notations.tex

\newglossaryentry{not:i}{
  name={$i$},
  description={$\sqrt{-1}$},
}

\newglossaryentry{not:real}{
  name={$\Re z$},
  description={real part},
}

\newglossaryentry{not:imag}{
  name={$\Im z$},
  description={imaginary part},
}

\newglossaryentry{not:C}{
  name={$\C$},
  description={complex numbers},
}

\newglossaryentry{not:R}{
  name={$\R$},
  description={real numbers numbers},
}

\newglossaryentry{not:Z}{
  name={$\Z$},
  description={integers},
}

\newglossaryentry{not:N}{
  name={$\N$},
  description={natural numbers $\{ 1,2,3,\ldots\}$},
}

\newglossaryentry{not:conj}{
  name={$\bar{z}$},
  description={complex conjugate},
}

\newglossaryentry{not:mod}{
  name={$\abs{z}$},
  description={modulus},
}

\newglossaryentry{not:wirt}{
  name={$\frac{\partial}{\partial z}$, $\frac{\partial}{\partial \bar{z}}$,
$\frac{\partial}{\partial z_j}$, $\frac{\partial}{\partial \bar{z}_j}$},
  description={Wirtinger operators},
}

\newglossaryentry{not:D}{
  name={$\D$},
  description={unit disc},
}

\newglossaryentry{not:laplacian}{
  name={$\nabla^2$},
  description={Laplacian},
}

\newglossaryentry{not:disc}{
  name={$\Delta_\rho(a)$},
  description={disc, polydisc},
}

\newglossaryentry{not:identeq}{
  name={$\equiv$},
  description={identically equal, equal at all points},
}

\newglossaryentry{not:dz}{
  name={$dz$, $dz_j$},
  description={$dz = dx+i\,dy$, $dz_j = dx_j+i\,dy_j$},
}

\newglossaryentry{not:dzbar}{
  name={$d\bar{z}$, $d\bar{z}_j$},
  description={$d\bar{z} = dx-i\,dy$, $d\bar{z}_j = dx_j-i\,dy_j$},
}

\newglossaryentry{not:supnorm}{
  name={$\snorm{f}_K$},
  description={supremum norm over $K$},
}

\newglossaryentry{not:hermprod}{
  name={$\linnprod{z}{w}$},
  description={Euclidean inner product},
}

\newglossaryentry{not:eucnorm}{
  name={$\snorm{z}$},
  description={Euclidean norm},
}

\newglossaryentry{not:ball}{
  name={$B_{\rho}(a)$},
  description={ball},
}

\newglossaryentry{not:unitball}{
  name={$\bB$, $\bB_n$},
  description={unit ball},
}

\newglossaryentry{not:N0}{
  name={$\N_0$},
  description={nonnegative integers $\{ 0,1,2,\ldots\}$},
}

\newglossaryentry{not:zalpha}{
  name={$z^\alpha$},
  description={$z_1^{\alpha_1}z_2^{\alpha_2} \cdots z_n^{\alpha_n}$},
}

\newglossaryentry{not:absalpha}{
  name={$\sabs{\alpha}$},
  description={$\alpha_1+\alpha_2+\cdots+\alpha_n$},
}

\newglossaryentry{not:alphabang}{
  name={$\alpha!$},
  description={$\alpha_1!\alpha_2!\cdots\alpha_n!$},
}

\newglossaryentry{not:alphader}{
  name={$\displaystyle \frac{\partial^{\sabs{\alpha}}}{\partial z^\alpha}$},
  description={$\displaystyle \frac{\partial^{\alpha_1}}{\partial z_1^{\alpha_1}}
\frac{\partial^{\alpha_2}}{\partial z_2^{\alpha_2}}
\cdots
\frac{\partial^{\alpha_n}}{\partial z_n^{\alpha_n}}$},
}

\newglossaryentry{not:zalphasum}{
  name={$\displaystyle \sum_{\alpha} c_\alpha z^\alpha$},
  description={multiindex power series},
}

\newglossaryentry{not:dzmulti}{
  name={$dz$},
  description={$dz_1 \wedge dz_2 \wedge \cdots \wedge dz_n$},
}

\newglossaryentry{not:O}{
  name={$\sO(U)$},
  description={the set/ring of holomorphic functions on $U$},
}

\newglossaryentry{not:composition}{
  name={$f \circ g$},
  description={composition, $p \mapsto f\bigl(g(p)\bigr)$},
}

\newglossaryentry{not:Df}{
  name={$Df$, $Df(a)$},
  description={Jacobian matrix / derivative (at $a$)},
}

