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31—40 of 199 matching pages

31: 13.5 Continued Fractions

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32: 13.17 Continued Fractions

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33: 18.15 Asymptotic Approximations

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18.15.5 $$\rho =n+\frac{1}{2}(\alpha +\beta +1).$$

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Defines:

$\rho$ (locally)

Symbols:

$n$: nonnegative integer

Referenced by:

§18.16(ii)

Permalink:

http://dlmf.nist.gov/18.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

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18.15.11 $${\theta }_{n,m}=(n+m+\lambda )\theta -\frac{1}{2}(m+\lambda )\pi .$$

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Defines:

${\theta }_{n,m}$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(ii), §18.15 and Ch.18

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18.15.17 $$\nu =4n+2\alpha +2,$$

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Defines:

$\nu$ (locally)

Symbols:

$n$: nonnegative integer

Referenced by:

§18.15(iv), §18.16(iv)

Permalink:

http://dlmf.nist.gov/18.15.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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18.15.18 $$\xi =\frac{1}{2}\left(\sqrt{x-x^2}+\arcsin (\sqrt{x})\right),$$ $0\le x\le 1$.

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Defines:

$\xi$ (locally)

Symbols:

$\arcsin z$: arcsine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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18.15.26 $${\omega }_{n,m}(x)={\mu }^{\frac{1}{2}}x-\frac{1}{2}(m+n)\pi .$$

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Defines:

${\omega }_{n,m}(x)$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(v), §18.15 and Ch.18

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34: 10.20 Uniform Asymptotic Expansions for Large Order

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10.20.10 $$A_k(\zeta )=\sum _{j=0}^{2k}{(\frac{3}{2})}^j{v}_j{\zeta }^{-3j/2}{U}_{2k-j}\left({(1-z^2)}^{-\frac{1}{2}}\right),$$

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Defines:

$A_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $v_k$: constants and $U_k(p)$: polynomial coefficient

A&S Ref:

9.3.40

Referenced by:

§10.20(i), §10.41(v), §10.74(i)

Permalink:

http://dlmf.nist.gov/10.20.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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10.20.11 $$B_k(\zeta )=-{\zeta }^{-\frac{1}{2}}\sum _{j=0}^{2k+1}{(\frac{3}{2})}^j{u}_j{\zeta }^{-3j/2}{U}_{2k-j+1}\left({(1-z^2)}^{-\frac{1}{2}}\right),$$

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Defines:

$B_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $u_k$: constants and $U_k(p)$: polynomial coefficient

A&S Ref:

9.3.40

Referenced by:

§10.21(viii), §10.41(v)

Permalink:

http://dlmf.nist.gov/10.20.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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10.20.12 $$C_k(\zeta )=-{\zeta }^{\frac{1}{2}}\sum _{j=0}^{2k+1}{(\frac{3}{2})}^j{v}_j{\zeta }^{-3j/2}{V}_{2k-j+1}\left({(1-z^2)}^{-\frac{1}{2}}\right),$$

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Defines:

$C_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $v_k$: constants and $V_k(p)$: polynomial coefficient

A&S Ref:

9.3.46

Referenced by:

§10.21(viii)

Permalink:

http://dlmf.nist.gov/10.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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10.20.13 $$D_k(\zeta )=\sum _{j=0}^{2k}{(\frac{3}{2})}^j{u}_j{\zeta }^{-3j/2}{V}_{2k-j}\left({(1-z^2)}^{-\frac{1}{2}}\right).$$

ⓘ

Defines:

$D_k(\zeta )$: coefficients (locally)

Symbols:

$k$: nonnegative integer, $z$: complex variable, $\zeta (z)$: solution, $u_k$: constants and $V_k(p)$: polynomial coefficient

A&S Ref:

9.3.46

Referenced by:

§10.20(i), §10.74(i)

Permalink:

http://dlmf.nist.gov/10.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

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10.20.18 $$c={({\tau }_0^2-1)}^{\frac{1}{2}}=0.66274\mathrm{\dots }.$$

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Defines:

$c$ (locally)

Referenced by:

§10.41(iii)

Permalink:

http://dlmf.nist.gov/10.20.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(ii), §10.20 and Ch.10

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35: 19.8 Quadratic Transformations

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36: 29.2 Differential Equations

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29.2.3 $$\xi ={\mathrm{sn}}^2(z,k).$$

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Defines:

$\xi$: variable (locally)

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.2.E3

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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29.2.7 $$g=(e_1-e_3)h+\nu (\nu +1)e_3,$$

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Defines:

$g$ (locally)

Symbols:

