# complex conjugates

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##### 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 $\Phi_{n+1}(z)=z\Phi_{n}(z)-\overline{\alpha_{n}}\,\Phi_{n}^{*}(z),$
18.33.27 $z\Phi_{n}(z)=\rho_{n}^{-2}\left(\Phi_{n+1}(z)+\overline{\alpha_{n}}\Phi_{n+1}^% {*}(z)\right),$
##### 2: 1.1 Special Notation
 $x,y$ real variables. … complex conjugate of the matrix $\mathbf{A}$ …
In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{*}$, $a$ being a complex number or a matrix; the Hermitian conjugate of $\mathbf{A}$ is usually being denoted $\mathbf{A}^{{\dagger}}$.
##### 3: 4.3 Graphics
Corresponding points share the same letters, with bars signifying complex conjugates. …
##### 4: 18.19 Hahn Class: Definitions
18.19.1 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
18.19.2 $w(z;a,b,\overline{a},\overline{b})=\Gamma\left(a+iz\right)\Gamma\left(b+iz% \right)\Gamma\left(\overline{a}-iz\right)\Gamma\left(\overline{b}-iz\right),$
18.19.3 $w(x)=w(x;a,b,\overline{a},\overline{b})=|\Gamma\left(a+\mathrm{i}x\right)% \Gamma\left(b+\mathrm{i}x\right)|^{2},$
18.19.4 $h_{n}=\frac{2\pi\Gamma\left(n+a+\overline{a}\right)\Gamma\left(n+b+\overline{b% }\right)|\Gamma\left(n+a+\overline{b}\right)|^{2}}{\left(2n+2\Re\left(a+b% \right)-1\right)\Gamma\left(n+2\Re\left(a+b\right)-1\right)n!},$
##### 5: 27.8 Dirichlet Characters
If $\chi$ is a character (mod $k$), so is its complex conjugate $\overline{\chi}$. …
27.8.6 $\sum_{r=1}^{\phi\left(k\right)}\chi_{r}\left(m\right)\overline{\chi}_{r}(n)=% \begin{cases}\phi\left(k\right),&m\equiv n\pmod{k},\\ 0,&\mbox{otherwise}.\end{cases}$
##### 6: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.4 $q_{n}(x)=\ifrac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{p_{n}\left(% \mathrm{i}a;a,b,\overline{a},\overline{b}\right)},$
18.22.13 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
18.22.27 $\delta_{x}\left(p_{n}\left(x;a,b,\overline{a},\overline{b}\right)\right)=(n+2% \Re\left(a+b\right)-1)\*p_{n-1}\left(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline% {a}+\tfrac{1}{2},\overline{b}+\tfrac{1}{2}\right),$
18.22.28 $\delta_{x}\left(w(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+\tfrac{1}{2},% \overline{b}+\tfrac{1}{2})p_{n}(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+% \tfrac{1}{2},\overline{b}+\tfrac{1}{2})\right)=-(n+1)w(x;a,b,\overline{a},% \overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).$
##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
1.18.12 $\left\langle f,g\right\rangle=\int_{a}^{b}f(x)\overline{g(x)}\,\mathrm{d}% \alpha(x),$
1.18.18 $K(x,y)=\sum_{n=0}^{\infty}\phi_{n}(x)\overline{\phi_{n}(y)}.$
1.18.20 $\delta_{n,m}=\int_{a}^{b}\phi_{n}(x)\overline{\phi_{m}(x)}\,\mathrm{d}x.$
##### 8: 27.10 Periodic Number-Theoretic Functions
27.10.10 $G\left(n,\chi\right)=\overline{\chi}(n)G\left(1,\chi\right).$
27.10.12 $\chi\left(n\right)=\frac{G\left(1,\chi\right)}{k}\sum_{m=1}^{k}\overline{\chi}% (m)e^{-2\pi\mathrm{i}mn/k}.$
##### 9: 1.2 Elementary Algebra
the complex conjugate is
1.2.29 $\overline{\mathbf{A}}=[\overline{a_{ij}}],$
1.2.30 ${\mathbf{A}}^{{\rm H}}=[\overline{a_{ji}}].$
If $\mathbf{u}$, $\mathbf{v}$, $\alpha$ and $\beta$ are real the complex conjugate bars can be omitted in (1.2.40)–(1.2.42). …
##### 10: 14.30 Spherical and Spheroidal Harmonics
14.30.6 $Y_{{l},{-m}}\left(\theta,\phi\right)=(-1)^{m}\overline{Y_{{l},{m}}\left(\theta% ,\phi\right)}.$
14.30.8 $\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.$
14.30.9 $\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}% \overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)}Y_{{l},{m}}\left(\theta_% {2},\phi_{2}\right).$