Euler–Maclaurin

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11: 31.3 Basic Solutions

… ►If the other exponent is not a positive integer, that is, if $\gamma \ne 0,-1,-2,\mathrm{\dots }$, then from §2.7(i) it follows that $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ exists, is analytic in the disk , and has the Maclaurin expansion …

12: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►

8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin \left(\pi a\right)}Q(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)-\mathrm{i}T(a,\eta ),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $T(a,\eta )$: function

Referenced by:

Erratum (V1.0.16) for Equation (8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E5

Encodings:

pMML, png, TeX

Change (effective with 1.0.16):

To be consistent with the notation used in (8.12.16), the original equation,

$$\mathrm{\Gamma }\left(a+1\right)\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\pi \mathrm{i}}\mathrm{\Gamma }(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\mp \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)+\mathrm{i}T(a,\eta ),$$

was rewritten.

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.6 $$z^{-a}{\gamma }^{\ast }(-a,-z)=\cos \left(\pi a\right)-2\sin \left(\pi a\right)\left(\frac{{\mathrm{e}}^{\frac{1}{2}a{\eta }^2}}{\sqrt{\pi }}F\left(\eta \sqrt{a/2}\right)+T(a,\eta )\right),$$

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $T(a,\eta )$: function

Permalink:

http://dlmf.nist.gov/8.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►where $g_k$, $k=0,1,2,\mathrm{\dots }$, are the coefficients that appear in the asymptotic expansion (5.11.3) of $\mathrm{\Gamma }\left(z\right)$. The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at $\eta =0$, and the Maclaurin series expansion of $c_k(\eta )$ is given by … ►Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a). …

13: 9.12 Scorer Functions

… ►

9.12.6 $$\mathrm{Gi}\left(0\right)=\frac{1}{2}\mathrm{Hi}\left(0\right)=\frac{1}{3}\mathrm{Bi}\left(0\right)=1/\left(3^{7/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)\right)=\mathrm{0.20497\hspace{0.33em}55424}\mathrm{\dots },$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function) and $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Source:

Olver (1997b, p. 431)

Permalink:

http://dlmf.nist.gov/9.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(iii), §9.12 and Ch.9

… ►

§9.12(vi) Maclaurin Series

… ►

9.12.24 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{2{\pi }^2\mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)\mathrm{\Gamma }\left(-t\right){(3^{1/3}{\mathrm{e}}^{\pi \mathrm{i}}z)}^{t}dt,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Keywords:

Mellin transform

Source:

Exton (1983, (2.1), p. 100)

Permalink:

http://dlmf.nist.gov/9.12.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12(vii), §9.12 and Ch.9

►where the integration contour separates the poles of $\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$. … ►where $\gamma$ is Euler’s constant (§5.2(ii)). …

14: 15.2 Definitions and Analytical Properties

… ►

§15.2(i) Gauss Series

… ►

15.2.1 $$F(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}{z}^2+\mathrm{\cdots }=\frac{\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+s\right)\mathrm{\Gamma }\left(b+s\right)}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$

ⓘ

Defines:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

A&S Ref:

15.1.1

Referenced by:

§15.19(i), §15.4(iii), §18.30(v), §18.35(i), §18.35(ii), §19.19

Permalink:

http://dlmf.nist.gov/15.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

… ►

15.2.2 $$𝐅(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$ ,

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Defines:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.12(i), §16.2(v)

Permalink:

http://dlmf.nist.gov/15.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

… ►

15.2.3 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};x+\mathrm{i}0)-𝐅(\genfrac{}{}{0pt}{}{a,b}{c};x-\mathrm{i}0)=\frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{(x-1)}^{c-a-b}𝐅(\genfrac{}{}{0pt}{}{c-a,c-b}{c-a-b+1};1-x),$$ $x>1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.2(i)

Permalink:

http://dlmf.nist.gov/15.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

… ►In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F(-m,b;-m;z)$ is equal to the first $m+1$ terms of the Maclaurin series for ${(1-z)}^{-b}$.

