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11: 26.3 Lattice Paths: Binomial Coefficients

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§26.3(v) Limiting Form

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26.3.12 $$\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)\sim \frac{4^n}{\sqrt{\pi n}},$$ $n\to \mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Referenced by:

§26.5(iv)

Permalink:

http://dlmf.nist.gov/26.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(v), §26.3 and Ch.26

12: 4.4 Special Values and Limits

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4.4.16 $$\underset{z\to \mathrm{\infty }}{lim}{z}^a{\mathrm{e}}^{-z}=0,$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ (),

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $a$: real or complex constant, $z$: complex variable and $\delta$: constant

A&S Ref:

4.2.20

Permalink:

http://dlmf.nist.gov/4.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(iii), §4.4 and Ch.4

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13: 33.18 Limiting Forms for Large $\mathrm{\ell }$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

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14: 10.28 Wronskians and Cross-Products

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10.28.1 $$𝒲\left\{I_{\nu }\left(z\right),I_{-\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)I_{-\nu -1}\left(z\right)-I_{\nu +1}\left(z\right)I_{-\nu }\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

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Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.14

Permalink:

http://dlmf.nist.gov/10.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

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15: 10.45 Functions of Imaginary Order

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$${\tilde{K}}_{\nu }\left(x\right)={(\pi /(2x))}^{\frac{1}{2}}{\mathrm{e}}^{-x}\left(1+O\left(x^{-1}\right)\right).$$

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10.45.6 $${\tilde{I}}_{\nu }\left(x\right)={\left(\frac{\sinh \left(\pi \nu \right)}{\pi \nu }\right)}^{\frac{1}{2}}\cos \left(\nu \ln \left(\frac{1}{2}x\right)-{\gamma }_{\nu }\right)+O\left(x^2\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sinh z$: hyperbolic sine function, ${\tilde{I}}_{\nu }\left(x\right)$: modified Bessel function of the first kind of imaginary order, $\ln z$: principal branch of logarithm function, $x$: real variable, $\nu$: complex parameter and ${\gamma }_{\nu }$: parameter

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.45.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.45 and Ch.10

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10.45.7 $${\tilde{K}}_{\nu }\left(x\right)=-{\left(\frac{\pi }{\nu \sinh \left(\pi \nu \right)}\right)}^{\frac{1}{2}}\sin \left(\nu \ln \left(\frac{1}{2}x\right)-{\gamma }_{\nu }\right)+O\left(x^2\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\tilde{K}}_{\nu }\left(x\right)$: modified Bessel function fo the second kind of imaginary order, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $x$: real variable, $\nu$: complex parameter and ${\gamma }_{\nu }$: parameter

Referenced by:

§10.30(i), §10.45

Permalink:

http://dlmf.nist.gov/10.45.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.45 and Ch.10

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16: 29.5 Special Cases and Limiting Forms

§29.5 Special Cases and Limiting Forms

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29.5.4 $$\underset{k\to 1-}{lim}{a}_{\nu }^m\left(k^2\right)=\underset{k\to 1-}{lim}{b}_{\nu }^{m+1}\left(k^2\right)=\nu (\nu +1)-{\mu }^2,$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

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29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

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$lim{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)={\mathrm{ce}}_m(\frac{1}{2}\pi -z,\theta ),$

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$lim{\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)={\mathrm{se}}_m(\frac{1}{2}\pi -z,\theta ),$

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17: 20.5 Infinite Products and Related Results

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20.5.14 $${\theta }_1\left(z\right|\tau )=z{\theta }_1^{\prime }\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}\prod _{n=-N}^N\underset{M\to \mathrm{\infty }}{lim}\prod _{\begin{array}{c}m=-M\\ \left|m\right|+\left|n\right|\ne 0\end{array}}^M\left(1+\frac{z}{(m+n\tau )\pi }\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.5.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(iii), §20.5 and Ch.20

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20.5.15 $${\theta }_2\left(z\right|\tau )={\theta }_2\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}\prod _{n=-N}^N\underset{M\to \mathrm{\infty }}{lim}\prod _{m=1-M}^M\left(1+\frac{z}{(m-\frac{1}{2}+n\tau )\pi }\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.5.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(iii), §20.5 and Ch.20

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20.5.16 $${\theta }_3\left(z\right|\tau )={\theta }_3\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}\prod _{n=1-N}^N\underset{M\to \mathrm{\infty }}{lim}\prod _{m=1-M}^M\left(1+\frac{z}{(m-\frac{1}{2}+(n-\frac{1}{2})\tau )\pi }\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.5.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(iii), §20.5 and Ch.20

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20.5.17 $${\theta }_4\left(z\right|\tau )={\theta }_4\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}\prod _{n=1-N}^N\underset{M\to \mathrm{\infty }}{lim}\prod _{m=-M}^M\left(1+\frac{z}{(m+(n-\frac{1}{2})\tau )\pi }\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(iii), §20.5 and Ch.20

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18: 1.17 Integral and Series Representations of the Dirac Delta

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1.17.6 $$\underset{n\to \mathrm{\infty }}{lim}\sqrt{\frac{n}{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx=\varphi (a),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $n$: nonnegative integer and $\varphi (x)$: continuous function

Referenced by:

§1.17(i), §1.17(ii), §7.19(iii)

Permalink:

http://dlmf.nist.gov/1.17.E6

Encodings:

pMML, png, TeX

See also:

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

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1.17.7 $$\underset{n\to \mathrm{\infty }}{lim}\sqrt{\frac{n}{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx=\frac{1}{2}\varphi (a-)+\frac{1}{2}\varphi (a+).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $n$: nonnegative integer and $\varphi (x)$: continuous function

Referenced by:

§1.17(i)

Permalink:

http://dlmf.nist.gov/1.17.E7

Encodings:

pMML, png, TeX

See also:

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

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

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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

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19: 7.2 Definitions

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7.2.1 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}{\int }_0^z{\mathrm{e}}^{-t^2}dt,$$

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Defines:

$\mathrm{erf}z$: error function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.1

Referenced by:

§7.5, §7.6(ii)

Permalink:

http://dlmf.nist.gov/7.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(i), §7.2 and Ch.7

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$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erf}z=1,$

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$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erfc}z=0$ , .

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$\underset{x\to \mathrm{\infty }}{lim}C\left(x\right)=\frac{1}{2},$

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$\underset{x\to \mathrm{\infty }}{lim}S\left(x\right)=\frac{1}{2}.$

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20: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.7 $$C=\underset{n\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^{n}k\ln k-\left(\frac{1}{2}{n}^2+\frac{1}{2}n+\frac{1}{12}\right)\ln n+\frac{1}{4}{n}^2\right)=\frac{\gamma +\ln \left(2\pi \right)}{12}-\frac{{\zeta }^{\prime }\left(2\right)}{2{\pi }^2}=\frac{1}{12}-{\zeta }^{\prime }\left(-1\right),$$

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $k$: nonnegative integer and $C$: log of Glaisher’s Constant

Referenced by:

§5.17

Permalink:

http://dlmf.nist.gov/5.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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