real%20and%20imaginary%20parts

SearchAdvancedHelp

(0.005 seconds)

11—19 of 19 matching pages

11: 36.2 Catastrophes and Canonical Integrals

… ►

36.2.6 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2\sqrt{\pi /3}\exp \left(\mathrm{i}\left(\frac{4}{27}{z}^3+\frac{1}{3}xz-\frac{1}{4}\pi \right)\right){\int }_{\mathrm{\infty }\exp \left(-7\pi \mathrm{i}/12\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/12\right)}\exp \left(\mathrm{i}\left(u^6+2z{u}^4+(z^2+x)u^2+\frac{y^2}{12u^2}\right)\right)du,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.2(i), §36.2(ii), §36.5(iii)

Permalink:

http://dlmf.nist.gov/36.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

… ►

36.2.9 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)=\frac{2\pi }{3^{1/3}}{\int }_{\mathrm{\infty }\exp \left(5\pi \mathrm{i}/6\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/6\right)}\exp \left(\mathrm{i}(s^3+xs)\right)\mathrm{Ai}\left(\frac{zs+y}{3^{1/3}}\right)ds.$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real parameter, $z$: real parameter, $s$: variable and $x$: real parameter

Referenced by:

§36.2(i), §36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

… ►

36.2.20 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,0\right)=2{\pi }^2{(\frac{2}{3})}^{2/3}\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{x+\mathrm{i}y}{{12}^{1/3}}\right)\mathrm{Bi}\left(\frac{x-\mathrm{i}y}{{12}^{1/3}}\right)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter and $x$: real parameter

Referenced by:

§36.2(ii), §36.7(iii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

… ►

36.2.28 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-z\right)}=2\pi \sqrt{\frac{\pi z}{27}}\exp \left(\frac{2}{27}\mathrm{i}{z}^3\right)\left(J_{-1/6}\left(\frac{2}{27}{z}^3\right)+\mathrm{i}{J}_{1/6}\left(\frac{2}{27}{z}^3\right)\right),$$ $z\ge 0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E28

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

►

36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

ⓘ

Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

12: 14.30 Spherical and Spheroidal Harmonics

… ►Sometimes $Y_{l,m}(\theta ,\varphi )$ is denoted by ${\mathrm{i}}^{-l}{𝔇}_{lm}(\theta ,\varphi )$; also the definition of $Y_{l,m}(\theta ,\varphi )$ can differ from (14.30.1), for example, by inclusion of a factor ${(-1)}^m$. … ► $P_n^m\left(\mathrm{i}x\right)$ and $Q_n^m\left(\mathrm{i}x\right)$ ($x>0$) are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics $R_n^m(x)={\mathrm{e}}^{-\mathrm{i}\pi n/2}{P}_n^m\left(\mathrm{i}x\right)$ and $T_n^m(x)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\pi n/2}{Q}_n^m\left(\mathrm{i}x\right)$ which are real when $x>0$ and $n=0,1,2,\mathrm{\dots }$. … ►where $𝐚=(\frac{1}{2\lambda }-\frac{\lambda }{2},-\frac{\mathrm{i}}{2\lambda }-\frac{\mathrm{i}\lambda }{2},1)$ and $𝐱=(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )$. … ►

14.30.13 $${\mathrm{L}}_z=-\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial \varphi };$$

ⓘ

Defines:

${\mathrm{L}}_z$: $z$ component of the angular momentum operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: complex variable

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

…

13: 5.22 Tables

… ►

§5.22(ii) Real Variables

►Abramowitz and Stegun (1964, Chapter 6) tabulates $\mathrm{\Gamma }\left(x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, and ${\psi }^{\prime }\left(x\right)$ for $x=1(.005)2$ to 10D; ${\psi }^{\prime \prime }\left(x\right)$ and ${\psi }^{(3)}\left(x\right)$ for $x=1(.01)2$ to 10D; $\mathrm{\Gamma }\left(n\right)$, $1/\mathrm{\Gamma }\left(n\right)$, $\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, $\psi \left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{3}\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, and ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{2}{3}\right)$ for $n=1(1)101$ to 8–11S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=100(100)1000$ to 20S. … ►Abramov (1960) tabulates $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$ for $x=1$ ($.01$) $2$, $y=0$ ($.01$) $4$ to 6D. …This reference also includes $\psi \left(x+\mathrm{i}y\right)$ for the same arguments to 5D. Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of $\mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, and $\psi \left(x+\mathrm{i}y\right)$ for $x=0.5,1,5,10$, $y=0(.5)10$ to 8S.

14: 12.19 Tables

… ►

Kireyeva and Karpov (1961) includes $D_p\left(x(1+\mathrm{i})\right)$ for $\pm x=0(.1)5$, $p=0(.1)2$, and $\pm x=5(.01)10$, $p=0(.5)2$, 7D.

… ►

Karpov and Čistova (1968) includes ${\mathrm{e}}^{-\frac{1}{4}{x}^2}{D}_p\left(-x\right)$ and ${\mathrm{e}}^{-\frac{1}{4}{x}^2}{D}_p\left(\mathrm{i}x\right)$ for $x=0(.01)5$ and $x^{-1}$ = 0(.001 or .0001)5, $p=-1(.1)1$, 7D or 8S.

