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1: 7.9 Continued Fractions

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7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.1.14 (in different form)

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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7.9.2 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{2z}{2z^2+1-}\frac{1\cdot 2}{2z^2+5-}\frac{3\cdot 4}{2z^2+9-}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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7.9.3 $$w\left(z\right)=\frac{\mathrm{i}}{\sqrt{\pi }}\frac{1}{z-}\frac{\frac{1}{2}}{z-}\frac{1}{z-}\frac{\frac{3}{2}}{z-}\frac{2}{z-}\mathrm{\cdots },$$ $\mathrm{\Im }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.15 (in different form)

Permalink:

http://dlmf.nist.gov/7.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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2: 1.12 Continued Fractions

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§1.12(iv) Contraction and Extension

… ►If $C_n^{\prime }=C_{2n}$, $n=0,1,2,\mathrm{\dots }$, then $C^{\prime }$ is called the even part of $C$. The even part of $C$ exists iff $b_{2k}\ne 0$, $k=1,2,\mathrm{\dots }$, and up to equivalence is given by … ►and the even and odd parts of the continued fraction converge to finite values. …

3: 18.17 Integrals

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18.17.41 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{\mathrm{𝐻𝑒}}_n\left(x\right)x^{z-1}dx=\mathrm{\Gamma }\left(z+n\right)a^{-n-2}{}_2{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{-\frac{1}{2}z-\frac{1}{2}n,-\frac{1}{2}z-\frac{1}{2}n+\frac{1}{2}};-\frac{1}{2}{a}^2),$$ $\mathrm{\Re }a>0$. Also, $\mathrm{\Re }z>0$, $n$ even; $\mathrm{\Re }z>-1$, $n$ odd.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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4: 28.23 Expansions in Series of Bessel Functions

… ►When $j=2,3,4$ the series in the even-numbered equations converge for $\mathrm{\Re }z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\mathrm{\Re }z>0$ and $|\sinh z|>1$. …

5: 20.1 Special Notation

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$m$, $n$	integers.
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$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
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6: 19.38 Approximations

… ►They are valid over parts of the complex $k$ and $\varphi$ planes. The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

7: 4.37 Inverse Hyperbolic Functions

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4.37.11 $$\mathrm{arccosh}\left(-z\right)=\pm \pi \mathrm{i}+\mathrm{arccosh}z,$$ $\mathrm{\Im }z\gtrless 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.6.12 (has an error even in the tenth printing.)

Referenced by:

§4.37(iii)

Permalink:

http://dlmf.nist.gov/4.37.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

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8: 14.17 Integrals

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14.17.15 $${\int }_{-1}^1{𝖯}_{\nu }\left(x\right){𝖰}_{\nu }\left(x\right)dx=-\frac{\sin \left(2\nu \pi \right){\psi }^{\prime }\left(\nu +1\right)}{\pi (2\nu +1)},$$ $\mathrm{\Re }\nu >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.9

Permalink:

http://dlmf.nist.gov/14.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

… ►(When $l+m+n$ is even the condition is not needed.) … ►

14.17.18 $${\int }_1^{\mathrm{\infty }}{P}_{\nu }\left(x\right)Q_{\lambda }\left(x\right)dx=\frac{1}{(\lambda -\nu )(\nu +\lambda +1)},$$ $\mathrm{\Re }\lambda >\mathrm{\Re }\nu >0$.

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.1

Permalink:

http://dlmf.nist.gov/14.17.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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14.17.19 $${\int }_1^{\mathrm{\infty }}{Q}_{\nu }\left(x\right)Q_{\lambda }\left(x\right)dx=\frac{\psi \left(\lambda +1\right)-\psi \left(\nu +1\right)}{(\lambda -\nu )(\lambda +\nu +1)},$$ $\mathrm{\Re }\left(\lambda +\nu \right)>-1$, $\lambda \ne \nu$, $\lambda$ and $\nu \ne -1,-2,-3,\mathrm{\dots }$.

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Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.2

Permalink:

http://dlmf.nist.gov/14.17.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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14.17.20 $${\int }_1^{\mathrm{\infty }}{(Q_{\nu }\left(x\right))}^{2}dx=\frac{{\psi }^{\prime }\left(\nu +1\right)}{2\nu +1},$$ $\mathrm{\Re }\nu >-\frac{1}{2}$.

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Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.3

Permalink:

http://dlmf.nist.gov/14.17.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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9: 18.32 OP’s with Respect to Freud Weights

… ►where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $x{Q}^{\prime }(x)$ increases for $x>0$, and $Q^{\prime }(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$, see Freud (1969). …See the early survey by Nevai (1986, Part 2). …

10: 37.15 Orthogonal Polynomials on the Ball

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37.15.20 $${𝒱}_{2n}^{\alpha }(𝐱;{𝔹}^d)=\underset{\mathit{ϵ}\in {\{0,1\}}^d;|\mathit{ϵ}|\mathrm{even}}{⨁}{𝐱}^{\mathit{ϵ}}{𝒱}_{n-\frac{1}{2}|\mathit{ϵ}|}^{-\frac{\mathrm{𝟏}}{\mathrm{𝟐}}+\mathit{ϵ},\alpha }({𝐱}^2;{\mathrm{△}}^d),$$

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Symbols:

$\in$: element of, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${\mathrm{△}}^d$: simplex and ${𝔹}^d$: unit ball

Permalink:

http://dlmf.nist.gov/37.15.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(vii), §37.15, Part 37.III and Ch.37

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