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41: 9.17 Methods of Computation

… ►In the case of $\mathrm{Ai}\left(z\right)$, for example, this means that in the sectors we may integrate along outward rays from the origin with initial values obtained from §9.2(ii). …

42: 10.61 Definitions and Basic Properties

… ►In general, Kelvin functions have a branch point at $x=0$ and functions with arguments $x{\mathrm{e}}^{\pm \pi \mathrm{i}}$ are complex. The branch point is absent, however, in the case of ${\mathrm{ber}}_{\nu }$ and ${\mathrm{bei}}_{\nu }$ when $\nu$ is an integer. … ►

$${\mathrm{ber}}_{\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left({\mathrm{e}}^x\cos \left(x+\frac{\pi }{8}\right)-{\mathrm{e}}^{-x}\cos \left(x-\frac{\pi }{8}\right)\right),$$

►

$${\mathrm{bei}}_{\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left({\mathrm{e}}^x\sin \left(x+\frac{\pi }{8}\right)+{\mathrm{e}}^{-x}\sin \left(x-\frac{\pi }{8}\right)\right).$$

►

$${\mathrm{ber}}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left({\mathrm{e}}^x\sin \left(x+\frac{\pi }{8}\right)-{\mathrm{e}}^{-x}\sin \left(x-\frac{\pi }{8}\right)\right),$$

…

43: 8.2 Definitions and Basic Properties

… ►However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ take their principal values; compare §4.2(i). … ►

8.2.8 $$\gamma (a,z{\mathrm{e}}^{2\pi m\mathrm{i}})={\mathrm{e}}^{2\pi m\mathrm{i}a}\gamma (a,z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$,

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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, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/8.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(ii), §8.2 and Ch.8

►

8.2.9 $$\mathrm{\Gamma }(a,z{\mathrm{e}}^{2\pi m\mathrm{i}})={\mathrm{e}}^{2\pi m\mathrm{i}a}\mathrm{\Gamma }(a,z)+(1-{\mathrm{e}}^{2\pi m\mathrm{i}a})\mathrm{\Gamma }\left(a\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $m$: integer

Referenced by:

§8.2(ii)

Permalink:

http://dlmf.nist.gov/8.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(ii), §8.2 and Ch.8

… ►For example, in the case $m=-1$ we have ►

8.2.10 $${\mathrm{e}}^{-\pi \mathrm{i}a}\mathrm{\Gamma }(a,z{\mathrm{e}}^{\pi \mathrm{i}})-{\mathrm{e}}^{\pi \mathrm{i}a}\mathrm{\Gamma }(a,z{\mathrm{e}}^{-\pi \mathrm{i}})=-\frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(1-a\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(ii), §8.2 and Ch.8

…

44: 25.12 Polylogarithms

… ►When $z={\mathrm{e}}^{\mathrm{i}\theta }$, $0\le \theta \le 2\pi$, (25.12.1) becomes … ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►valid when $\mathrm{\Re }s>0$ and , or $\mathrm{\Re }s>1$ and $z=1$. (In the latter case (25.12.11) becomes (25.5.1)). … ►When $s=2$ and ${\mathrm{e}}^{2\pi \mathrm{i}a}=z$, (25.12.13) becomes (25.12.4). …

45: 1.12 Continued Fractions

… ►In this case $\left|\mathrm{ph}C\right|\le \frac{1}{2}\pi$. …

46: 4.43 Cubic Equations

… ►

4.43.2 $$z^3+pz+q=0$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

…

47: 23.12 Asymptotic Approximations

… ►If $q(={\mathrm{e}}^{\pi \mathrm{i}{\omega }_3/{\omega }_1})\to 0$ with ${\omega }_1$ and $z$ fixed, then … ►

23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12, Erratum (V1.0.23) for Equation (23.12.2)

Permalink:

http://dlmf.nist.gov/23.12.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.

