complex conjugates

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11: 25.15 Dirichlet $L$-functions

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25.15.5 $$L(1-s,\chi )=\frac{k^{s-1}\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}+\chi (-1){\mathrm{e}}^{\pi \mathrm{i}s/2}\right)G(\chi )L(s,\overline{\chi }),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

connection formula

Source:

Apostol (1976, (16), p. 263)

Referenced by:

§25.15(ii)

Permalink:

http://dlmf.nist.gov/25.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

►where $\overline{\chi }$ is the complex conjugate of $\chi$, and …

12: 4.15 Graphics

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

13: 10.34 Analytic Continuation

… ►For complex $\nu$ replace $\nu$ by $\overline{\nu }$ on the right-hand sides.

14: 25.10 Zeros

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25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

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15: 1.9 Calculus of a Complex Variable

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Complex Conjugate

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1.9.11 $$\overline{z}=x-\mathrm{i}y,$$

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Defines:

$\overline{z}$: complex conjugate

Symbols:

$\mathrm{i}$: imaginary unit and $z$: variable

A&S Ref:

3.7.7

Permalink:

http://dlmf.nist.gov/1.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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1.9.12 $$\left|\overline{z}\right|=\left|z\right|,$$

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Symbols:

$\overline{z}$: complex conjugate, $z$: variable and $\left|x\right|$: absolute value of $x$

A&S Ref:

3.7.8

Permalink:

http://dlmf.nist.gov/1.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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1.9.13 $$\mathrm{ph}\overline{z}=-\mathrm{ph}z.$$

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Symbols:

$\overline{z}$: complex conjugate, $\mathrm{ph}$: phase and $z$: variable

A&S Ref:

3.7.9

Permalink:

http://dlmf.nist.gov/1.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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1.9.16 $$\frac{z_1}{z_2}=\frac{z_1{\overline{z}}_2}{{\left|z_2\right|}^2}=\frac{x_1{x}_2+y_1{y}_2+\mathrm{i}(x_2{y}_1-x_1{y}_2)}{x_2^2+y_2^2},$$

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Symbols:

$\overline{z}$: complex conjugate, $\mathrm{i}$: imaginary unit, $z$: variable and $\left|x\right|$: absolute value of $x$

A&S Ref:

3.7.13

Permalink:

http://dlmf.nist.gov/1.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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16: 13.7 Asymptotic Expansions for Large Argument

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13.7.7 $$z\in {\text{R}}_1,z\in {\text{R}}_2\cup {\overline{\text{R}}}_2,z\in {\text{R}}_3\cup {\overline{\text{R}}}_3,$$

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Symbols:

$\overline{z}$: complex conjugate, $\in$: element of, $\cup$: union and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(ii), §13.7 and Ch.13

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17: 18.37 Classical OP’s in Two or More Variables

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

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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

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

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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

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18: 18.20 Hahn Class: Explicit Representations

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18.20.3 $$w(x;a,b,\overline{a},\overline{b})p_n(x;a,b,\overline{a},\overline{b})=\frac{1}{n!}{\delta }_x^n\left(w(x;a+\frac{1}{2}n,b+\frac{1}{2}n,\overline{a}+\frac{1}{2}n,\overline{b}+\frac{1}{2}n)\right).$$

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Symbols:

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

Referenced by:

§18.20(i)

Permalink:

http://dlmf.nist.gov/18.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

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18.20.9 $$p_n(x;a,b,\overline{a},\overline{b})=\frac{{\mathrm{i}}^n{\left(a+\overline{a}\right)}_n{\left(a+\overline{b}\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+2\mathrm{\Re }\left(a+b\right)-1,a+\mathrm{i}x}{a+\overline{a},a+\overline{b}};1).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19, §18.20(ii), §18.21(ii), §18.22(iii), §18.26(ii)

Permalink:

http://dlmf.nist.gov/18.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(ii), §18.20 and Ch.18

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19: 28.7 Analytic Continuation of Eigenvalues

… ►For real $q$ with , $a_0\left(\mathrm{i}q\right)$ and $a_2\left(\mathrm{i}q\right)$ are real-valued, whereas for real $q$ with $|q|>|q_0|$, $a_0\left(\mathrm{i}q\right)$ and $a_2\left(\mathrm{i}q\right)$ are complex conjugates. …

20: 18.23 Hahn Class: Generating Functions

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18.23.6 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a+\mathrm{i}x}{2\mathrm{\Re }a};-\mathrm{i}z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\overline{b}-\mathrm{i}x}{2\mathrm{\Re }b};\mathrm{i}z)=\sum _{n=0}^{\mathrm{\infty }}\frac{p_n(x;a,b,\overline{a},\overline{b})}{{\left(2\mathrm{\Re }a\right)}_n{\left(2\mathrm{\Re }b\right)}_n}{z}^n.$$

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Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

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