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21—30 of 209 matching pages

21: 7.17 Inverse Error Functions

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7.17.2 $$\mathrm{inverf}x=t+\frac{1}{3}{t}^3+\frac{7}{30}{t}^5+\frac{127}{630}{t}^7+\mathrm{\cdots }=\sum _{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1},$$ ,

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Symbols:

$\mathrm{inverf}x$: inverse error function and $x$: real variable

Referenced by:

Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The equality “$={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1}$” was added to this equation.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

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7.17.2_5 $$a_{m+1}=\frac{1}{2m+3}\sum _{n=0}^m\frac{2n+1}{m-n+1}{a}_n{a}_{m-n},$$ $m=0,1,2,\mathrm{\dots }$.

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Symbols:

$n$: nonnegative integer

Referenced by:

§7.17(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

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22: 14.32 Methods of Computation

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23: 28.36 Software

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24: 18.32 OP’s with Respect to Freud Weights

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18.32.2 $$w(x)={\left|x\right|}^{\alpha }\exp \left(-Q(x)\right),$$ $x\in ℝ$, $\alpha >-1$,

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Symbols:

$\in$: element of, $\exp z$: exponential function, $ℝ$: real line, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.32, Erratum (V1.2.0) Section 18.32

Permalink:

http://dlmf.nist.gov/18.32.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.32 and Ch.18

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25: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. … ►

18.36.2 $$L_n^{(-k)}\left(x\right)={(-1)}^k\frac{(n-k)!}{n!}{x}^k{L}_{n-k}^{(k)}\left(x\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(v), §18.36 and Ch.18

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18.36.3 $${\widehat{W}}_k(x)=\frac{x^k{\mathrm{e}}^{-x}}{{(x+k)}^2},$$ $k>0,x\in [0,\mathrm{\infty })$.

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Symbols:

$[a,b)$: half-closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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18.36.5 $${\widehat{L}}_n^{(k)}\left(x\right)=-(x+k+1)L_{n-1}^{(k)}\left(x\right)+L_{n-2}^{(k)}\left(x\right),$$ $n\ge 1$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\widehat{L}}_n^{(k)}\left(x\right)$: exceptional Laguerre polynomial, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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18.36.7 $$T_k(y)\equiv -x{y}^{\prime \prime }+\frac{x-k}{x+k}((x+k+1)y^{\prime }-y)=(n-1)y.$$

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Symbols:

$\equiv$: equals by definition, $y$: real variable, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.36(vi)

Permalink:

http://dlmf.nist.gov/18.36.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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26: 4.13 Lambert $W$-Function

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4.13.1_1 $$W_k\left(z\right)={\ln }_k(z)-\ln \left({\ln }_k(z)\right)+o\left(1\right),$$ $\left|z\right|\to \mathrm{\infty }$,

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function, $k$: integer and $z$: complex variable

Notes:

See Corless et al. (1996, §4).

Permalink:

http://dlmf.nist.gov/4.13.E1_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.1_3 $$T{\mathrm{e}}^{-T}=z.$$

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Symbols:

$T\left(z\right)$: Tree $T$-function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E1_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.5_3 $${\left(1+W_0\left(z\right)\right)}^2=1-2\sum _{n=1}^{\mathrm{\infty }}\frac{n^{n-2}}{n!}{\left(-z\right)}^n,$$ .

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

Notes:

See https://vixra.org/abs/2308.0054.

Referenced by:

§4.13, Erratum (V1.1.11) for Additions

Permalink:

http://dlmf.nist.gov/4.13.E5_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.11):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.12 $$\int W\left(z\right)dz=\frac{z}{W\left(z\right)}+zW\left(z\right)-z,$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.13.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.13 $$\int \frac{W\left(z\right)}{z}dz=\frac{1}{2}{W\left(z\right)}^2+W\left(z\right),$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.13.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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27: 7.8 Inequalities

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7.8.7 $x>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

Referenced by:

Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/7.8.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.1.5):

A new inequality was added.

