change of modulus

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51: 26.7 Set Partitions: Bell Numbers

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26.7.7 $$B\left(n\right)=\frac{N^n{\mathrm{e}}^{N-n-1}}{{(1+\ln N)}^{1/2}}\left(1+O\left(\frac{{(\ln n)}^{1/2}}{n^{1/2}}\right)\right),$$ $n\to \mathrm{\infty }$,

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Symbols:

$B\left(n\right)$: Bell number, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $N$

Permalink:

http://dlmf.nist.gov/26.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.7(iv), §26.7 and Ch.26

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52: 12.11 Zeros

… ►When $a=\frac{1}{2}$ these zeros are the same as the zeros of the complementary error function $\mathrm{erfc}(z/\sqrt{2})$; compare (12.7.5). … ►

12.11.5 $$p_0(\zeta )=t(\zeta ),$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.6 $$p_1(\zeta )=\frac{t^3-6t}{24{(t^2-1)}^2}+\frac{5}{48{((t^2-1){\zeta }^3)}^{\frac{1}{2}}}.$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.8 $$q_0(\zeta )=t(\zeta ).$$

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Symbols:

$\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/12.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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53: 19.28 Integrals of Elliptic Integrals

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19.28.9 $${\int }_0^{\pi /2}{R}_F({\sin }^2\theta {\cos }^2\left(x+y\right),{\sin }^2\theta {\cos }^2\left(x-y\right),1)d\theta =R_F(0,{\cos }^{2}x,1)R_F(0,{\cos }^{2}y,1),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $\sin z$: sine function

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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54: 10.21 Zeros

… ► $j_{\nu ,m}/\nu$ and $j_{\nu ,m}^{\prime }/\nu$ are decreasing functions of $\nu$ when $\nu >0$ for $m=1,2,3,\mathrm{\dots }$. … ►Let ${𝒞}_{\nu }\left(x\right)$, ${\rho }_{\nu }(t)$, and ${\sigma }_{\nu }(t)$ be defined as in §10.21(ii) and $M\left(x\right)$, $\theta \left(x\right)$, $N\left(x\right)$, and $\varphi \left(x\right)$ denote the modulus and phase functions for the Airy functions and their derivatives as in §9.8. … ►(Note: If the term $z(\zeta ){(h(\zeta ))}^2{C}_0(\zeta )/(2\zeta \nu )$ in (10.21.43) is omitted, then the uniform character of the error term $O(1/\nu )$ is destroyed.) … ►Secondly, there is a conjugate pair of infinite strings of zeros with asymptotes $\mathrm{\Im }z=\pm \mathrm{i}a/n$, where … ►Higher coefficients in the asymptotic expansions in this subsection can be obtained by expressing the cross-products in terms of the modulus and phase functions (§10.18), and then reverting the asymptotic expansion for the difference of the phase functions. …

55: 2.6 Distributional Methods

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2.6.51 $$ℳf\left(s\right)={(-1)}^s\pi /\sin \left(\pi \alpha \right),$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $f(t)$: locally integrable function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.6.E51

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §2.6(iii), §2.6(iii), §2.6 and Ch.2

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56: 9.12 Scorer Functions

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9.12.29 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k}+\frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$.

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Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Permalink:

http://dlmf.nist.gov/9.12.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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57: 29.2 Differential Equations

… ►This equation has regular singularities at the points $2pK+(2q+1)\mathrm{i}{K}^{\prime }$, where $p,q\in \mathbb{Z}$, and $K$, $K^{\prime }$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime }(={(1-k^2)}^{1/2})$, respectively; see §19.2(ii). … ►

29.2.4 $$(1-k^2{\cos }^2\varphi )\frac{d^{2}w}{{d\varphi }^2}+k^2\cos \varphi \sin \varphi \frac{dw}{d\varphi }+(h-\nu (\nu +1)k^2{\cos }^2\varphi )w=0,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $\varphi$: change of variable

Permalink:

http://dlmf.nist.gov/29.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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29.2.5 $$\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k).$$

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Defines:

$\varphi$: change of variable (locally)

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: real parameter

Referenced by:

§29.15(i), §29.15(ii), §29.6(i)

Permalink:

http://dlmf.nist.gov/29.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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$(e_2-e_3)/(e_1-e_3)=k^2.$