15: 4.19 Maclaurin Series and Laurent Series

§4.19 Maclaurin Series and Laurent Series

… ►In (4.19.3)–(4.19.9), $B_n$ are the Bernoulli numbers and $E_n$ are the Euler numbers (§§24.2(i)–24.2(ii)). … ►

4.19.5 $$\sec z=1+\frac{z^2}{2}+\frac{5}{24}{z}^4+\frac{61}{720}{z}^6+\mathrm{\cdots }+\frac{{(-1)}^n{E}_{2n}}{(2n)!}{z}^{2n}+\mathrm{\cdots },$$ ,

ⓘ

Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sec z$: secant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.69

Referenced by:

§24.1, §24.19(i), §24.2(ii)

Permalink:

http://dlmf.nist.gov/4.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

…

16: 18.17 Integrals

… ►

18.17.14 $$\frac{x^{\alpha +\mu }{L}_n^{(\alpha +\mu )}\left(x\right)}{\mathrm{\Gamma }\left(\alpha +\mu +n+1\right)}={\int }_0^x\frac{y^{\alpha }{L}_n^{(\alpha )}\left(y\right)}{\mathrm{\Gamma }\left(\alpha +n+1\right)}\frac{{(x-y)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$, $x>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.13.13

Referenced by:

§18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

… ►For the beta function $\mathrm{B}(a,b)$ see §5.12, and for the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. … ►

18.17.16_5 $${\int }_{-1}^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_n^{(\lambda )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx=\frac{2\pi {\mathrm{i}}^n\mathrm{\Gamma }\left(n+2\lambda \right)J_{n+\lambda }\left(y\right)}{n!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v), Erratum (V1.2.0) Section 18.17

Permalink:

http://dlmf.nist.gov/18.17.E16_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

►

18.17.17 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n}^{(\lambda )}\left(x\right)\cos \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda \right)J_{\lambda +2n}\left(y\right)}{(2n)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

… ►

18.17.37 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_n^{(\lambda )}\left(x\right)x^{z-1}dx=\frac{\pi {\mathrm{\hspace{0.17em}2}}^{1-2\lambda -z}\mathrm{\Gamma }\left(n+2\lambda \right)\mathrm{\Gamma }\left(z\right)}{n!\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}n+\lambda +\frac{1}{2}z\right)\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n\right)},$$ $\mathrm{\Re }z>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

…

17: 13.9 Zeros

… ►

13.9.9 $$z=\pm (2n+a)\pi \mathrm{i}+\ln \left(-\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(b-a\right)}{\left(\pm 2n\pi \mathrm{i}\right)}^{b-2a}\right)+O\left(n^{-1}\ln n\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(i)

Permalink:

http://dlmf.nist.gov/13.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(i), §13.9 and Ch.13

… ►The same results apply for the $n$th partial sums of the Maclaurin series (13.2.2) of $M(a,b,z)$. … ►

13.9.11 $$T(a,b)=\lfloor -a\rfloor +1,$$ , $\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\lfloor x\rfloor$: floor of $x$ and $T(a,b)$: number of zeros

Permalink:

http://dlmf.nist.gov/13.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

►

13.9.12 $$T(a,b)=\lfloor -a\rfloor ,$$ , ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\lfloor x\rfloor$: floor of $x$ and $T(a,b)$: number of zeros

Permalink:

http://dlmf.nist.gov/13.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

…

18: Errata

… ►

In previous versions of the DLMF, in §8.18(ii), the notation $\tilde{\mathrm{\Gamma }}(z)$ was used for the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$. Now in §8.18(ii), we adopt the notation which was introduced in Version 1.1.7 (October 15, 2022) and correspondingly, Equation (8.18.13) has been removed. In place of Equation (8.18.13), it is now mentioned to see (5.11.3).

