\\subsection{Holomorphic Map}
\n\\begin{defn}
\n A complex-valued function $f(z)$ defined on a connected open domain $W\\subseteq\\C^n$ is called \\iemph{holomorphic} if for each $a=(a_1,\\ldots, a_n)\\in W$, $f(z)$ can be represented as a power series
\n \\[
\n \\sum_{k_1\\geq0,\\cdots,k_n\\geq0}^\\infty C_{k_1\\cdots k_n}(z_1-a_1)^{k_1}\\cdots (z_n-a_n)^k_n
\n \\]
\n in some neighbourhood of $a$.
\n\\end{defn}
\nWe have the following equivalent definition
\n\\begin{defn}
\n Let $f(z)$ be a (continuously) differentiable function on an open set $W\\subseteq\\C^n$. Then $f(z)$ is \\emph{holomorphic} iff $\\frac{\\pt{f}}{\\pt\\bar z_\\nu}=0$, $1\\leq\\nu\\leq n$.
\n\\end{defn}
\n\\begin{proof}
\n The proof is based on the Cauchy integral theorem and called “\\iemph{Osgood Theorem}”, cf. \\cite{morrow1971complex} for detail.
\n\\end{proof}
\nRecall that a complex-valued function of $n$ complex variables can be considered as a function of $2n$ real variables, since $\\C^n\\simeq \\R^{2n}$, related by $z_\\nu=x_\\nu+i y_\\nu$, $i=\\sqrt{-1}$, $x_\\nu,y_\\nu\\in\\R$. We have the following
\n\\begin{alignat*}{2}
\n\\rd z_\\nu&=\\rd x_\\nu+i\\rd y_\\nu,
\n&\\qquad
\n\\rd \\bar z_\\nu&=\\rd x_\\nu-i\\rd y_\\nu\\\\
\n\\ppt{z_\\nu}&=\\frac{1}{2}\\left(\\ppt{x_\\nu}-i\\ppt{y_\\nu}\\right),
\n&\\qquad
\n\\ppt{\\bar z_\\nu}&=\\frac{1}{2}\\left(\\ppt{x_\\nu}+i\\ppt{y_\\nu}\\right)\\\\
\n\\rd x_\\nu&=\\frac{1}{2}(\\rd z_\\nu+\\rd\\bar z_\\nu),
\n&\\qquad
\n\\rd y_\\nu&=\\frac{1}{2i}(\\rd z_\\nu-\\rd\\bar z_\\nu)\\\\
\n\\ppt{x_\\nu}&=\\ppt{z_\\nu}+\\ppt{\\bar z_\\nu},
\n&\\qquad
\n\\ppt{y_\\nu}&=i\\left(\\ppt{z_\\nu}-\\ppt{\\bar z_\\nu}\\right).
\n\\end{alignat*}
\nWith the help of the above relation, the \\iemph{total differential} of a complex-$n$ valued function $f$ is
\n\\begin{align*}
\n \\rd f &=\\frac{\\pt f}{\\pt x_\\nu}\\rd x_\\nu+\\frac{\\pt f}{\\pt y_\\nu}\\rd y_\\nu\\\\
\n &=\\left(
\n \\frac{\\pt f}{\\pt z_\\nu}+\\frac{\\pt f}{\\pt \\bar z_\\nu}\\right)\\frac{1}{2}(\\rd z_\\nu+\\rd\\bar z_\\nu)+
\n i\\left(
\n \\frac{\\pt f}{\\pt z_\\nu}-\\frac{\\pt f}{\\pt \\bar z_\\nu}\\right)
\n \\frac{1}{2i}(\\rd z_\\nu-\\rd\\bar z_\\nu)\\\\
\n &=\\frac{\\pt f}{\\pt z_\\nu}\\rd z_\\nu+\\frac{\\pt f}{\\pt \\bar z_\\nu}\\rd \\bar z_\\nu\\\\
\n &\\eqdef \\pt f+\\bar\\pt f.
\n\\end{align*}
\nWe shall call $\\pt f$ and $\\bar\\pt f$ the \\emph{holomorphic part}\\index{holomorphic!part} and \\emph{Anti-holomorphic part}\\index{holomorphic!Anti-!part} of $f$, respectively. \\begin{rem}
\n $\\bar\\pt f=0$ is the \\iemph{Cauchy-Riemann Equation}.
\n\\end{rem} <\/p>\n","protected":false},"excerpt":{"rendered":"
\\subsection{Holomorphic Map} \\begin{defn …<\/p>\n