Ces%C3%A0ro%20means

SearchAdvancedHelp

(0.002 seconds)

21—30 of 176 matching pages

21: Guide to Searching the DLMF

… ►

The following standard special functions: si, Si, ci, Ci, shi, Shi, ce, Ce, se, Se, ln, Ln, Lommels, LommelS, Jacobiphi, and the list is still growing.

…

22: 29.12 Definitions

… ►

29.12.3 $${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)={\mathrm{𝐸𝑠}}_{2n+1}^{2m+1}(z,k^2),$$

ⓘ

Defines:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial

Symbols:

${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.12(i), §29.12 and Ch.29

… ►

Table 29.12.1: Lamé polynomials.

► ►►►

$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
…
$2n+1$	$b_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑐𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{cn}P({\mathrm{sn}}^2)$	$4K$	$4\mathrm{i}{K}^{\prime }$	even	odd	even
…

►

…

23: 29.13 Graphics

… ►

►►

Figure 29.13.9: ${\mathrm{𝑐𝐸}}_5^m(x,0.1)$ for $-2K\le x\le 2K$, $m=0,1,2$. … Magnify

►

►►

Figure 29.13.10: ${\mathrm{𝑐𝐸}}_5^m(x,0.9)$ for $-2K\le x\le 2K$, $m=0,1,2$. … Magnify

…

24: 28.4 Fourier Series

… ►

28.4.1 $${\mathrm{ce}}_{2n}(z,q)=\sum _{m=0}^{\mathrm{\infty }}{A}_{2m}^{2n}(q)\cos 2mz,$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.2.3 (in slightly different form) 20.2.4 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(i), §28.4 and Ch.28

►

28.4.2 $${\mathrm{ce}}_{2n+1}(z,q)=\sum _{m=0}^{\mathrm{\infty }}{A}_{2m+1}^{2n+1}(q)\cos (2m+1)z,$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(i), §28.4 and Ch.28

…

25: 29.15 Fourier Series and Chebyshev Series

… ►

Polynomial ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$

… ►

29.15.13 $${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)=\sum _{p=0}^n{B}_{2p+1}\sin \left((2p+1)\varphi \right).$$

ⓘ

Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $B_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.45 $${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)=\mathrm{cn}(z,k)\sum _{p=0}^n{B}_{2p+1}{U}_{2p}\left(\mathrm{sn}(z,k)\right),$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $B_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

…

26: 28.20 Definitions and Basic Properties

… ►

§28.20(ii) Solutions ${\mathrm{Ce}}_{\nu }$, ${\mathrm{Se}}_{\nu }$, ${\mathrm{Me}}_{\nu }$, ${\mathrm{Fe}}_n$, ${\mathrm{Ge}}_n$

►

28.20.3 $${\mathrm{Ce}}_{\nu }(z,q)={\mathrm{ce}}_{\nu }(\pm \mathrm{i}z,q),$$ $\nu \ne -1,-2,\mathrm{\dots }$,

ⓘ

Defines:

${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(ii), §28.20 and Ch.28

…

27: 29.1 Special Notation

… ►The main functions treated in this chapter are the eigenvalues $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$, and the Lamé polynomials ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$, ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$. …

28: 28.31 Equations of Whittaker–Hill and Ince

… ►

$C_p^m(x,\xi )\to {\mathrm{ce}}_m(x,q),$

…

29: 20 Theta Functions

Chapter 20 Theta Functions

…

30: Errata

… ►

The factor on the right-hand side containing $\cos (\mu -\nu )\theta$ has been been replaced with $\cos \left((\mu -\nu )\theta \right)$ to clarify the meaning.

… ►

The second and the fourth lines containing $k^{\prime }/ik$ have both been replaced with $-i{k}^{\prime }/k$ to clarify the meaning.

… ►

28.28.21 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\cos \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{ce}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Mc}}_{2m+1}^{(j)}(z,h)$$

28.28.22 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{se}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Ms}}_{2m+1}^{(j)}(z,h),$$

Originally the prefactor $\frac{4}{\pi }$ and upper limit of integration $\pi /2$ in these two equations were given incorrectly as $\frac{2}{\pi }$ and $\pi$.

Reported 2015-05-20 by Ruslan Kabasayev

… ►

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

… ►

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

…