conjugate

SearchAdvancedHelp

(0.001 seconds)

31—40 of 78 matching pages

31: 10.63 Recurrence Relations and Derivatives

…

32: 18.37 Classical OP’s in Two or More Variables

… ►

18.37.3 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}{c}_j{z}^{m-j}{\overline{z}}^{n-j},$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►

18.37.6 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}\frac{{(-1)}^j{\left(\alpha +1\right)}_{m+n-j}{\left(-m\right)}_j{\left(-n\right)}_j}{{\left(\alpha +1\right)}_m{\left(\alpha +1\right)}_{n}j!}{z}^{m-j}{\overline{z}}^{n-j}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $!$: factorial (as in $n!$), $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

…

33: 28.12 Definitions and Basic Properties

… ►

28.12.10 $$\overline{{\mathrm{me}}_{\nu }(z,q)}={\mathrm{me}}_{\overline{\nu }}(-\overline{z},\overline{q}).$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\overline{z}$: complex conjugate, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.12(ii)

Permalink:

http://dlmf.nist.gov/28.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

…

34: 14.15 Uniform Asymptotic Approximations

… ►Here we introduce the envelopes of the parabolic cylinder functions $U(-c,x)$, $\overline{U}(-c,x)$, which are defined in §12.2. For $U(-c,x)$ or $\overline{U}(-c,x)$, with $c$ and $x$ nonnegative, … ►

►where $x=X_c$ denotes the largest positive root of the equation $U(-c,x)=\overline{U}(-c,x)$. … ►

14.15.25 $${𝖰}_{\nu }^{-\mu }\left(x\right)=\frac{\pi }{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu +2)/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2-{\alpha }^2}{x^2-a^2}\right)}^{1/4}\left(\overline{U}(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )+O\left({\nu }^{-2/3}\right)\mathrm{env}\overline{U}(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$ , $\overline{U}(a,x)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

…

35: 36.2 Catastrophes and Canonical Integrals

… ►

36.2.23 $${\mathrm{\Psi }}_{2K+1}\left({𝐱}^{\prime }\right)=\overline{{\mathrm{\Psi }}_{2K+1}\left(𝐱\right)},$$ $x_{2m+1}^{\prime }=x_{2m+1}$, $x_{2m}^{\prime }=-x_{2m}$.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\overline{z}$: complex conjugate, $n$: integer, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E23

Encodings:

pMML, png, TeX

See also:

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

►

36.2.24 $${\mathrm{\Psi }}^{(\mathrm{U})}\left(x,y,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{U})}\left(x,y,-z\right)},$$ $\mathrm{U}=\mathrm{E},\mathrm{H}$.

ⓘ

Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$: umbilic canonical integral function, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §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

36: 36.11 Leading-Order Asymptotics

… ►

36.11.5 $${\mathrm{\Psi }}_3\left(0,y,0\right)=\overline{{\mathrm{\Psi }}_3(0,-y,0)}=\exp \left(\frac{1}{4}\mathrm{i}\pi \right)\sqrt{\pi /y}\left(1-(\mathrm{i}/\sqrt{3})\exp \left(\frac{3}{2}\mathrm{i}{(2y/5)}^{5/3}\right)+o\left(1\right)\right),$$ $y\to +\mathrm{\infty }$.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

…

37: Guide to Searching the DLMF

… ►

Table 2: Wildcard Examples

► ►►►

Query	What it stands for
…
co$te	Coordinate, conjugate, correlate,…
…

►

… ►

Table 4: Font and Accent Examples

► ►►►

Query	Matches
…
U bar	$\overline{U}$
…

►

…

38: 12.12 Integrals

… ►

12.12.4 $${(U(a,z))}^2+{(\overline{U}(a,z))}^2=\frac{2^{\frac{3}{2}}}{\pi }\mathrm{\Gamma }\left(\frac{1}{2}-a\right){\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2at+\frac{1}{2}{z}^2\tanh t}}{\sqrt{\sinh \left(2t\right)}}dt,$$ .

ⓘ

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, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Notes:

See Durand (1975).

Referenced by:

§12.12

Permalink:

http://dlmf.nist.gov/12.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12, §12.12 and Ch.12

…

39: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►

13.20.17 $$M_{\kappa ,\mu }\left(x\right)={\left(8\mu \right)}^{\frac{1}{4}}{\left(\frac{2\mu }{\mathrm{e}}\right)}^{2\mu }{\left(\frac{\mathrm{e}}{\kappa +\mu }\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\left(U(\mu -\kappa ,-\zeta \sqrt{2\mu })+\mathrm{env}\overline{U}(\mu -\kappa ,\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function

Referenced by:

§13.20(iv), §13.20(iv)

Permalink:

http://dlmf.nist.gov/13.20.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

… ►

13.20.18 $$W_{\kappa ,\mu }\left(x\right)={\left(\frac{1}{2}\mu \right)}^{-\frac{1}{4}}{\left(\frac{\kappa +\mu }{\mathrm{e}}\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\left(U(\mu -\kappa ,\zeta \sqrt{2\mu })+\mathrm{env}\overline{U}(\mu -\kappa ,-\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function

Referenced by:

§13.20(iv), §13.20(iv)

Permalink:

http://dlmf.nist.gov/13.20.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

… ►For the parabolic cylinder functions $U$ and $\overline{U}$ see §12.2, and for the $\mathrm{env}$ functions associated with $U$ and $\overline{U}$ see §14.15(v). …

40: 18.26 Wilson Class: Continued

… ►

18.26.6 $$\underset{t\to \mathrm{\infty }}{lim}\frac{W_n({(x+t)}^2;a-it,b-it,\overline{a}+it,\overline{b}+it)}{{(-2t)}^{n}n!}=p_n(x;a,b,\overline{a},\overline{b}).$$

ⓘ

Symbols:

$W_n(x;a,b,c,d)$: Wilson polynomial, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(i), §18.22(ii), §18.23, §18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

…