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11: 25.14 Lerch’s Transcendent

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25.14.5 $$\mathrm{\Phi }(z,s,a)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1-z{\mathrm{e}}^{-x}}dx,$$ $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $z=1$; $\mathrm{\Re }s>0$, $\mathrm{\Re }a>0$ if $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $[a,b)$: half-closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $\setminus$: set subtraction, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.3), p. 27)

Referenced by:

Erratum (V1.1.4) for Equation (25.14.5)

Permalink:

http://dlmf.nist.gov/25.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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12: 26.9 Integer Partitions: Restricted Number and Part Size

… ►equivalently, partitions into at most $k$ parts either have exactly $k$ parts, in which case we can subtract one from each part, or they have strictly fewer than $k$ parts. …

13: 26.15 Permutations: Matrix Notation

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26.15.5 $$R(x,B)=xR(x,B\setminus [j,k])+R(x,B\setminus (j,k)).$$

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Symbols:

$[a,b]$: closed interval, $(a,b)$: open interval, $\setminus$: set subtraction, $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $B$: subset and $R(x,B)$: rook polynomial

Permalink:

http://dlmf.nist.gov/26.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

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14: 19.26 Addition Theorems

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19.26.14 $$(p-y)R_C(x,p)+(q-y)R_C(x,q)=(\eta -\xi )R_C(\xi ,\eta ),$$ $x\ge 0$, $y\ge 0$; $p,q\in ℝ\setminus \{0\}$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\in$: element of, $ℝ$: real line, $\setminus$: set subtraction, $x$: positive, $y$: positive, $\xi$: coordinate and $\eta$: coordinate

Referenced by:

§19.26(i)

Permalink:

http://dlmf.nist.gov/19.26.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

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19.26.17 $$\sqrt{\alpha }{R}_C(\beta ,\alpha +\beta )+\sqrt{\beta }{R}_C(\alpha ,\alpha +\beta )=\pi /2,$$ $\alpha ,\beta \in ℂ\setminus (-\mathrm{\infty },0)$, $\alpha +\beta >0$.

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b)$: open interval, $\setminus$: set subtraction and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.26(ii)

Permalink:

http://dlmf.nist.gov/19.26.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(ii), §19.26 and Ch.19

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15: 19.23 Integral Representations

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19.23.10 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^1{u}^{a-1}{(1-u)}^{a^{\prime }-1}\prod _{j=1}^n{(1-u+u{z}_j)}^{-b_j}du,$$ $a,a^{\prime }>0$; $a+a^{\prime }={\sum }_{j=1}^n{b}_j$; $z_j\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.19, §19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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16: 22.14 Integrals

… ►Corresponding results for the subsidiary functions follow by subtraction; compare (22.2.10).

17: 2.1 Definitions and Elementary Properties

… ►These include addition, subtraction, multiplication, and division. …

18: 3.2 Linear Algebra

… ►By repeatedly subtracting multiples of each row from the subsequent rows we obtain a matrix of the form …

19: 19.21 Connection Formulas

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19.21.1 $$R_F(0,z+1,z)R_D(0,z+1,1)+R_D(0,z+1,z)R_F(0,z+1,1)=3\pi /(2z),$$ $z\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval and $\setminus$: set subtraction

Referenced by:

§19.21(i), §19.21(i), §19.7(i)

Permalink:

http://dlmf.nist.gov/19.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(i), §19.21 and Ch.19

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20: 21.7 Riemann Surfaces

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21.7.12 $$T_1\ominus T_2=(T_1\cup T_2)\setminus (T_1\cap T_2).$$

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Symbols:

$\cap$: intersection, $\setminus$: set subtraction, $\cup$: union and $T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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