHelp

(0.002 seconds)

11—20 of 105 matching pages

11: 12.4 Power-Series Expansions

… ►where the initial values are given by (12.2.6)–(12.2.9), and $u_1(a,z)$ and $u_2(a,z)$ are the even and odd solutions of (12.2.2) given by …

12: 29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … ►

Table 29.3.1: Eigenvalues of Lamé’s equation.

► ►►►►

eigenvalue $h$	parity	period
…
$a_{\nu }^{2m+1}\left(k^2\right)$	odd	$4K$
…
$b_{\nu }^{2m+2}\left(k^2\right)$	odd	$2K$

►

… ►

Table 29.3.2: Lamé functions.

► ►►►►

boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
…
$w(0)={dw/dz|}_{z=K}=0$	$a_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$	odd	even	$4K$
${dw/dz|}_{z=0}=w(K)=0$	$b_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$	even	odd	$4K$
…

►

…

13: 1.12 Continued Fractions

… ►

§1.12(iv) Contraction and Extension

… ►The even part of $C$ exists iff $b_{2k}\ne 0$, $k=1,2,\mathrm{\dots }$, and up to equivalence is given by …If $C_n^{\prime }=C_{2n+1}$, $n=0,1,2,\mathrm{\dots }$, then $C^{\prime }$ is called the odd part of $C$. The odd part of $C$ exists iff $b_{2k+1}\ne 0$, $k=0,1,2,\mathrm{\dots }$, and up to equivalence is given by … ►and the even and odd parts of the continued fraction converge to finite values. …

14: 28.3 Graphics

… ►

Even $\pi$-Periodic Solutions

… ►

Even $\pi$-Antiperiodic Solutions

… ►

Odd $\pi$-Antiperiodic Solutions

… ►

Odd $\pi$-Periodic Solutions

…

15: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

… ►

28.5.3

ⓘ

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

… ►

28.5.4

ⓘ

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

… ►

Odd Second Solutions

… ►

Even Second Solutions

…

16: 9.13 Generalized Airy Functions

… ►

9.13.6

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $A_n\left(z\right)$: generalized Airy function, $B_n\left(z\right)$: generalized Airy function, $z$: complex variable, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (5a)–(5b))

Permalink:

http://dlmf.nist.gov/9.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.7

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $A_n\left(z\right)$: generalized Airy function, $B_n\left(z\right)$: generalized Airy function, $z$: complex variable, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (6a)–(6b))

Permalink:

http://dlmf.nist.gov/9.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►where $m=3,4,5,\mathrm{\dots }.$ For real variables the solutions of (9.13.13) are denoted by $U_m(t)$, $U_m(-t)$ when $m$ is even, and by $V_m(t)$, ${\overline{V}}_m(t)$ when $m$ is odd. … ►

9.13.15

ⓘ

Symbols:

$A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $z$: complex variable, $n$: parameter, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $m$: index and $V_m$: Olver’s generalized Airy function

Permalink:

http://dlmf.nist.gov/9.13.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.16

ⓘ

Symbols:

$B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $z$: complex variable, $n$: parameter, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $m$: index and $V_m$: Olver’s generalized Airy function

Permalink:

http://dlmf.nist.gov/9.13.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

…

17: 27.9 Quadratic Characters

… ►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. … ►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that ►

27.9.3 $$(p|q)(q|p)={(-1)}^{(p-1)(q-1)/4}.$$

ⓘ

Symbols:

$(n|p)$: Legendre symbol, $p$: odd prime and $q$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.9 and Ch.27

►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …Both (27.9.1) and (27.9.2) are valid with $p$ replaced by $P$; the reciprocity law (27.9.3) holds if $p,q$ are replaced by any two relatively prime odd integers $P,Q$.

18: 12.14 The Function $W(a,x)$

… ►

12.14.8 $$W(a,x)=W(a,0)w_1(a,x)+W^{\prime }(a,0)w_2(a,x).$$

ⓘ

Symbols:

$W(a,x)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $w_1(a,x)$: even solution and $w_2(a,x)$: odd solution

A&S Ref:

19.17.1 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(v), §12.14 and Ch.12

►Here $w_1(a,x)$ and $w_2(a,x)$ are the even and odd solutions of (12.2.3): ►

12.14.9 $$w_1(a,x)=\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n(a)\frac{x^{2n}}{(2n)!},$$

ⓘ

Defines:

$w_1(a,x)$: even solution (locally)

Symbols:

$!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer, $a$: real or complex parameter and ${\alpha }_n(a)$: recursion

A&S Ref:

19.16.1 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(v), §12.14 and Ch.12

►

12.14.10 $$w_2(a,x)=\sum _{n=0}^{\mathrm{\infty }}{\beta }_n(a)\frac{x^{2n+1}}{(2n+1)!},$$

ⓘ

Defines:

$w_2(a,x)$: odd solution (locally)

Symbols:

$!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer, $a$: real or complex parameter and ${\beta }_n(a)$: recursion

A&S Ref:

19.16.2 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(v), §12.14 and Ch.12

… ►The even and odd solutions of (12.2.3) (see §12.14(v)) are given by …

19: 30.1 Special Notation

… ►

$S_{mn}^{(1)}(\gamma ,0)={(-1)}^m{𝖯}_n^m\left(0\right),$ $n-m$ even,

►

${{\displaystyle \frac{d}{dx}}{S}_{mn}^{(1)}(\gamma ,x)|}_{x=0}={{(-1)}^m{\displaystyle \frac{d}{dx}}{𝖯}_n^m\left(x\right)|}_{x=0},$ $n-m$ odd.

…

20: 10.74 Methods of Computation

… ►Furthermore, the attainable accuracy can be increased substantially by use of the exponentially-improved expansions given in §10.17(v), even more so by application of the hyperasymptotic expansions to be found in the references in that subsection. … ►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when $\nu$ is half an odd positive integer. …