complex conjugates

SearchAdvancedHelp

(0.003 seconds)

1—10 of 67 matching pages

1: 18.33 Polynomials Orthogonal on the Unit Circle

… ►where the bar signifies complex conjugate. … ►where the bar again signifies complex conjugate. … ►where the bar signifies complex conjugate and ${\kappa }_n>0$, ${\kappa }_0=1$. … ►

18.33.23 $${\mathrm{\Phi }}_{n+1}(z)=z{\mathrm{\Phi }}_n(z)-\overline{{\alpha }_n}{\mathrm{\Phi }}_n^{\ast }(z),$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer

Proved:

Simon (2005a, Theorem 1.5.2)(proved)

Referenced by:

§18.33(vi), §18.33(vi), §18.33(vi), §18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

… ►

18.33.27 $$z{\mathrm{\Phi }}_n(z)={\rho }_n^{-2}\left({\mathrm{\Phi }}_{n+1}(z)+\overline{{\alpha }_n}{\mathrm{\Phi }}_{n+1}^{\ast }(z)\right),$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer

Proved:

Simon (2005a, Theorem 1.5.4)(proved)

Referenced by:

§18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

…

2: 4.3 Graphics

… ►Corresponding points share the same letters, with bars signifying complex conjugates. …

3: 1.1 Special Notation

… ► ►►►

$x,y$	real variables.
…
$\overline{𝐀}$	complex conjugate of the matrix $𝐀$
…

►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

4: 18.19 Hahn Class: Definitions

… ►

18.19.1 $$p_n(x)=p_n(x;a,b,\overline{a},\overline{b}),$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19

Permalink:

http://dlmf.nist.gov/18.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

►

18.19.2 $$w(z;a,b,\overline{a},\overline{b})=\mathrm{\Gamma }\left(a+iz\right)\mathrm{\Gamma }\left(b+iz\right)\mathrm{\Gamma }\left(\overline{a}-iz\right)\mathrm{\Gamma }\left(\overline{b}-iz\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\overline{z}$: complex conjugate, $w(x)$: weight function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/18.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

►

18.19.3 $$w(x)=w(x;a,b,\overline{a},\overline{b})={|\mathrm{\Gamma }\left(a+\mathrm{i}x\right)\mathrm{\Gamma }\left(b+\mathrm{i}x\right)|}^2,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\overline{z}$: complex conjugate, $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

►

18.19.4 $$h_n=\frac{2\pi \mathrm{\Gamma }\left(n+a+\overline{a}\right)\mathrm{\Gamma }\left(n+b+\overline{b}\right){|\mathrm{\Gamma }\left(n+a+\overline{b}\right)|}^2}{\left(2n+2\mathrm{\Re }\left(a+b\right)-1\right)\mathrm{\Gamma }\left(n+2\mathrm{\Re }\left(a+b\right)-1\right)n!},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $h_n$

Referenced by:

§18.19

Permalink:

http://dlmf.nist.gov/18.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

…

5: 27.8 Dirichlet Characters

… ►If $\chi$ is a character (mod $k$), so is its complex conjugate $\overline{\chi }$. … ►

27.8.6

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\varphi \left(n\right)$: Euler’s totient, $\overline{z}$: complex conjugate, $\equiv$: modular equivalence, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

…

6: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

18.22.4 $$q_n(x)=p_n(x;a,b,\overline{a},\overline{b})/p_n(\mathrm{i}a;a,b,\overline{a},\overline{b}),$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(i)

Permalink:

http://dlmf.nist.gov/18.22.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

… ►

18.22.13 $$p_n(x)=p_n(x;a,b,\overline{a},\overline{b}),$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(ii)

Permalink:

http://dlmf.nist.gov/18.22.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

… ►

18.22.27 $${\delta }_x\left(p_n(x;a,b,\overline{a},\overline{b})\right)=(n+2\mathrm{\Re }\left(a+b\right)-1)p_{n-1}(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2}),$$

ⓘ

Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

►

18.22.28 $${\delta }_x\left(w(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})p_n(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})\right)=-(n+1)w(x;a,b,\overline{a},\overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).$$