… ►

§4.13 has been enlarged. The Lambert $W$-function is multi-valued and we use the notation $W_k\left(x\right)$, $k\in \mathbb{Z}$, for the branches. The original two solutions are identified via $\mathrm{Wp}\left(x\right)=W_0\left(x\right)$ and $\mathrm{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

Other changes are the introduction of the Wright $\omega$-function and tree $T$-function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{d^{n}W}{{dz}^n}$, additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$-functions in the end of the section.

… ►

It is now mentioned that (25.2.5), defines the Stieltjes constants ${\gamma }_n$. Consequently, ${\gamma }_n$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

… ►

The table of extrema for the Euler gamma function $\mathrm{\Gamma }$ had several entries in the $x_n$ column that were wrong in the last 2 or 3 digits. These have been corrected and 10 extra decimal places have been included.

$n$	$x_n$	$\mathrm{\Gamma }\left(x_n\right)$
0	$\mathrm{1.46163\hspace{0.33em}21449\hspace{0.33em}68362\hspace{0.33em}34126}$	$\mathrm{0.88560\hspace{0.33em}31944\hspace{0.33em}10888\hspace{0.33em}70028}$
1	$-\mathrm{0.50408\hspace{0.33em}30082\hspace{0.33em}64455\hspace{0.33em}40926}$	$-\mathrm{3.54464\hspace{0.33em}36111\hspace{0.33em}55005\hspace{0.33em}08912}$
2	$-\mathrm{1.57349\hspace{0.33em}84731\hspace{0.33em}62390\hspace{0.33em}45878}$	$\mathrm{2.30240\hspace{0.33em}72583\hspace{0.33em}39680\hspace{0.33em}13582}$
3	$-\mathrm{2.61072\hspace{0.33em}08684\hspace{0.33em}44144\hspace{0.33em}65000}$	$-\mathrm{0.88813\hspace{0.33em}63584\hspace{0.33em}01241\hspace{0.33em}92010}$
4	$-\mathrm{3.63529\hspace{0.33em}33664\hspace{0.33em}36901\hspace{0.33em}09784}$	$\mathrm{0.24512\hspace{0.33em}75398\hspace{0.33em}34366\hspace{0.33em}25044}$
5	$-\mathrm{4.65323\hspace{0.33em}77617\hspace{0.33em}43142\hspace{0.33em}44171}$	$-\mathrm{0.05277\hspace{0.33em}96395\hspace{0.33em}87319\hspace{0.33em}40076}$
6	$-\mathrm{5.66716\hspace{0.33em}24415\hspace{0.33em}56885\hspace{0.33em}53585}$	$\mathrm{0.00932\hspace{0.33em}45944\hspace{0.33em}82614\hspace{0.33em}85052}$
7	$-\mathrm{6.67841\hspace{0.33em}82130\hspace{0.33em}73426\hspace{0.33em}74283}$	$-\mathrm{0.00139\hspace{0.33em}73966\hspace{0.33em}08949\hspace{0.33em}76730}$
8	$-\mathrm{7.68778\hspace{0.33em}83250\hspace{0.33em}31626\hspace{0.33em}03744}$	$\mathrm{0.00018\hspace{0.33em}18784\hspace{0.33em}44909\hspace{0.33em}40419}$
9	$-\mathrm{8.69576\hspace{0.33em}41638\hspace{0.33em}16401\hspace{0.33em}26649}$	$-\mathrm{0.00002\hspace{0.33em}09252\hspace{0.33em}90446\hspace{0.33em}52667}$
10	$-\mathrm{9.70267\hspace{0.33em}25400\hspace{0.33em}01863\hspace{0.33em}73608}$	$\mathrm{0.00000\hspace{0.33em}21574\hspace{0.33em}16104\hspace{0.33em}52285}$

Reported 2018-02-17 by David Smith.

… ►

17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}$$

Originally the differential was identified incorrectly as $d_{q}t$; the correct differential is $dt$.

Reported 2011-04-08.

