circular cases

SearchAdvancedHelp

(0.003 seconds)

31—40 of 181 matching pages

31: 19.9 Inequalities

… ►The lower bound in (19.9.4) is sharper than $2/\pi$ when $0\le k^2\le 0.9960$. … ►Ramanujan’s approximation and its leading error term yield the following approximation to $L(a,b)/(\pi (a+b))$: … ►Throughout this subsection we assume that , $0\le \varphi \le \pi /2$, and $\mathrm{\Delta }=\sqrt{1-k^2{\sin }^2\varphi }>0$. … ►

19.9.13 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,0)\le \mathrm{\Pi }(\varphi ,{\alpha }^2,k)\le \min (\mathrm{\Pi }(\varphi ,{\alpha }^2,0)/\mathrm{\Delta },\mathrm{\Pi }(\varphi ,{\alpha }^2,1)).$$

ⓘ

Symbols:

$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $\mathrm{\Delta }$

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

… ►Inequalities for both $F(\varphi ,k)$ and $E(\varphi ,k)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). …

32: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

Example 1: Three Simple Cases where $q(x)=0$, $X=[0,\pi ]$

… ►Case 1: $\varphi (0)=\varphi (\pi )=0$. … ►Case 2: ${\varphi }^{\prime }(0)={\varphi }^{\prime }(\pi )=0$. … ►Case 3: Periodic Boundary Conditions: $\varphi (0)=\varphi (\pi )$ and ${\varphi }^{\prime }(0)={\varphi }^{\prime }(\pi )$. …

33: 18.19 Hahn Class: Definitions

… ►A special case of (18.19.8) is $w^{(1/2)}(x;\pi /2)=\frac{\pi }{\cosh \left(\pi x\right)}$.

34: 6.7 Integral Representations

… ►The first integrals on the right-hand sides apply when ; the second ones when $\mathrm{\Re }z\ge 0$ and (in the case of (6.7.14)) $z\ne 0$. …

35: 2.11 Remainder Terms; Stokes Phenomenon

… ►In the transition through $\theta =\pi$, $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case $\rho =100$. …

36: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►in each case uniformly with respect to $\lambda$ in the sector $|\mathrm{ph}\lambda |\le 2\pi -\delta$ (). …

37: 1.17 Integral and Series Representations of the Dirac Delta

… ►In this case … ►provided that $\varphi (x)$ is continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$ (as in the case of (1.17.6)). … ►

1.17.19 $$\underset{n\to \mathrm{\infty }}{lim}{\int }_{-\pi }^{\pi }{\delta }_n\left(x-a\right)\varphi (x)dx=\varphi (a),$$

ⓘ

Symbols:

${\delta }_n\left(x\right)$: Dirac delta sequence, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\varphi (x)$: continuous function

Permalink:

http://dlmf.nist.gov/1.17.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17 and Ch.1

… ►provided that $\varphi (x)$ is continuous and of period $2\pi$; see §1.8(ii). … ►(1.17.22)–(1.17.24) are special cases of Morse and Feshbach (1953a, Eq. (6.3.11)). …

38: 10.43 Integrals

… ►

10.43.2 $$\int z^{\nu }{𝒵}_{\nu }\left(z\right)dz={\pi }^{\frac{1}{2}}{2}^{\nu -1}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)z\left({𝒵}_{\nu }\left(z\right){𝐋}_{\nu -1}\left(z\right)-{𝒵}_{\nu -1}\left(z\right){𝐋}_{\nu }\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

11.1.8 (Case $\nu =0$ only)

Referenced by:

§10.43(i)

Permalink:

http://dlmf.nist.gov/10.43.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(i), §10.43 and Ch.10

… ►

10.43.25 $${\int }_0^{\mathrm{\infty }}{K}_{\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt=\frac{\sqrt{\pi }}{4p}\sec \left(\frac{1}{2}\pi \nu \right)\exp \left(\frac{b^2}{8p^2}\right)K_{\frac{1}{2}\nu }\left(\frac{b^2}{8p^2}\right),$$ , $\mathrm{\Re }\left(p^2\right)>0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function and $\nu$: complex parameter

A&S Ref:

11.4.32 (Case $\nu =0$)

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

…

39: 4.15 Graphics

… ►they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by $-\frac{1}{2}\pi$ parallel to the $x$-axis, and adjusting the phase coloring in the case of Figure 4.15.10. …

40: 5.11 Asymptotic Expansions

… ►As $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$, … ►In the case $K=1$ the factor $1+\zeta \left(K\right)$ is replaced with 4. For this result and a similar bound for the sector $\frac{1}{2}\pi \le \mathrm{ph}z\le \pi$ see Boyd (1994). … ►If $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$, then … ►For the error term in (5.11.19) in the case $z=x(>0)$ and $c=1$, see Olver (1995). …