$h$: real parameter, $\nu$: real parameter and $e_j$: constants

Permalink:

http://dlmf.nist.gov/29.2.E7

Encodings:

pMML, png, TeX

See also:

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

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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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29.2.11 $$\zeta =\mathrm{\wp }(\eta ;g_2,g_3)=\mathrm{\wp }\left(\eta \right),$$

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Defines:

$\zeta$ (locally)

Symbols:

$\mathrm{\wp }(z;g_2,g_3)$: Weierstrass $\mathrm{\wp }$-function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\eta$ , $g_2$ and $g_3$

Permalink:

http://dlmf.nist.gov/29.2.E11

Encodings:

pMML, png, TeX

See also:

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

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37: 30.3 Eigenvalues

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38: 10.41 Asymptotic Expansions for Large Order

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10.41.7 $$\eta ={(1+z^2)}^{\frac{1}{2}}+\ln \frac{z}{1+{(1+z^2)}^{\frac{1}{2}}},$$

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Defines:

$\eta$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.7.11

Permalink:

http://dlmf.nist.gov/10.41.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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39: 19.7 Connection Formulas

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19.7.8 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+\mathrm{\Pi }(\varphi ,{\omega }^2,k)=F(\varphi ,k)+\sqrt{c}{R}_C((c-1)(c-k^2),(c-{\alpha }^2)(c-{\omega }^2)),$$ ${\alpha }^2{\omega }^2=k^2$.

ⓘ

Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: 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 $c$: change of variable

Referenced by:

§19.25(i), §19.6(i)

Permalink:

http://dlmf.nist.gov/19.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.9 $$(k^2-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+(k^2-{\omega }^2)\mathrm{\Pi }(\varphi ,{\omega }^2,k)=k^{2}F(\varphi ,k)-{\alpha }^2{\omega }^2\sqrt{c-1}{R}_C(c(c-k^2),(c-{\alpha }^2)(c-{\omega }^2)),$$ $(1-{\alpha }^2)(1-{\omega }^2)=1-k^2$.

ⓘ

Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: 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 $c$: change of variable

Permalink:

http://dlmf.nist.gov/19.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.10 $$(1-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+(1-{\omega }^2)\mathrm{\Pi }(\varphi ,{\omega }^2,k)=F(\varphi ,k)+(1-{\alpha }^2-{\omega }^2)\sqrt{c-k^2}{R}_C(c(c-1),(c-{\alpha }^2)(c-{\omega }^2)),$$ $(k^2-{\alpha }^2)(k^2-{\omega }^2)=k^2(k^2-1)$.

ⓘ

Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: 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 $c$: change of variable

Permalink:

http://dlmf.nist.gov/19.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

40: 28.8 Asymptotic Expansions for Large $q$

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28.8.6 $${\widehat{C}}_m\sim {\left(\frac{\pi h}{2{(m!)}^2}\right)}^{1/4}{\left(1+\frac{2m+1}{8h}+\frac{m^4+2m^3+263m^2+262m+108}{2048h^2}+\mathrm{\cdots }\right)}^{-1/2},$$

ⓘ

Defines:

${\widehat{C}}_m$: coefficient (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer and $h$: parameter

A&S Ref:

20.9.19 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

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28.8.7 $${\widehat{S}}_m\sim {\left(\frac{\pi h}{2{(m!)}^2}\right)}^{1/4}{\left(1-\frac{2m+1}{8h}+\frac{m^4+2m^3-121m^2-122m-84}{2048h^2}+\mathrm{\cdots }\right)}^{-1/2}.$$

ⓘ

Defines:

${\widehat{S}}_m$: coefficient (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer and $h$: parameter

A&S Ref:

20.9.20 (in slightly different form)

Referenced by:

§28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

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28.8.9 $$W_m^{\pm }(x)=\frac{{\mathrm{e}}^{\pm 2h\sin x}}{{(\cos x)}^{m+1}}\{\begin{array}{cc}{\left(\cos \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \\ {\left(\sin \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \end{array}$$

ⓘ

Defines:

$W_m^{\pm }$: function (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.13 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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28.8.11 $$P_m(x)\sim 1+\frac{s}{2^{3}h{\cos }^{2}x}+\frac{1}{h^2}\left(\frac{s^4+86s^2+105}{2^{11}{\cos }^{4}x}-\frac{s^4+22s^2+57}{2^{11}{\cos }^{2}x}\right)+\mathrm{\cdots },$$

ⓘ

Defines:

$P_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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28.8.12 $$Q_m(x)\sim \frac{\sin x}{{\cos }^{2}x}\left(\frac{1}{2^{5}h}(s^2+3)+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cos }^{2}x}\right)\right)+\mathrm{\cdots }.$$

ⓘ

Defines:

$Q_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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