ⓘ

Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

…

7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

1.18.9 $$⟨v,w⟩=\sum _{n=0}^{\mathrm{\infty }}{c}_n\overline{d_n}.$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $w$: variable and $n$: nonnegative integer

Referenced by:

§1.18(ii)

Permalink:

http://dlmf.nist.gov/1.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(i), §1.18 and Ch.1

… ►

1.18.12 $$⟨f,g⟩={\int }_a^{b}f(x)\overline{g(x)}d\alpha (x),$$

ⓘ

Defines:

$⟨f,g⟩$: inner product over functions

Symbols:

$\overline{z}$: complex conjugate, $dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

… ►

1.18.18 $$K(x,y)=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_n(x)\overline{{\varphi }_n(y)}.$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $(a,b)$: open interval and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.18.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

… ►

1.18.19 $$\delta \left(x-y\right)=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_n(x)\overline{{\varphi }_n(y)},$$

ⓘ

Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(vi), §1.18(ii)

Permalink:

http://dlmf.nist.gov/1.18.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

… ►

1.18.20 $${\delta }_{n,m}={\int }_a^b{\varphi }_n(x)\overline{{\varphi }_m(x)}dx.$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(vi), §1.18(vi)

Permalink:

http://dlmf.nist.gov/1.18.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

…

8: 27.10 Periodic Number-Theoretic Functions

… ►

27.10.10 $$G(n,\chi )=\overline{\chi }(n)G(1,\chi ).$$

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $\overline{z}$: complex conjugate, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

… ►

27.10.12 $$\chi \left(n\right)=\frac{G(1,\chi )}{k}\sum _{m=1}^k\overline{\chi }(m){\mathrm{e}}^{-2\pi \mathrm{i}mn/k}.$$

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

9: 1.2 Elementary Algebra

… ►the complex conjugate is ►

1.2.29 $$\overline{𝐀}=[\overline{a_{ij}}],$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate and $j$: integer

Permalink:

http://dlmf.nist.gov/1.2.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.30 $${𝐀}^{\mathrm{H}}=[\overline{a_{ji}}].$$

ⓘ

Defines:

${𝐀}^{\mathrm{H}}$: Hermitian conjugate of matrix

Symbols:

$\overline{z}$: complex conjugate and $j$: integer

Permalink:

http://dlmf.nist.gov/1.2.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►If $𝐮$, $𝐯$, $\alpha$ and $\beta$ are real the complex conjugate bars can be omitted in (1.2.40)–(1.2.42). … ►

1.2.56 $$a_{ji}=\overline{a_{ij}},a_{ij}\in ℂ,$$

ⓘ

Symbols:

$ℂ$: complex plane, $\overline{z}$: complex conjugate, $\in$: element of and $j$: integer

Permalink:

http://dlmf.nist.gov/1.2.E56

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

…

10: 14.30 Spherical and Spheroidal Harmonics

… ►

14.30.6 $$Y_{l,-m}(\theta ,\varphi )={(-1)}^m\overline{Y_{l,m}(\theta ,\varphi )}.$$

ⓘ

Symbols:

$\overline{z}$: complex conjugate, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E6

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the spherical harmonic on the right-hand side has been explicity marked up as a complexconjugate.

See also:

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

… ►

14.30.8 $${\int }_0^{2\pi }{\int }_0^{\pi }\overline{Y_{l_1,m_1}(\theta ,\varphi )}{Y}_{l_2,m_2}(\theta ,\varphi )\sin \theta d\theta d\varphi ={\delta }_{l_1,l_2}{\delta }_{m_1,m_2}.$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E8

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complexconjugate.

See also:

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

… ►

14.30.9 $${𝖯}_l\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \left({\varphi }_1-{\varphi }_2\right)\right)=\frac{4\pi }{2l+1}\sum _{m=-l}^l\overline{Y_{l,m}({\theta }_1,{\varphi }_1)}{Y}_{l,m}({\theta }_2,{\varphi }_2).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§18.18(ii)

Permalink:

http://dlmf.nist.gov/14.30.E9

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the sum over $m$ has been explicity marked up as a complexconjugate.

See also:

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

…