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11: 12.2 Differential Equations

… ►Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where … ►

§12.2(vi) Solution $\overline{U}(a,x)$; Modulus and Phase Functions

►When $z$ $(=x)$ is real the solution $\overline{U}(a,x)$ is defined by …unless $a=\frac{1}{2},\frac{3}{2},\mathrm{\dots }$, in which case $\overline{U}(a,x)$ is undefined. …Properties of $\overline{U}(a,x)$ follow immediately from those of $V(a,x)$ via (12.2.21). …

12: 26.1 Special Notation

… ►Many combinatorics references use the rising and falling factorials: ►

$$x^{\overline{n}}=x(x+1)(x+2)\mathrm{\cdots }(x+n-1),$$

►

$$x^{\underset{¯}{n}}=x(x-1)(x-2)\mathrm{\cdots }(x-n+1).$$

…

13: 27.8 Dirichlet Characters

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

27.8.6

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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

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14: 10.11 Analytic Continuation

… ►

$J_{\nu }\left(\overline{z}\right)=\overline{J_{\nu }\left(z\right)},$ $Y_{\nu }\left(\overline{z}\right)=\overline{Y_{\nu }\left(z\right)},$

►

$H_{\nu }^{(1)}\left(\overline{z}\right)=\overline{H_{\nu }^{(2)}\left(z\right)},$ $H_{\nu }^{(2)}\left(\overline{z}\right)=\overline{H_{\nu }^{(1)}\left(z\right)}.$

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

15: 18.23 Hahn Class: Generating Functions

… ►

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:

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See also:

Annotations for §18.23, §18.23 and Ch.18

…

16: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

…

Figure 26.9.1: Ferrers graph of the partition $7+4+3+3+2+1$.

►The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots. The conjugate to the example in Figure 26.9.1 is $6+5+4+2+1+1+1$. Conjugation establishes a one-to-one correspondence between partitions of $n$ into at most $k$ parts and partitions of $n$ into parts with largest part less than or equal to $k$. …

17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space $V$ is called an inner product space if an inner product $⟨u,v⟩\in ℂ$ is defined for all $u,v\in V$ with the properties: (i) $⟨u,v⟩$ is complex linear in $u$; (ii) $⟨u,v⟩=\overline{⟨v,u⟩}$; (iii) $⟨v,v⟩\ge 0$; (iv) if $⟨v,v⟩=0$ then $v=0$. … ►

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:

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See also:

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

… ►

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

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

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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:

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See also:

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

… ►Also, because $q$ is real-valued, $f\in N_z$ iff $\overline{f}\in N_{\overline{z}}$. …

18: 13.7 Asymptotic Expansions for Large Argument

… ►

►►

Figure 13.7.1: Regions ${\text{R}}_1$, ${\text{R}}_2$, ${\overline{\text{R}}}_2$, ${\text{R}}_3$, and ${\overline{\text{R}}}_3$ are the closures of the indicated unshaded regions bounded by the straight lines and circular arcs centered at the origin, with $r=|b-2a|$. Magnify

… ►

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:

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See also:

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

… ►Also, when $z\in {\text{R}}_1\cup {\text{R}}_2\cup {\overline{\text{R}}}_2$ …and when $z\in {\text{R}}_3\cup {\overline{\text{R}}}_3$ $\sigma$ is replaced by $\nu \sigma$ and ${|z|}^{-1}$ is replaced by $\nu {|z|}^{-1}$ everywhere in (13.7.9). …

19: 1.2 Elementary Algebra

… ►the complex conjugate is ►

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

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Symbols:

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

Permalink:

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

Encodings:

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See also:

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

►the Hermitian conjugate is ►

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). …

20: 4.15 Graphics

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

►

… ►

►►

	A	B	C	$\overline{\mathrm{C}}$	D	$\overline{\mathrm{D}}$	E	$\overline{\mathrm{E}}$	F
…

►

Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify

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