Theorem 1   If $f \colon D \to \mathbb{C}$ and $g \colon D \to \mathbb{C}$ satisfy the Cauchy-Riemann equations, so does $f + g$ . Furthermore, if $z\in \mathbb{C}$ , then $zf$ satisfies the Cauchy-Riemann equations.

Proof. This is an immediate consequence of the linearity of derivatives.

Theorem 2   If $f \colon D \to \mathbb{C}$ and $g \colon D \to \mathbb{C}$ satisfy the Cauchy-Riemann equations, so does $f \cdot g$ .

Proof. Letting $h$ and $k$ denote the real and imaginary parts of $f \cdot g$ respectively, we have
and

Theorem 3   If $f \colon D \to D'$ and $g \colon D' \to \mathbb{C}$ satisfy the Cauchy-Riemann equations, so does $f \circ g$ .

Proof. Letting $h$ and $k$ denote the real and imaginary parts of $f \circ g$ respectively, we have
and

Theorem 4   If $f \colon D \to \mathbb{C}$ satisfies the Cauchy-Riemann equations, and has non-vanishing Jacobian, then $f^{-1}$ also satisfies the Cauchy-Riemann equations.

Proof. Let us denote the real and imaginary parts of $f^{-1}$ as $h$ and $k$ , respectively. Then, by definition of inverse function, we have
Taking derivatives,
By the Cauchy-Riemann equations, $\partial u / \partial h = \partial v / \partial k$ and $\partial u / \partial k = - \partial v / \partial h$ . Using these relations to re-express the derivatives of $u$ as derivatives of $v$ , then subtracting the fourth equation form the first equation and adding the second and third equations, we obtain
With a little algebraic manipulation, we may conclude
Note that, by the Cauchy-Riemann equations, the Jacobian of $f$ equals the common prefactor of these equations: $$ {\partial (u,v) \over \partial (h,k)} = {\partial u \over \partial h} {\partial v \over \partial k} - {\partial u \over \partial k} {\partial v \over \partial h} = \left( {\partial u \over \partial h} \right)^2 + \left( {\partial u \over \partial k} \right)^2 $$ Hence, by assumptions, this quantity differs from zero and we may cancel it to obtain the Cauchy-Riemann equations for $f^{-1}$ .