►

Murzewski and Sowa (1972) includes $D_{-n}\left(x\right)$ $\left(=U(n-\frac{1}{2},x)\right)$ for $n=1(1)20$, $x=0(.05)3$, 7S.

►

Zhang and Jin (1996, pp. 455–473) includes $U(\pm n-\frac{1}{2},x)$, $V(\pm n-\frac{1}{2},x)$, $U(\pm \nu -\frac{1}{2},x)$, $V(\pm \nu -\frac{1}{2},x)$, and derivatives, $\nu =n+\frac{1}{2}$, $n=0(1)10(10)30$, $x=0.5,1,5,10,30,50$, 8S; $W(a,\pm x)$, $W(-a,\pm x)$, and derivatives, $a=h(1)5+h$, $x=0.5,1$ and $a=h(1)5+h$, $x=5$, $h=0,0.5$, 8S. Also, first zeros of $U(a,x)$, $V(a,x)$, and of derivatives, $a=-6(.5)-1$, 6D; first three zeros of $W(a,-x)$ and of derivative, $a=0(.5)4$, 6D; first three zeros of $W(-a,\pm x)$ and of derivative, $a=0.5(.5)5.5$, 6D; real and imaginary parts of $U(a,z)$, $a=-1.5(1)1.5$, $z=x+\mathrm{i}y$, $x=0.5,1,5,10$, $y=0(.5)10$, 8S.

…

15: Bibliography D

… ►

Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

ⓘ

Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

BibTeX

►

Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

ⓘ

Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

BibTeX

… ►

Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Linear operators. Part II. Wiley Classics Library, John Wiley & Sons, Inc., New York.

ⓘ

Notes:

Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication

External Links:

ISBN 0-471-60847-5, MathReview Entry

Referenced by:

(1.18.68), §1.18(ix), §1.18(ix), §1.18(ix), §1.18(x).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

…

16: Bibliography B

… ►

Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

ⓘ

External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

… ►

A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

ⓘ

Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

… ►

Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

BibTeX

… ►

Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

ⓘ

External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

…

17: 1.11 Zeros of Polynomials

… ►A polynomial of degree $n$ with real or complex coefficients has exactly $n$ real or complex zeros counting multiplicity. Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term. … ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. … ►are $1$, ${\mathrm{e}}^{2\pi \mathrm{i}/n}$, ${\mathrm{e}}^{4\pi \mathrm{i}/n},\mathrm{\dots },{\mathrm{e}}^{(2n-2)\pi \mathrm{i}/n}$, and of $z^n+1=0$ they are ${\mathrm{e}}^{\pi \mathrm{i}/n},{\mathrm{e}}^{3\pi \mathrm{i}/n},\mathrm{\dots },{\mathrm{e}}^{(2n-1)\pi \mathrm{i}/n}$. … ►with real coefficients, is called stable if the real parts of all the zeros are strictly negative. …

18: 25.6 Integer Arguments

… ►

25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

… ►

25.6.7 $$\zeta \left(2\right)={\int }_0^1{\int }_0^1\frac{1}{1-xy}dxdy.$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Keywords:

double integral

Source:

Apostol (1983, p. 59)

Permalink:

http://dlmf.nist.gov/25.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

… ►

25.6.13 $${(-1)}^k{\zeta }^{(k)}\left(-2n\right)=\frac{2{(-1)}^n}{{(2\pi )}^{2n+1}}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\mathrm{\Im }\left(c^{k-m}\right){\mathrm{\Gamma }}^{(r)}\left(2n+1\right){\zeta }^{(m-r)}\left(2n+1\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $c$

Source:

Apostol (1985a, p. 225)

Permalink:

http://dlmf.nist.gov/25.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(ii), §25.6 and Ch.25

►

25.6.14 $${(-1)}^k{\zeta }^{(k)}\left(1-2n\right)=\frac{2{(-1)}^n}{{(2\pi )}^{2n}}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\mathrm{\Re }\left(c^{k-m}\right){\mathrm{\Gamma }}^{(r)}\left(2n\right){\zeta }^{(m-r)}\left(2n\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $c$

Source:

Apostol (1985a, p. 225)

Permalink:

http://dlmf.nist.gov/25.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(ii), §25.6 and Ch.25

…

19: Errata

… ►

The general constraint $\mathrm{\Re }s>2$ has been extended to $\mathrm{\Re }s>1$ for (20.10.1), (20.10.2) and to $\mathrm{\Re }s>0$ for (20.10.3).

… ►

The constraint which originally read “$\mathrm{\Re }s>1$, ” has been extended to be “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $a=1$”.

Reported by Gergő Nemes on 2021-08-23

►

The constraint which originally read “, $\mathrm{\Re }s>0$” has been extended to be “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$”.

Reported by Gergő Nemes on 2021-09-14

… ►

In the first sentence of this subsection, the constraint $\sinh \left|\beta \right|\le \mathrm{\ell }$ has been replaced with $\left|\mathrm{\Re }\beta \right|+\left|\mathrm{\Im }\beta \right|\le \mathrm{\ell }$.

… ►

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

…