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21: 7.17 Inverse Error Functions

… ►

§7.17(ii) Power Series

… ►

7.17.2 $$\mathrm{inverf}x=t+\frac{1}{3}{t}^3+\frac{7}{30}{t}^5+\frac{127}{630}{t}^7+\mathrm{\cdots }=\sum _{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1},$$ ,

ⓘ

Symbols:

$\mathrm{inverf}x$: inverse error function and $x$: real variable

Referenced by:

Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The equality “$={\sum }_{m=0}^{\mathrm{\infty }}{a}_m{t}^{2m+1}$” was added to this equation.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

… ►

7.17.2_5 $$a_{m+1}=\frac{1}{2m+3}\sum _{n=0}^m\frac{2n+1}{m-n+1}{a}_n{a}_{m-n},$$ $m=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$n$: nonnegative integer

Referenced by:

§7.17(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/7.17.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §7.17(ii), §7.17 and Ch.7

…

22: 2.1 Definitions and Elementary Properties

… ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, … ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. …

23: 20.6 Power Series

§20.6 Power Series

… ►

20.6.7 $${\alpha }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m-\frac{1}{2}+n\tau )}^{-2j},$$

ⓘ

Defines:

${\alpha }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

►

20.6.8 $${\beta }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m-\frac{1}{2}+(n-\frac{1}{2})\tau )}^{-2j},$$

ⓘ

Defines:

${\beta }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

►

20.6.9 $${\gamma }_{2j}(\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{(m+(n-\frac{1}{2})\tau )}^{-2j},$$

ⓘ

Defines:

${\gamma }_{2j}(\tau )$: coefficient (locally)

Symbols:

$m$: integer, $n$: integer and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

… ►For further information on ${\delta }_{2j}$ see §23.9: since the double sums in (20.6.6) and (23.9.1) are the same, we have ${\delta }_{2n}=c_n/(2n-1)$ when $n\ge 2$.

24: 5.19 Mathematical Applications

… ►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. … ►

$S={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{a}_k,$

… ►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …

25: 6.6 Power Series

§6.6 Power Series

►

6.6.1 $$\mathrm{Ei}\left(x\right)=\gamma +\ln x+\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n!n},$$ $x>0$.

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

5.1.10

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

►

6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.1.11

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

… ►

6.6.4 $$\mathrm{Ein}\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{z}^n}{n!n},$$

ⓘ

Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

►

6.6.5 $$\mathrm{Si}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{(2n+1)!(2n+1)},$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{Si}\left(z\right)$: sine integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.14

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

…

26: 10.53 Power Series

§10.53 Power Series

►

10.53.1 $${𝗃}_n\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(-\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.1.2

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.2 $${𝗒}_n\left(z\right)=-\frac{1}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(\frac{1}{2}{z}^2)}^k}{k!}+\frac{{(-1)}^{n+1}}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(-\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.1.3

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.3 $${𝗂}_n^{(1)}\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.5

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.4 $${𝗂}_n^{(2)}\left(z\right)=\frac{{(-1)}^n}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(-\frac{1}{2}{z}^2)}^k}{k!}+\frac{1}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

ⓘ

Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.6

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

…

27: 10.69 Uniform Asymptotic Expansions for Large Order

… ►

10.69.2 $${\mathrm{ber}}_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{bei}}_{\nu }\left(\nu x\right)\sim \frac{{\mathrm{e}}^{\nu \xi }}{{(2\pi \nu \xi )}^{1/2}}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.37

Referenced by:

§10.69

Permalink:

http://dlmf.nist.gov/10.69.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

►

10.69.3 $${\ker }_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }\left(\nu x\right)\sim {\mathrm{e}}^{-\nu \xi }{\left(\frac{\pi }{2\nu \xi }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.38

Permalink:

http://dlmf.nist.gov/10.69.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

►

10.69.4 $${\mathrm{ber}}_{\nu }^{\prime }\left(\nu x\right)+\mathrm{i}{\mathrm{bei}}_{\nu }^{\prime }\left(\nu x\right)\sim \frac{{\mathrm{e}}^{\nu \xi }}{x}{\left(\frac{\xi }{2\pi \nu }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{V_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $V_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.39

Referenced by:

§10.69

Permalink:

http://dlmf.nist.gov/10.69.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

►

10.69.5 $${\ker }_{\nu }^{\prime }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }^{\prime }\left(\nu x\right)\sim -\frac{{\mathrm{e}}^{-\nu \xi }}{x}{\left(\frac{\pi \xi }{2\nu }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $V_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.40

Permalink:

http://dlmf.nist.gov/10.69.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

… ►All fractional powers take their principal values. …

28: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. Cancellation errors increase with increases in $\rho$ and $|r|$, and may be estimated by comparing the final sum of the series with the largest partial sum. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

29: 36.8 Convergent Series Expansions

… ►

${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\exp \left(\mathrm{i}{\displaystyle \frac{\pi (2n+1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{2n+1}{K+2}}\right)a_{2n}(𝐱),$ $K$ even,

►

${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\mathrm{i}}^n\cos \left({\displaystyle \frac{\pi (n(K+1)-1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{n+1}{K+2}}\right)a_n(𝐱),$ $K$ odd,

… ►

$a_{n+1}(𝐱)={\displaystyle \frac{\mathrm{i}}{n+1}}{\displaystyle \sum _{p=0}^{\min (n,K-1)}}(p+1)x_{p+1}{a}_{n-p}(𝐱),$ $n=0,1,2,\mathrm{\dots }$.

►For multinomial power series for ${\mathrm{\Psi }}_K\left(𝐱\right)$, see Connor and Curtis (1982). ►

36.8.3 $$\frac{3^{2/3}}{4{\pi }^2}{\mathrm{\Psi }}^{(\mathrm{H})}\left(3^{1/3}𝐱\right)=\mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=0}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)c_n(y)}{n!}+\mathrm{Ai}\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)d_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)c_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=1}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)d_n(y)}{n!},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter, $c_n(t)$: coefficients and $d_n(t)$: coefficients

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

…

30: 7.6 Series Expansions

… ►

§7.6(i) Power Series

►

7.6.1 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{n!(2n+1)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.5

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E1

Encodings:

pMML, png, TeX

See also:

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

►

7.6.2 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}\sum _{n=0}^{\mathrm{\infty }}\frac{2^n{z}^{2n+1}}{1\cdot 3\mathrm{\cdots }(2n+1)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.6

Referenced by:

§7.6(i), §7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

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

… ►

7.6.7 $$S\left(z\right)=-\cos \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n+1}}{1\cdot 3\mathrm{\cdots }(4n+3)}{z}^{4n+3}+\sin \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n}}{1\cdot 3\mathrm{\cdots }(4n+1)}{z}^{4n+1}.$$

ⓘ

Symbols:

$S\left(z\right)$: 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:

7.3.14

Referenced by:

§11.10(vi), §7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E7

Encodings:

pMML, png, TeX

See also:

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

…