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21: 33.17 Recurrence Relations and Derivatives

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33.17.1 $$(\mathrm{\ell }+1)rf(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)f(ϵ,\mathrm{\ell };r)+\mathrm{\ell }\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r)=0,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

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33.17.2 $$(\mathrm{\ell }+1)\left(1+{\mathrm{\ell }}^2ϵ\right)rh(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)h(ϵ,\mathrm{\ell };r)+\mathrm{\ell }rh(ϵ,\mathrm{\ell }+1;r)=0,$$

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Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

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33.17.3 $$(\mathrm{\ell }+1)r{f}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)f(ϵ,\mathrm{\ell };r)-\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r),$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

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33.17.4 $$(\mathrm{\ell }+1)r{h}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)h(ϵ,\mathrm{\ell };r)-rh(ϵ,\mathrm{\ell }+1;r).$$

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Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

22: 6.8 Inequalities

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6.8.1

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

5.1.20

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

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6.8.2

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

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6.8.3

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

23: 4.3 Graphics

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§4.3(i) Real Arguments

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Figure 4.3.1: $\ln x$ and ${\mathrm{e}}^x$. … Magnify

… ►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). …Lines parallel to the real axis in the $z$-plane map onto rays in the $w$-plane, and lines parallel to the imaginary axis in the $z$-plane map onto circles centered at the origin in the $w$-plane. In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\mathrm{\infty })$. …

24: 13.13 Addition and Multiplication Theorems

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13.13.1 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{y}^n}{{\left(b\right)}_{n}n!}M(a+n,b+n,x),$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

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13.13.3 $${\left(\frac{x}{x+y}\right)}^a\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{y}^n}{n!{(x+y)}^n}M(a+n,b,x),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

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13.13.5 $${\mathrm{e}}^y{\left(\frac{x}{x+y}\right)}^{b-a}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(b-a\right)}_n{y}^n}{n!{(x+y)}^n}M(a-n,b,x),$$ $\mathrm{\Re }\left((y+x)/x\right)>\frac{1}{2}$,

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

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13.13.10 $${\mathrm{e}}^y\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!}U(a,b+n,x),$$ ,

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

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13.13.11 $${\mathrm{e}}^y{\left(\frac{x}{x+y}\right)}^{b-a}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!{(x+y)}^n}U(a-n,b,x),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

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25: 13.26 Addition and Multiplication Theorems

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13.26.2 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{y}{\sqrt{x}}\right)}^n{M}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.3 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n{y}^n}{n!{(x+y)}^n}{M}_{\kappa -n,\mu }\left(x\right),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§13.26(i)

Permalink:

http://dlmf.nist.gov/13.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.5 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{-y}{\sqrt{x}}\right)}^n{M}_{\kappa +\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.6 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x}{x+y}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n{y}^n}{n!{(x+y)}^n}{M}_{\kappa +n,\mu }\left(x\right),$$ $\mathrm{\Re }\left((y+x)/x\right)>\frac{1}{2}$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§13.26(i)

Permalink:

http://dlmf.nist.gov/13.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.12 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x}{x+y}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\left(\frac{-y}{x+y}\right)}^n{W}_{\kappa +n,\mu }\left(x\right),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$.

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(ii), §13.26 and Ch.13

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26: 4.40 Integrals

… ►Throughout this section the variables are assumed to be real. … ►

4.40.1 $$\int \sinh xdx=\cosh x,$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.77

Permalink:

http://dlmf.nist.gov/4.40.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.2 $$\int \cosh xdx=\sinh x,$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.78

Permalink:

http://dlmf.nist.gov/4.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.3 $$\int \tanh xdx=\ln \left(\cosh x\right).$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.79

Permalink:

http://dlmf.nist.gov/4.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.5 $$\int \mathrm{sech}xdx=\mathrm{gd}\left(x\right).$$

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Symbols:

$\mathrm{gd}x$: Gudermannian function, $dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.81

Permalink:

http://dlmf.nist.gov/4.40.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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27: 7.8 Inequalities

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7.8.2 $x\ge 0$,

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real variable

A&S Ref:

7.1.13

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

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7.8.3 $x\ge 0$,

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

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7.8.4 $x>-\frac{1}{2}\sqrt{2}$,

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

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7.8.6 $a,x>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E6

Encodings:

pMML, png, TeX

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: 14.4 Graphics

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29: 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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30: 33.1 Special Notation

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$k,\mathrm{\ell }$	nonnegative integers.
$r,x$	real variables.
$\rho$	nonnegative real variable.
…

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