sums and integrals

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21: 7.6 Series Expansions

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7.6.4 $$C\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n}}{(2n)!(4n+1)}{z}^{4n+1},$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.11

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.5 $$C\left(z\right)=\cos \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n}}{1\cdot 3\mathrm{\cdots }(4n+1)}{z}^{4n+1}+\sin \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n+1}}{1\cdot 3\mathrm{\cdots }(4n+3)}{z}^{4n+3}.$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.12

Referenced by:

§11.10(vi), §7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.6 $$S\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n+1}}{(2n+1)!(4n+3)}{z}^{4n+3},$$

ⓘ

Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.13

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.10 $$C\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n}\left(\frac{1}{2}\pi z^2\right),$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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7.6.11 $$S\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n+1}\left(\frac{1}{2}\pi z^2\right).$$

ⓘ

Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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22: 19.5 Maclaurin and Related Expansions

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19.5.1 $$K\left(k\right)=\frac{\pi }{2}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}{k}^{2m}=\frac{\pi }{2}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $!$: factorial (as in $n!$), $m$: nonnegative integer and $k$: real or complex modulus

Source:

Carlson (1961b, (2.4))

Referenced by:

§19.36(iv), §19.5

Permalink:

http://dlmf.nist.gov/19.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.2 $$E\left(k\right)=\frac{\pi }{2}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}{k}^{2m}=\frac{\pi }{2}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2},\frac{1}{2}}{1};k^2),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $!$: factorial (as in $n!$), $m$: nonnegative integer and $k$: real or complex modulus

Source:

Carlson (1961b, (2.5))

Referenced by:

§19.36(iv)

Permalink:

http://dlmf.nist.gov/19.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.4 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{\pi }{2}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_n}{n!}\sum _{m=0}^n\frac{{\left(\frac{1}{2}\right)}_m}{m!}{k}^{2m}{\alpha }^{2n-2m}=\frac{\pi }{2}{F}_1(\frac{1}{2};\frac{1}{2},1;1;k^2,{\alpha }^2),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Source:

Carlson (1961b, (2.10))

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.8 $$K\left(k\right)=\frac{\pi }{2}{\left(1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\right)}^2,$$ ,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $q$: nome, $n$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.9 $$E\left(k\right)=K\left(k\right)+\frac{2{\pi }^2}{K\left(k\right)}\frac{{\sum }_{n=1}^{\mathrm{\infty }}{(-1)}^n{n}^2{q}^{n^2}}{1+2{\sum }_{n=1}^{\mathrm{\infty }}{(-1)}^n{q}^{n^2}},$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $q$: nome, $n$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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23: 36.2 Catastrophes and Canonical Integrals

§36.2 Catastrophes and Canonical Integrals

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§36.2(i) Definitions

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Canonical Integrals

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§36.2(iii) Symmetries

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24: 19.18 Derivatives and Differential Equations

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19.18.8 $$\sum _{j=1}^n{\partial }_j{R}_{-a}(𝐛;𝐳)=-a{R}_{-a-1}(𝐛;𝐳).$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\partial x$: partial differential of $x$ and $n$: nonnegative integer

Referenced by:

§19.18(i), §19.18(ii)

Permalink:

http://dlmf.nist.gov/19.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

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25: 6.12 Asymptotic Expansions

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6.12.5 $$\mathrm{f}\left(z\right)=\frac{1}{z}\sum _{m=0}^{n-1}{(-1)}^m\frac{(2m)!}{z^{2m}}+R_n^{(\mathrm{f})}(z),$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

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6.12.6 $$\mathrm{g}\left(z\right)=\frac{1}{z^2}\sum _{m=0}^{n-1}{(-1)}^m\frac{(2m+1)!}{z^{2m}}+R_n^{(\mathrm{g})}(z),$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{g})}(z)$: remainder term

Permalink:

http://dlmf.nist.gov/6.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

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26: 1.15 Summability Methods

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1.15.50 $${𝐼}^{\alpha }f(x)=\sum _{k=0}^{\mathrm{\infty }}\frac{k!}{\mathrm{\Gamma }\left(k+\alpha +1\right)}{a}_k{x}^{k+\alpha }.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: integer and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)

Permalink:

http://dlmf.nist.gov/1.15.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vi), §1.15 and Ch.1

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27: 19.22 Quadratic Transformations

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19.22.9 $$\frac{4}{\pi }{R}_G(0,a_0^2,g_0^2)=\frac{1}{M(a_0,g_0)}\left(a_0^2-\sum _{n=0}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right)=\frac{1}{M(a_0,g_0)}\left(a_1^2-\sum _{n=2}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right),$$

ⓘ

Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $a_n$: iterate and $g_n$: iterate

Permalink:

http://dlmf.nist.gov/19.22.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(ii), §19.22 and Ch.19

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19.22.10 $$R_D(0,g_0^2,a_0^2)=\frac{3\pi }{4M(a_0,g_0)a_0^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n,$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $Q_n$ , $a_n$: iterate and $g_n$: iterate

Referenced by:

§19.22(ii)

Permalink:

http://dlmf.nist.gov/19.22.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(ii), §19.22 and Ch.19

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19.22.12 $$R_J(0,g_0^2,a_0^2,p_0^2)=\frac{3\pi }{4M(a_0,g_0)p_0^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n,$$

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Symbols:

$R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $Q_n$ , $p_j$: coefficient, $a_n$: iterate and $g_n$: iterate

Referenced by:

§19.8(i)

Permalink:

http://dlmf.nist.gov/19.22.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(ii), §19.22 and Ch.19

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19.22.14 $$R_J(0,g_0^2,a_0^2,-q_0^2)=\frac{-3\pi }{4M(a_0,g_0)(q_0^2+a_0^2)}\left(2+\frac{a_0^2-g_0^2}{q_0^2+g_0^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n\right),$$

ⓘ

Symbols:

$R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $Q_n$ , $a_n$: iterate and $g_n$: iterate

Permalink:

http://dlmf.nist.gov/19.22.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(ii), §19.22 and Ch.19

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28: 19.20 Special Cases

§19.20 Special Cases

… ►The general lemniscatic case is … ►where $x,y,z$ may be permuted. … ►Define $c={\sum }_{j=1}^n{b}_j$. …

29: 19.23 Integral Representations

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19.23.9 $$R_{-a}(𝐛;𝐳)=\frac{4\mathrm{\Gamma }\left(b_1+b_2+b_3\right)}{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\Gamma }\left(b_3\right)}{\int }_0^{\pi /2}{\int }_0^{\pi /2}{\left(\sum _{j=1}^3{z}_j{l}_j^2\right)}^{-a}\prod _{j=1}^3{l}_j^{2b_j-1}\sin \theta d\theta d\varphi ,$$ $b_j>0$, $\mathrm{\Re }{z}_j>0$.

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $l$: nonnegative integer and $\varphi$: real or complex argument

Referenced by:

§19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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

ⓘ

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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30: 11.7 Integrals and Sums

§11.7 Integrals and Sums

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