A complex function $f\!: D \to \mathbb{C}$ , where $D$ is a domain of the complex plane, having the derivative $$\frac{df}{d \overline{z}}$$ in each point $z$ of $D$ , is said to be antiholomorphic in $D$ .

The following conditions are equivalent:

• $f(z)$ is antiholomorphic in $D$ .
• $\overline{f(z)}$ is holomorphic in $D$ .
• $f(\overline{z})$ is holomorphic in $\overline{D} \,:=\, \{\overline{z}\;\vdots\;\, z \in D\}$ .
• $f(z)$ may be expanded to a power series $\sum_{n=0}^\infty a_n(\overline{z}-u)^n$ at each $u \in D$ .
• The real part $u(x,\,y)$ and the imaginary part $v(x,\,y)$ of the function $f$ satisfy the equations $$\frac{\partial u}{\partial x} \;=\; -\frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} \;=\; \frac{\partial v}{\partial x}.$$ N.B. the place of minus; cf. the Cauchy-Riemann equations.
  

Example. The function $\displaystyle z \mapsto \frac{1}{\overline{z}}$ is antiholomorphic in $\mathbb{C}\!\smallsetminus\!\{0\}$ . One has $$f(z) \,=\, \frac{z}{|z|^2} \,=\, \underbrace{\frac{x}{x^2+y^2}}_{u}+i\underbrace{\frac{y}{x^2+y^2}}_{v},$$ $$\frac{\partial u}{\partial x} \;=\; \frac{y^2\!-\!x^2}{(x^2\!+\!y^2)^2}, \quad \frac{\partial v}{\partial y} \;=\; \frac{x^2\!-\!y^2}{(x^2\!+\!y^2)^2}, \quad \frac{\partial u}{\partial y} \;=\; -\frac{2xy}{(x^2\!+\!y^2)^2}, \quad \frac{\partial v}{\partial x} \;=\; -\frac{2xy}{(x^2\!+\!y^2)^2}.$$