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1: 26.11 Integer Partitions: Compositions

… ► $c\left(n\right)$ denotes the number of compositions of $n$, and $c_m\left(n\right)$ is the number of compositions into exactly $m$ parts. $c(\in T,n)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\mathrm{\dots }\}$. The integer 0 is considered to have one composition consisting of no parts: … ►

26.11.2 $$c_m\left(0\right)={\delta }_{0,m},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts and $m$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

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26.11.3 $$c_m\left(n\right)=\left(\genfrac{}{}{0.0pt}{}{n-1}{m-1}\right),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

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2: 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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27.4.10 $$\sum _{n=1}^{\mathrm{\infty }}{d}_k\left(n\right)n^{-s}={(\zeta \left(s\right))}^k,$$ $\mathrm{\Re }s>1$,

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Symbols:

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

Referenced by:

§27.4

Permalink:

http://dlmf.nist.gov/27.4.E10

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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3: 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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4: 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, $\mathrm{e}$: 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, $\mathrm{e}$: 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, $\gamma (a,z)$: 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, $\mathrm{\Gamma }(a,z)$: 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, $\mathrm{e}$: 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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5: Sidebar 21.SB1: Periodic Surface Waves

… ►Part 2. …

6: 26.21 Tables

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …

7: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.4 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le 2\pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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28.25.5 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-2\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le \pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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8: 19.3 Graphics

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Figure 19.3.9: $\mathrm{\Re }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The real part is symmetric under reflection in the real axis. … Magnify 3D Help

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Figure 19.3.10: $\mathrm{\Im }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The imaginary part is 0 for , and is antisymmetric under reflection in the real axis. … Magnify 3D Help

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Figure 19.3.11: $\mathrm{\Re }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The real part is symmetric under reflection in the real axis. … Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The imaginary part is 0 for $k^2\le 1$ and is antisymmetric under reflection in the real axis. … Magnify 3D Help

9: 13.10 Integrals

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13.10.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,c,kt)dt=\mathrm{\Gamma }\left(b\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;c;k/z),$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>\max (\mathrm{\Re }k,0)$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential, $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,b,t)dt=z^{-b}{\left(1-\frac{1}{z}\right)}^{-a},$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>1$,

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Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{b-1}𝐌(a,c,t)dt=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)},$$ $\mathrm{\Re }\left(c-a\right)>\mathrm{\Re }b>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Permalink:

http://dlmf.nist.gov/13.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}U(a,c,t)dt=\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(b-c+1\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;a+b-c+1;1-\frac{1}{z}),$$ $\mathrm{\Re }b>\max (\mathrm{\Re }c-1,0)$, $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential, $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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13.10.10 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}𝐌(a,b,-t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-\lambda \right)},$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iii), §13.10 and Ch.13

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10: 7.14 Integrals

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7.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\left(bt\right)dt=\frac{1}{a}{\mathrm{e}}^{a^2/(4b^2)}\mathrm{erfc}\left(\frac{a}{2b}\right),$$ $\mathrm{\Re }a>0$, ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.17

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\sqrt{bt}dt=\frac{1}{a}\sqrt{\frac{b}{a+b}},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$,

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Symbols:

$dx$: differential, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.19

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{(a-b)t}\mathrm{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t}}\right)dt=\frac{{\mathrm{e}}^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+\sqrt{b})},$$ , $\mathrm{\Re }b>0$, $\mathrm{\Re }c\ge 0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.21

Referenced by:

§3.5(ix), §7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(t\right)dt=\frac{1}{a}\mathrm{g}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.28 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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