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29.2.8 $$\eta ={(e_1-e_3)}^{-1/2}(z-\mathrm{i}{K}^{\prime }),$$

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Defines:

$\eta$ (locally)

Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable, $k$: real parameter and $e_j$: constants

Permalink:

http://dlmf.nist.gov/29.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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58: 22.15 Inverse Functions

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22.15.3 $$\mathrm{dn}(\zeta ,k)=x,$$ $k^{\prime }\le x\le 1$,

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $x$: real, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: solution

Referenced by:

§22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

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22.15.13 $$\mathrm{arccn}(x,k)={\int }_x^1\frac{dt}{\sqrt{(1-t^2)(k_{}^{\prime }{}_{}{}^2+k^2{t}^2)}},$$ $-1\le x\le 1$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\mathrm{arccn}(x,k)$: inverse Jacobian elliptic function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

17.4.52

Referenced by:

§22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(ii), §22.15 and Ch.22

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22.15.16 $$\mathrm{arcsd}(x,k)={\int }_0^x\frac{dt}{\sqrt{(1-k_{}^{\prime }{}_{}{}^2{t}^2)(1+k^2{t}^2)}},$$ $-1/k^{\prime }\le x\le 1/k^{\prime }$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\mathrm{arcsd}(x,k)$: inverse Jacobian elliptic function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

17.4.51

Permalink:

http://dlmf.nist.gov/22.15.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(ii), §22.15 and Ch.22

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22.15.17 $$\mathrm{arcnd}(x,k)={\int }_1^x\frac{dt}{\sqrt{(t^2-1)(1-k_{}^{\prime }{}_{}{}^2{t}^2)}},$$ $1\le x\le 1/k^{\prime }$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\mathrm{arcnd}(x,k)$: inverse Jacobian elliptic function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

17.4.43

Permalink:

http://dlmf.nist.gov/22.15.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(ii), §22.15 and Ch.22

… ►can be transformed into normal form by elementary change of variables. …

59: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where $f(x)$ is continuous, with convergence to $(f(x_0-)+f(x_0+))/2$ if $x_0$ is an isolated point of discontinuity. … ►with ${\lambda }_{\pm n}=4n^2$, $n=0,1,2,\mathrm{\dots }$, with all eigenvalues, for $\left|n\right|>0$, having multiplicity two, as changing the sign of $n$ changes the eigenfunction but not the eigenvalue, and multiplicity one for $n=0$. Letting $n$ run from $-\mathrm{\infty }$ to $\mathrm{\infty }$ this multiplicity change is automatically included: … ►Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies ${\lambda }_{\mathrm{res}}-\mathrm{i}{\mathrm{\Gamma }}_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/{\mathrm{\Gamma }}_{\mathrm{res}}$. …This is accomplished by the variable change $x\to x{\mathrm{e}}^{\mathrm{i}\theta }$, in $ℒ$, which rotates the continuous spectrum ${𝝈}_c\to {𝝈}_c{\mathrm{e}}^{-2\mathrm{i}\theta }$ and the branch cut of (1.18.66) into the lower half complex plain by the angle $-2\theta$, with respect to the unmoved branch point at $\lambda =0$; thus, providing access to resonances on the higher Riemann sheet should $\theta$ be large enough to expose them. …

60: 4.13 Lambert $W$-Function

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4.13.6 $$W\left(-{\mathrm{e}}^{-1-(t^2/2)}\right)=\sum _{n=0}^{\mathrm{\infty }}{(-1)}^{n-1}{c}_n{t}^n,$$ ,

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $n$: integer and $c_{\mathrm{i}}$: coefficients

Referenced by:

§4.13, §4.13, §4.45(iii)

Permalink:

http://dlmf.nist.gov/4.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.13 and Ch.4

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4.13.9_1 $$W_0\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{d}_n{\left(\mathrm{e}z+1\right)}^{n/2},$$ , ,

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $n$: integer and $z$: complex variable

Proof sketch:

Substitute (4.13.9_1) into $z(1+W)\frac{dW}{dz}=W$.

Referenced by:

(4.13.9_1), §4.13, §4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E9_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

… ►where $\eta =\ln \left(-1/x\right)$. …