\newglossaryentry{not:DRf}{
  name={$D_{\R}f$, $D_{\R}f(a)$},
  description={real Jacobian matrix / real derivative (at $a$)},
}

\newglossaryentry{not:DCf}{
  name={$D_{\C}f$, $D_{\C}f(a)$},
  description={complexified real derivative},
}

\newglossaryentry{not:inverse}{
  name={$f^{-1}$},
  description={function inverse},
}

\newglossaryentry{not:pullback}{
  name={$f^{-1}(S)$},
  description={pullback of a set, $\{ q : f(q) \in S \}$},
}

\newglossaryentry{not:compact}{
  name={$\subset \subset$},
  description={compact or relatively compact subset},
}

\newglossaryentry{not:unitsphere}{
  name={$S^{2n-1}$},
  description={unit sphere in $\C^n$, $S^{2n-1} = \partial \bB_n$},
}

\newglossaryentry{not:Ck}{
  name={$C^k$},
  description={$k$ times continuously differentiable},
}

\newglossaryentry{not:Cinfty}{
  name={$C^\infty$},
  description={infinitely differentiable},
}
\newglossaryentry{not:Comega}{
  name={$C^\omega$},
  description={real-analytic},
}

\newglossaryentry{not:realtangentspace}{
  name={$T_p \R^n$, $T_p M$},
  description={tangent space},
}

\newglossaryentry{not:realtangentbundle}{
  name={$T \R^n$, $T M$},
  description={tangent bundle},
}

\newglossaryentry{not:evalpartial}{
  name={$\displaystyle \frac{\partial}{\partial x_j}\Big|_p$},
  description={tangent vector},
}

\newglossaryentry{not:gradient}{
  name={$\nabla r$},
  description={gradient of $r$},
}

\newglossaryentry{not:Ol}{
  name={$O(\ell)$},
  description={big-oh notation},
}

\newglossaryentry{not:complexifiedtangentspace}{
  name={$\C T_p \R^n$, $\C T_p M$},
  description={complexified tangent space},
}

\newglossaryentry{not:holtangentspace}{
  name={$T^{(1,0)}_p \R^n$, $T^{(1,0)}_p M$},
  description={holomorphic tangent vectors},
}

\newglossaryentry{not:aholtangentspace}{
  name={$T^{(0,1)}_p \R^n$, $T^{(0,1)}_p M$},
  description={antiholomorphic tangent vectors},
}

\newglossaryentry{not:Leviform}{
  name={$\sL(X_p,X_p)$},
  description={Levi form},
}

\newglossaryentry{not:star}{
  name={$v^*$, $A^*$},
  description={conjugate transpose},
}

\newglossaryentry{not:Pr}{
  name={$P_r(\theta)$},
  description={Poisson kernel for the unit disc},
}

\newglossaryentry{not:convolution}{
  name={$f * g$},
  description={convolution},
}

\newglossaryentry{not:dV}{
  name={$dV$, $dV(w)$},
  description={volume measure, euclidean volume form},
}

\newglossaryentry{not:dA}{
  name={$dA$, $dA(w)$},
  description={area form, $dA = dx \wedge dy$},
}

\newglossaryentry{not:dpsi}{
  name={$d\psi$},
  description={exterior derivative},
}

\newglossaryentry{not:d}{
  name={$\partial \psi$},
  description={holomorphic part of exterior derivative},
}

\newglossaryentry{not:dbar}{
  name={$\bar{\partial} \psi$},
  description={antiholomorphic part of exterior derivative},
}

\newglossaryentry{not:Khat}{
  name={$\widehat{K}$, $\widehat{K}_{\sF}$},
  description={hull of $K$ with respect to $\sF$},
}

\newglossaryentry{not:dist}{
  name={$\operatorname{dist}(x,y)$},
  description={distance of two points, sets, or a point and a set},
}

\newglossaryentry{not:KhatU}{
  name={$\widehat{K}_{U}$},
  description={holomorphic hull of $K$},
}

\newglossaryentry{not:kronecker}{
  name={$\delta_j^k$},
  description={Kronecker delta, $\delta_j^j = 1$, $\delta_j^k = 0$ if $j \not= k$},
}

\newglossaryentry{not:bracketsquare}{
  name={$[v]^2$},
  description={$v_1^2 + \cdots + v_n^2$},
}

\newglossaryentry{not:hatremove}{
  name={$\widehat{dx_j}$},
  description={removed one-form, $dx_1 \wedge \widehat{dx_2} \wedge dx_3 =
dx_1 \wedge dx_2$},
}

\newglossaryentry{not:A2}{
  name={$A^2(U)$},
  description={Bergman space, $\sO(U) \cap L^2(U)$},
}