Suggested 2019-05-27 by Blagoje Oblak

See also:

Annotations for §23.12 and Ch.23

►

23.12.3 $$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\pi }^2{z}^2}{24{\omega }_1^2}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\left(1-\left(\frac{{\pi }^2{z}^2}{{\omega }_1^2}-4{\sin }^2\left(\frac{\pi z}{2{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Permalink:

http://dlmf.nist.gov/23.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

►provided that $z\notin 𝕃$ in the case of (23.12.1) and (23.12.2). … ►

23.12.4 $${\eta }_1=\frac{{\pi }^2}{4{\omega }_1}\left(\frac{1}{3}-8q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $q$: nome, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.12

Permalink:

http://dlmf.nist.gov/23.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

…

48: 15.6 Integral Representations

… ►

15.6.2 $$𝐅(a,b;c;z)=\frac{\mathrm{\Gamma }\left(1+b-c\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b\right)}{\int }_0^{(1+)}\frac{t^{b-1}{(t-1)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $c-b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }b>0$.

ⓘ

Symbols:

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

Keywords:

Mellin transform

Referenced by:

§15.6

Permalink:

http://dlmf.nist.gov/15.6.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►

15.6.3 $$𝐅(a,b;c;z)={\mathrm{e}}^{-b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(c-b\right)}{\int }_{\mathrm{\infty }}^{(0+)}\frac{t^{b-1}{(t+1)}^{a-c}}{{(t-zt+1)}^a}dt,$$ ; $b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }\left(c-b\right)>0$.

ⓘ

Symbols:

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

Keywords:

Mellin transform

Referenced by:

§15.6, §15.6

Permalink:

http://dlmf.nist.gov/15.6.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. … ►In (15.6.3) the point $1/(z-1)$ lies outside the integration contour, the contour cuts the real axis between $t=-1$ and $0$, at which point $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1+t\right)=0$. ►In (15.6.4) the point $1/z$ lies outside the integration contour, and at the point where the contour cuts the negative real axis $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1-t\right)=0$. …

49: 19.17 Graphics

… ►The cases $x=0$ or $y=0$ correspond to the complete integrals. The case $y=1$ corresponds to elementary functions. … ►

►►

Figure 19.17.7: Cauchy principal value of $R_J(0.5,y,1,p)$ for $y=0,\mathrm{\hspace{0.17em}0.01},\mathrm{\hspace{0.17em}0.05},\mathrm{\hspace{0.17em}0.2},\mathrm{\hspace{0.17em}1}$, . $y=1$ corresponds to $3(R_C(0.5,p)-(\pi /\sqrt{8}))/(1-p)$. … Magnify

►

►

►

Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. …If $y=1$, then it has the value $\frac{3}{2}\pi /(p+\sqrt{p})$ when $p>0$, and $\frac{3}{2}\pi /(p-1)$ when . See (19.20.10), (19.20.11), and (19.20.8) for the cases $p\to 0\pm$, $y\to 0+$, and $y=1$, respectively. Magnify 3D Help

50: 18.35 Pollaczek Polynomials

… ►

18.35.6 $$w^{(\lambda )}(\cos \theta ;a,b)={\pi }^{-1}{\mathrm{e}}^{(2\theta -\pi ){\tau }_{a,b}(\theta )}{\left(2\sin \theta \right)}^{2\lambda -1}{\left|\mathrm{\Gamma }\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)\right|}^2,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $w(x)$: weight function and ${\tau }_{a,b}(\theta )$

Source:

Ismail (2009, (5.4.13))

Referenced by:

(18.35.5), §18.35(ii), §18.39(iv), §18.39(iv), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

The constraints in this equation have been moved to (18.35.5) except for .

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

… ►

18.35.6_1

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\ln z$: principal branch of logarithm function and $w(x)$: weight function

Referenced by:

§18.35(ii), Erratum (V1.2.0) Section 18.35

Permalink:

http://dlmf.nist.gov/18.35.E6_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

… ►Hence, only in the case $a=b=0$ does $\ln \left(w^{(\lambda )}(x;a,b)\right)$ satisfy the condition (18.2.39) for the Szegő class $𝒢$. … ►More generally, the $P_n^{(\lambda )}(x;a,b)$ are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)). ►

18.35.6_2

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Symbols:

$n$: nonnegative integer and $x$: real variable

Proved:

Ismail (2009, (5.5.8)); Case (iii) is overlooked in the given reference.(proved)

Proof sketch:

By Favard’s theorem (see §18.2(viii)) the necessary and sufficient condition for orthogonality is that in (18.35.2_4), for $c=0$, the coefficient of $Q_n^{(\lambda )}(x;a,b,0)$ is real for $n\ge 0$ and the coefficient of $Q_{n-1}^{(\lambda )}(x;a,b,0)$ is positive for $n\ge 1$.

Referenced by:

§18.35(ii), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

…