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11: 28.14 Fourier Series

… ►

28.14.1 $${\mathrm{me}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q){\mathrm{e}}^{\mathrm{i}(\nu +2m)z},$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $q=h^2$: parameter, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

A&S Ref:

20.3.8 (in slightly different form)

Referenced by:

§28.22(ii)

Permalink:

http://dlmf.nist.gov/28.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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12: 7.9 Continued Fractions

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7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

ⓘ

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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13: 28.29 Definitions and Basic Properties

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28.29.7 $$w(z+\pi )={\mathrm{e}}^{\pi \mathrm{i}\nu }w(z),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: complex parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

… ►It has the form ►

28.29.10 $$F_{\nu }(z)={\mathrm{e}}^{\mathrm{i}\nu z}{P}_{\nu }(z),$$

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Defines:

$F_{\nu }(z)$: Floquet solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.29.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

…

14: 27.10 Periodic Number-Theoretic Functions

… ►Every function periodic (mod $k$) can be expressed as a finite Fourier series of the form ►

27.10.2 $$f(n)=\sum _{m=1}^{k}g(m){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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27.10.3 $$g(m)=\frac{1}{k}\sum _{n=1}^{k}f(n){\mathrm{e}}^{-2\pi \mathrm{i}mn/k}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

… ►

27.10.4 $$c_k\left(n\right)=\sum _{m=1}^k{\chi }_1\left(m\right){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

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Defines:

$c_k\left(n\right)$: Ramanujan’s sum

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

… ►The finite Fourier expansion of a primitive Dirichlet character $\chi (modk)$ has the form …

15: 2.6 Distributional Methods

… ►This leads to integrals of the form … ►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form … ►The replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … ►It is easily seen that $K^{+}$ forms a commutative, associative linear algebra. … ►On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form …

16: 7.7 Integral Representations

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7.7.1 $$\mathrm{erfc}z=\frac{2}{\pi }{\mathrm{e}}^{-z^2}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-z^2{t}^2}}{t^2+1}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

ⓘ

Symbols:

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

A&S Ref:

7.4.11 (in different form)

Referenced by:

§7.11, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.6 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-(a{t}^2+2bt+c)}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{(b^2-ac)/a}\mathrm{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

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

A&S Ref:

7.4.32 (in different form)

Referenced by:

§7.14(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

►

7.7.7 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{4a}\left({\mathrm{e}}^{2ab}\mathrm{erfc}\left(ax+(b/x)\right)+{\mathrm{e}}^{-2ab}\mathrm{erfc}\left(ax-(b/x)\right)\right),$$ $x>0$, .

ⓘ

Symbols:

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

A&S Ref:

7.4.33 (in different form)

Referenced by:

§7.7(i), §7.8

Permalink:

http://dlmf.nist.gov/7.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

►

7.7.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{2a}{\mathrm{e}}^{-2ab},$$ , .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{ph}$: phase

A&S Ref:

7.4.3 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

►

7.7.9 $${\int }_0^x\mathrm{erf}tdt=x\mathrm{erf}x+\frac{1}{\sqrt{\pi }}\left({\mathrm{e}}^{-x^2}-1\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

A&S Ref:

7.4.35 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

…

17: 27.14 Unrestricted Partitions

… ►

27.14.6 $$p\left(n\right)=\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\left(p\left(n-\omega (k)\right)+p\left(n-\omega (-k)\right)\right)=p\left(n-1\right)+p\left(n-2\right)-p\left(n-5\right)-p\left(n-7\right)+\mathrm{\cdots },$$

ⓘ

Symbols:

$p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer, $n$: positive integer and $\omega (k)$: pentagonal numbers

A&S Ref:

24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►Logarithmic differentiation of the generating function $1/f\left(x\right)$ leads to another recursion: ►

27.14.7 $$np\left(n\right)=\sum _{k=1}^n{\sigma }_1\left(k\right)p\left(n-k\right),$$

ⓘ

Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer and $n$: positive integer

A&S Ref:

24.2.1 II.A (in slightly different form) 24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20, Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/27.14.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

The erroneous ${\sigma }_1\left(n\right)$ in
$np\left(n\right)={\sum }_{k=1}^n{\sigma }_1\left(n\right)p\left(n-k\right)$
was replaced.

Reported 2015-08-17 by Howard Cohl and Hannah Cohen

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

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27.14.8 $$p\left(n\right)\sim \frac{{\mathrm{e}}^{K\sqrt{n}}}{4n\sqrt{3}},$$

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Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $K$: change of variable

A&S Ref:

24.2.1 III

Permalink:

http://dlmf.nist.gov/27.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iii), §27.14 and Ch.27

… ►

27.14.12 $$\eta \left(\tau \right)={\mathrm{e}}^{\pi \mathrm{i}\tau /12}\prod _{n=1}^{\mathrm{\infty }}(1-{\mathrm{e}}^{2\pi \mathrm{i}n\tau }),$$ $\mathrm{\Im }\tau >0$.

ⓘ

Defines:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $n$: positive integer

Referenced by:

§23.15(ii), §27.14(vi)

Permalink:

http://dlmf.nist.gov/27.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iv), §27.14 and Ch.27

…

18: 13.6 Relations to Other Functions

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13.6.6 $$U(a,a,z)=z^{1-a}U(1,2-a,z)=z^{1-a}{\mathrm{e}}^z{E}_a\left(z\right)={\mathrm{e}}^z\mathrm{\Gamma }(1-a,z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.28 (in different form)

Permalink:

http://dlmf.nist.gov/13.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

…

19: 26.10 Integer Partitions: Other Restrictions

… ►

§26.10(v) Limiting Form

►

26.10.16 $$p(𝒟,n)\sim \frac{{\mathrm{e}}^{\pi \sqrt{n/3}}}{{(768n^3)}^{1/4}},$$ $n\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ and $n$: nonnegative integer

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(v), §26.10 and Ch.26

… ►

26.10.18

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $h$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $A_k(n)$: coefficient and $f(h,k)$: function

Permalink:

http://dlmf.nist.gov/26.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(vi), §26.10 and Ch.26

…

20: 27.2 Functions

… ►An equivalent form states that the $n$th prime $p_n$ (when the primes are listed in increasing order) is asymptotic to $n\ln n$ as $n\to \mathrm{\infty }$: …