\newglossaryentry{not:L2}{
  name={$L^2(U)$},
  description={square integrable functions},
}

\newglossaryentry{not:L2norm}{
  name={$\snorm{f}_{A^2(U)}$, $\snorm{f}_{L^2(U)}$},
  description={$L^2$ norm},
}

\newglossaryentry{not:L2innprod}{
  name={$\linnprod{f}{g}$},
  description={$L^2$ inner product},
}

\newglossaryentry{not:H2}{
  name={$H^2(\partial U)$},
  description={Hardy space},
}

\newglossaryentry{not:germf}{
  name={$(f,p)$},
  description={germ of $f$ at $p$},
}

\newglossaryentry{not:germA}{
  name={$(A,p)$},
  description={germ of a set $A$ at $p$},
}

\newglossaryentry{not:ringofgerms}{
  name={${}_n\sO_p$, $\sO_p$},
  description={ring of germs of holomorphic functions at $p$},
}

\newglossaryentry{not:ord}{
  name={$\ord_a f$},
  description={order of vanishing of $f$ at $a$},
}

\newglossaryentry{not:polyinOU}{
  name={$\sO(U)[z_n]$},
  description={polynomial in $z_n$ with coefficients in $\sO(U)$},
}

\newglossaryentry{not:polyinO0}{
  name={$\sO_0[z_n]$},
  description={polynomial in $z_n$ with coefficients in $\sO_0$},
}

\newglossaryentry{not:Zf}{
  name={$Z_f$},
  description={zero set of $f$, that is, $f^{-1}(0)$},
}

\newglossaryentry{not:ideal}{
  name={$(f)$, $(f_1,\ldots,f_n)$},
  description={ideal generated by $f$ or by $f_1,\ldots,f_n$},
}

\newglossaryentry{not:idealfromset}{
  name={$I_p(X)$},
  description={ideal of germs at $p$ vanishing on $X$},
}

\newglossaryentry{not:vanishingset}{
  name={$V_p(I)$},
  description={the common zero set of germs in $I$},
}

\newglossaryentry{not:Xreg}{
  name={$X_{\mathit{reg}}$},
  description={regular points of $X$},
}

\newglossaryentry{not:Xsing}{
  name={$X_{\mathit{sing}}$},
  description={singular points of $X$},
}

\newglossaryentry{not:dimpX}{
  name={$\dim_p X$},
  description={dimension of $X$ at $p$},
}
\newglossaryentry{not:dimX}{
  name={$\dim X$},
  description={dimension of $X$},
}

\newglossaryentry{not:Ustar}{
  name={$U^*$},
  description={complex conjugate of a domain},
}

\newglossaryentry{not:Segrevar}{
  name={$\Sigma_q(U,r)$, $\Sigma_q$},
  description={Segre variety},
}

\newglossaryentry{not:Bergmanker}{
  name={$K_U(z,\bar{\zeta})$},
  description={Bergman kernel of $U$},
}

\newglossaryentry{not:Szegoker}{
  name={$S_U(z,\bar{\zeta})$},
  description={Szeg{\"o} kernel of $U$},
}

\newglossaryentry{not:pathint}{
  name={$\displaystyle \int_\gamma f(z) \,dz$},
  description={path integral},
}

\newglossaryentry{not:graph}{
  name={$\Gamma_f$},
  description={graph of $f$},
}

\newglossaryentry{not:wedge}{
  name={$\omega \wedge \eta$},
  description={wedge product of differential forms},
}

\newglossaryentry{not:pairing}{
  name={$\langle \omega , v \rangle$},
  description={pairing of a form and a vector (evaluating $\omega$ on $v$)},
}

\newglossaryentry{not:boundary}{
  name={$\partial X$},
  description={boundary of $X$ (topological, or manifold)},
}

\newglossaryentry{not:function}{
  name={$f \colon X \to Y$},
  description={a function from $X$ to $Y$},
}


\newglossaryentry{not:mapsto}{
  name={$x \mapsto F(x)$},
  description={a function of $x$},
}

\newglossaryentry{not:restriction}{
  name={$f|_S$},
  description={restriction of $f$ to $S$},
}

\newglossaryentry{not:setminus}{
  name={$A \setminus B$},
  description={set subraction},
}

\newglossaryentry{not:closure}{
  name={$\overline{S}$},
  description={topological closure},
}

\newglossaryentry{not:definition}{
  name={$X \overset{\text{def}}{=} Y$},
  description={define $X$ to be $Y$},
}

\newglossaryentry{not:dolbeault}{
  name={$H^{(p,q)}(U)$},
  description={Dolbeault cohomology groups},
}