= Holomorphic function
{title2=$f\in\mathcal O(U)$}
{wiki}
= Holomorphy
{synonym}
= Holomorphically
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= Holomorphic
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= Analytic function
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A complex-valued <function> $f:U\to\mathbb C$ is holomorphic on an <open set> $U$ when it has a complex <derivative> at every point of $U$. Requiring a <derivative> throughout a neighbourhood is much stronger than <complex differentiability at a point>. It forces locally convergent <Taylor series> and <derivatives> of every order, as follows from the <Cauchy integral formula> and <Cauchy derivative formula>.
To see the restriction in real coordinates, write $z=x+iy$ and $f(z)=u(x,y)+iv(x,y)$. The real and imaginary increment quotients must agree, giving the <Cauchy-Riemann equations>
$$
u_x=v_y,\qquad u_y=-v_x,\qquad f'=u_x+iv_x.
$$
Conversely, if $u,v$ have continuous <partial derivatives> on $U$ and satisfy the <Cauchy-Riemann equations> there, then $f$ is a <holomorphic function>. The equations at a lone point, without a differentiability hypothesis for the real coordinate map, are insufficient. Their role is to force the <Jacobian matrix> to have the special form
$$
Df=\begin{pmatrix}u_x&u_y\\v_x&v_y\end{pmatrix}=\begin{pmatrix}p&-q\\q&p\end{pmatrix},\qquad f'=p+iq.
$$
Such a <matrix> represents multiplication by a <complex number>; it has only two free real entries. <Complex Variables with Applications, MIT 18.04, Spring 2018/Topic 2 Notes>[Complex Variables with Applications, MIT 18.04, Spring 2018; Orloff, MIT Topic 2, sections 2.7.2–2.7.4]{ref}
For $f(z)=z^2$, $u=x^2-y^2$ and $v=2xy$, so $u_x=v_y=2x$ and $u_y=-v_x=-2y$. Directly,
$$
f(z_0+h)-f(z_0)=2z_0h+h^2,\qquad f'(z_0)=2z_0.
$$
At $z_0=1+i$, multiplication by $2+2i$ scales lengths by its <modulus> $2\sqrt2$ and rotates by its <argument> $\pi/4$. A horizontal increment $h=0.01$ gives the exact displacement $0.0201+0.02i$; a vertical increment $h=0.01i$ gives $-0.0201+0.02i$. The first-order predictions $0.02+0.02i$ and $-0.02+0.02i$ are perpendicular, and the errors have <modulus> $10^{-4}$.
At every nonzero $z_0$, this local rotation and scaling makes $z^2$ a local <conformal map>. At zero, the first-order map vanishes: $re^{i\theta}$ becomes $r^2e^{2i\theta}$, so the <argument> doubles and the origin is a <critical point>. Globally $z$ and $-z$ have the same square. Local preservation of <angles> therefore coexists with global folding, a useful warning when using <holomorphic functions> as coordinate changes.
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