See also:

Annotations for §7.8 and Ch.7

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7.8.8 $x>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $x$: real variable

Source:

Pólya (1949, (1.5))

Referenced by:

§7.8, Erratum (V1.0.17) for Equation (7.8.8)

Permalink:

http://dlmf.nist.gov/7.8.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11).

Suggested by Roberto Iacono

See also:

Annotations for §7.8 and Ch.7

28: 15.19 Methods of Computation

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29: 12.21 Software

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30: 18.34 Bessel Polynomials

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18.34.5_5 $$\frac{2^{1-a}}{\mathrm{\Gamma }\left(1-a\right)}{\int }_0^{\mathrm{\infty }}{y}_n(x;a)y_m(x;a)x^{a-2}{\mathrm{e}}^{-2x^{-1}}dx=\frac{1-a}{1-a-2n}\frac{n!}{{\left(2-a-n\right)}_n}{\delta }_{n,m},$$ $m,n=0,1,\mathrm{\dots },N=\lceil -(1+a)/2\rceil$.

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Symbols:

$y_n(x;a)$: Bessel polynomial, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\lceil x\rceil$: ceiling of $x$ , $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Lesky (1996, p.183, case a.3))(proved)

Sources:

Romanovski (1929, Type V); Koekoek et al. (2010, (9.13.2))

Referenced by:

(18.34.7_3), §18.34(ii), §18.34(iii), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E5_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(ii), §18.34 and Ch.18

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18.34.7_1 $${\varphi }_n(x;\lambda )={\mathrm{e}}^{-\lambda {\mathrm{e}}^{-x}}{\left(2\lambda {\mathrm{e}}^{-x}\right)}^{\lambda -\frac{1}{2}}{y}_n({\lambda }^{-1}{\mathrm{e}}^x;2-2\lambda )/n!={(-1)}^n{\mathrm{e}}^{-\lambda {\mathrm{e}}^{-x}}{\left(2\lambda {\mathrm{e}}^{-x}\right)}^{\lambda -n-\frac{1}{2}}{L}_n^{(2\lambda -2n-1)}\left(2\lambda {\mathrm{e}}^{-x}\right)={(2\lambda )}^{-\frac{1}{2}}{\mathrm{e}}^{x/2}{W}_{\lambda ,n+\frac{1}{2}-\lambda }\left(2\lambda {\mathrm{e}}^{-x}\right)/n!,$$ $n=0,1,\mathrm{\dots },N=\lceil \lambda -\frac{3}{2}\rceil$, $\lambda >\frac{1}{2}$,

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Symbols:

$y_n(x;a)$: Bessel polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\lceil x\rceil$: ceiling of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Proof sketch:

The second and third equality follow from (18.34.1).

Referenced by:

(18.34.7_2), (18.34.7_3), §18.34(iii), (18.39.17), (18.39.18), §18.39(i), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E7_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

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18.34.7_2 $$\left(\frac{d^2}{{dx}^2}-{\lambda }^2\left({\mathrm{e}}^{-2x}-2{\mathrm{e}}^{-x}\right)-{\left(\lambda -\left(n+\frac{1}{2}\right)\right)}^2\right){\varphi }_n(x;\lambda )=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $x$: real variable

Proof sketch:

This follows by straightforward computation from (18.34.7) and (18.34.7_1).

Referenced by:

(18.39.17), (18.39.18)

Permalink:

http://dlmf.nist.gov/18.34.E7_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

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18.34.7_3 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\varphi }_n(x;\lambda ){\varphi }_m(x;\lambda )dx=\frac{\mathrm{\Gamma }\left(2\lambda -n\right)}{(2\lambda -2n-1)n!}{\delta }_{n,m},$$ $m,n=0,1,\mathrm{\dots },N=\lceil \lambda -\frac{3}{2}\rceil$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, $\lceil x\rceil$: ceiling of $x$ , $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proof sketch:

This follows by straightforward computation from (18.34.5_5) and (18.34.7_1).

Referenced by:

§18.34(iii), (18.39.17), §18.39(i), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E7_3

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

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