…

19: 11.10 Anger–Weber Functions

… ►

§11.10(iii) Maclaurin Series

… ►

11.10.10 $$S_1(\nu ,z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(\frac{1}{2}z)}^{2k}}{\mathrm{\Gamma }\left(k+\frac{1}{2}\nu +1\right)\mathrm{\Gamma }\left(k-\frac{1}{2}\nu +1\right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Referenced by:

(11.11.5), (11.11.6)

Permalink:

http://dlmf.nist.gov/11.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(iii), §11.10 and Ch.11

►

11.10.11 $$S_2(\nu ,z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(\frac{1}{2}z)}^{2k+1}}{\mathrm{\Gamma }\left(k+\frac{1}{2}\nu +\frac{3}{2}\right)\mathrm{\Gamma }\left(k-\frac{1}{2}\nu +\frac{3}{2}\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Referenced by:

§11.10(vi), (11.11.5), (11.11.6)

Permalink:

http://dlmf.nist.gov/11.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(iii), §11.10 and Ch.11

… ►

11.10.22 $${𝐄}_n\left(z\right)=-{𝐇}_n\left(z\right)+\frac{1}{\pi }\sum _{k=0}^{m_1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n+\frac{1}{2}-k\right)}{(\frac{1}{2}z)}^{n-2k-1},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: integer order and $k$: nonnegative integer

A&S Ref:

12.3.6 (with upper limit corrected in later printings.)

Referenced by:

§11.10(vi), §11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

… ►

11.10.23 $${𝐄}_{-n}\left(z\right)=-{𝐇}_{-n}\left(z\right)+\frac{{(-1)}^{n+1}}{\pi }\sum _{k=0}^{m_2}\frac{\mathrm{\Gamma }\left(n-k-\frac{1}{2}\right)}{\mathrm{\Gamma }\left(k+\frac{3}{2}\right)}{(\frac{1}{2}z)}^{-n+2k+1},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: integer order and $k$: nonnegative integer

A&S Ref:

12.3.7 (with upper limit corrected.)

Referenced by:

§11.10(vi), Ch.11

Permalink:

http://dlmf.nist.gov/11.10.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

…

20: 13.2 Definitions and Basic Properties

… ►The first two standard solutions are: … ►

13.2.4 $$M(a,b,z)=\mathrm{\Gamma }\left(b\right)𝐌(a,b,z).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.2(ii)

Permalink:

http://dlmf.nist.gov/13.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

… ►

13.2.18 $$U(a,b,z)=\frac{\mathrm{\Gamma }\left(b-1\right)}{\mathrm{\Gamma }\left(a\right)}{z}^{1-b}+\frac{\mathrm{\Gamma }\left(1-b\right)}{\mathrm{\Gamma }\left(a-b+1\right)}+O\left(z^{2-\mathrm{\Re }b}\right),$$ , $b\ne 1$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iii), §13.2 and Ch.13

►

13.2.19 $$U(a,1,z)=-\frac{1}{\mathrm{\Gamma }\left(a\right)}\left(\ln z+\psi \left(a\right)+2\gamma \right)+O\left(z\ln z\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $U(a,b,z)$: Kummer confluent hypergeometric function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

13.5.9 (as corrected in later editions)

Referenced by:

§13.2(iii)

Permalink:

http://dlmf.nist.gov/13.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iii), §13.2 and Ch.13

… ►

13.2.42 $$U(a,b,z)=\frac{\mathrm{\Gamma }\left(1-b\right)}{\mathrm{\Gamma }\left(a-b+1\right)}M(a,b,z)+\frac{\mathrm{\Gamma }\left(b-1\right)}{\mathrm{\Gamma }\left(a\right)}{z}^{1-b}M(a-b+1,2-b,z).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.1.3 (in different form)

Referenced by:

§13.14(vii), §13.2(ii), §13.6(v), §13.9(ii), §33.13

Permalink:

http://dlmf.nist.gov/13.2.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13