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1: 32.1 Special Notation

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$m,n$	integers.
$x$	real variable.
$z$	complex variable.
$k$	real parameter.

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2: 37.1 Notation

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$x$, $y$	real variables.
$(x,y)$	element of ${ℝ}^2$.
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$d$	positive integer, usually $\ge 2$.
${ℝ}_{+}$	positive real line.
${ℝ}^d$	$d$-dimensional Euclidean space.
$𝐱,𝐲$	$(x_1,\mathrm{\dots },x_d),(y_1,\mathrm{\dots },y_d)\in {ℝ}^d$.
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3: 5.24 Software

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§5.24(ii) $\mathrm{\Gamma }\left(x\right)$, $x\in ℝ$

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§5.24(iii) $\psi \left(x\right)$, ${\psi }^{(n)}\left(x\right)$, $x\in ℝ$

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§5.24(v) $\mathrm{B}(a,b)$, $a,b\in ℝ$

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4: 15.7 Continued Fractions

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15.7.1 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=t_0-\frac{u_{1}z}{t_1-}\frac{u_{2}z}{t_2-}\frac{u_{3}z}{t_3-}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $t_n$: coefficient and $u_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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15.7.3 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=v_0-\frac{w_1}{v_1-}\frac{w_2}{v_2-}\frac{w_3}{v_3-}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $v_n$: coefficient and $w_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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15.7.5 $$\frac{𝐅(a,b;c;z)}{𝐅(a+1,b+1;c+1;z)}=x_0+\frac{y_1}{x_1+}\frac{y_2}{x_2+}\frac{y_3}{x_3+}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $y$: real variable, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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5: 8.14 Integrals

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8.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}\frac{\gamma (b,x)}{\mathrm{\Gamma }\left(b\right)}dx=\frac{{(1+a)}^{-b}}{a},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14, §8.14

Permalink:

http://dlmf.nist.gov/8.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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8.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}\mathrm{\Gamma }(b,x)dx=\mathrm{\Gamma }\left(b\right)\frac{1-{(1+a)}^{-b}}{a},$$ $\mathrm{\Re }a>-1$, $\mathrm{\Re }b>-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

A&S Ref:

6.5.36

Referenced by:

§8.14, §8.14

Permalink:

http://dlmf.nist.gov/8.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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8.14.3 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}\gamma (b,x)dx=-\frac{\mathrm{\Gamma }\left(a+b\right)}{a},$$ , $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/8.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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8.14.4 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}\mathrm{\Gamma }(b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{a},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

A&S Ref:

6.5.37

Permalink:

http://dlmf.nist.gov/8.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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8.14.6 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}{\mathrm{e}}^{-sx}\mathrm{\Gamma }(b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{a{(1+s)}^{a+b}}F(1,a+b;1+a;s/(1+s)),$$ $\mathrm{\Re }s>-1$, $\mathrm{\Re }\left(a+b\right)>0$, $\mathrm{\Re }a>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/8.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.14 and Ch.8

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6: 27.4 Euler Products and Dirichlet Series

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27.4.3 $$\zeta \left(s\right)=\sum _{n=1}^{\mathrm{\infty }}{n}^{-s}=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

A&S Ref:

23.2.1 and 23.2.2

Permalink:

http://dlmf.nist.gov/27.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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27.4.5 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)n^{-s}=\frac{1}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mu \left(n\right)$: Möbius function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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27.4.6 $$\sum _{n=1}^{\mathrm{\infty }}\varphi \left(n\right)n^{-s}=\frac{\zeta \left(s-1\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>2$,

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.2 I.B

Permalink:

http://dlmf.nist.gov/27.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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27.4.7 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)n^{-s}=\frac{\zeta \left(2s\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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27.4.11 $$\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)n^{-s}=\zeta \left(s\right)\zeta \left(s-\alpha \right),$$ $\mathrm{\Re }s>\max (1,1+\mathrm{\Re }\alpha )$,

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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7: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. … Magnify

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8: 31.3 Basic Solutions

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31.3.2 $$a\gamma c_1-q{c}_0=0,$$

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Symbols:

$\gamma$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter and $c_j$: coefficients

Permalink:

http://dlmf.nist.gov/31.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(i), §31.3 and Ch.31

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31.3.5 $$z^{1-\gamma }H\mathrm{\ell }(a,(a\delta +ϵ)(1-\gamma )+q;\alpha +1-\gamma ,\beta +1-\gamma ,2-\gamma ,\delta ;z).$$

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Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.3(iii), §31.7(ii)

Permalink:

http://dlmf.nist.gov/31.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(i), §31.3 and Ch.31

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31.3.6 $$H\mathrm{\ell }(1-a,\alpha \beta -q;\alpha ,\beta ,\delta ,\gamma ;1-z),$$

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Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

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31.3.7 $${(1-z)}^{1-\delta }H\mathrm{\ell }(1-a,((1-a)\gamma +ϵ)(1-\delta )+\alpha \beta -q;\alpha +1-\delta ,\beta +1-\delta ,2-\delta ,\gamma ;1-z).$$

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Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

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31.3.8 $$H\mathrm{\ell }(\frac{a}{a-1},\frac{\alpha \beta a-q}{a-1};\alpha ,\beta ,ϵ,\delta ;\frac{a-z}{a-1}),$$

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Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

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9: 37.17 Hermite Polynomials on ${ℝ}^d$

§37.17 Hermite Polynomials on ${ℝ}^d$

… ►which is normalized such that $⟨1_{{ℝ}^d},1_{{ℝ}^d}⟩=1$. … ►The spaces ${𝒱}_n({ℝ}^d)$ are eigenspaces of a second order partial differential operator: … … ►The Poisson kernel (37.13.6) of ${𝒱}_n({ℝ}^d)$ is given explicitly by the Mehler formula …

10: 4.15 Graphics

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§4.15(i) Real Arguments

… ►Figure 4.15.7 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the real axis from $-\mathrm{\infty }$ to $-1$ and $1$ to $\mathrm{\infty }$, where $w=\sin z$ and $z=\arcsin w$ (principal value). …Lines parallel to the real axis in the $z$-plane map onto ellipses in the $w$-plane with foci at $w=\pm 1$, and lines parallel to the imaginary axis in the $z$-plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\mathrm{\infty })$. … ►

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Figure 4.15.13: $\mathrm{arccsc}\left(x+\mathrm{i}y\right)$ (principal value). There is a branch cut along the real axis from $-1$ to $1$. Magnify 3D Help

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