Complex derivative
Definition. Let \(U\subset\mathbb C\) be open. A function \(f:U\to\mathbb C\) is complex differentiable at \(z_0\in U\) if \[ f'(z_0)=\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h} \] exists with \(h\in\mathbb C\). The key point is that the limit must be independent of every complex direction.
Cauchy-Riemann equations
Theorem. Write \(f=u+iv\) with \(u,v\) real-valued. If \(f\) is complex differentiable and \(u,v\) are real differentiable at \((x,y)\), then \(u_x=v_y\) and \(u_y=-v_x\). Conversely, if the partial derivatives are continuous near the point and satisfy these equations, then \(f\) is holomorphic there.
Real-linear proof
Proof. The derivative of \(f\) as a real map has matrix \(\begin{pmatrix}u_x&u_y\\ v_x&v_y\end{pmatrix}\). Complex differentiability means this real-linear map commutes with multiplication by \(i\), whose matrix is \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\). Equating the two products gives the Cauchy-Riemann equations.
Holomorphic functions
Definition. A function is holomorphic on \(U\) if it is complex differentiable at every point of \(U\). It is analytic if it has a convergent power-series expansion near every point. Cauchy's theory will prove that these two notions coincide.
Basic examples
Example. Polynomials, rational functions away from poles, \(e^z\), \(\sin z\), and \(\cos z\) are holomorphic. The conjugation map \(z\mapsto\overline z\) is real differentiable but not complex differentiable at any point, since the real and imaginary directional quotients disagree.
Local rigidity
Remark. One complex derivative imposes two real first-order equations. This overdetermined nature is why holomorphic functions are much more rigid than general smooth functions of two real variables.
Real differentiability and Wirtinger derivatives
Write \(f=u+iv\), \(z=x+iy\). If \(f:\Omega\subset\mathbb C\to\mathbb C\) is differentiable as a map \(\mathbb R^2\to\mathbb R^2\), its differential has the decomposition \[ \mathrm{d}f =f_z\,\mathrm{d}z+f_{\bar z}\,\mathrm{d}\bar z, \] where \[ f_z=\frac12(f_x-if_y),\qquad f_{\bar z}=\frac12(f_x+if_y). \]
Theorem. A real-differentiable function \(f\) is complex differentiable at \(z_0\) if and only if \[ f_{\bar z}(z_0)=0, \] equivalently \[ u_x=v_y,\qquad u_y=-v_x \] at \(z_0\). In that case its complex derivative is \(f'(z_0)=f_z(z_0)=u_x+iv_x\).
Proof. Real differentiability gives \[ f(z_0+h)-f(z_0) =f_z(z_0)h+f_{\bar z}(z_0)\bar h+o(|h|). \] After division by \(h\), the second term becomes \(f_{\bar z}\bar h/h\), whose limit is independent of the direction of \(h\) only when \(f_{\bar z}=0\). Expanding that equation into real and imaginary parts gives the Cauchy--Riemann equations.
When the Cauchy--Riemann equations suffice
The equations at a single point do not imply complex differentiability unless real differentiability is known. A standard sufficient hypothesis is that \(u,v\) have first partial derivatives in a neighborhood, continuous at the point, and satisfy Cauchy--Riemann there.
Counterexample. Define \[ f(z)= \begin{cases} \dfrac{x^3}{x^2+y^2},&z\ne0,\\ 0,&z=0. \end{cases} \] Both partial derivatives at \(0\) vanish and hence satisfy Cauchy--Riemann, but \(f(t(1+i))/\{t(1+i)\}=1/\{2(1+i)\}\), while along the imaginary axis the quotient is \(0\). The complex derivative does not exist.
Conformality and the Jacobian
Proposition. If \(f\) is holomorphic near \(z_0\) and \(f'(z_0)\ne0\), its real differential is multiplication by \(f'(z_0)\); it scales lengths by \(|f'(z_0)|\) and rotates oriented angles by \(\arg f'(z_0)\). Moreover, \[ \det D_{\mathbb R}f(z_0)=|f'(z_0)|^2>0. \]
Proof. With \(f'=a+ib\), the real Jacobian is \[ \begin{pmatrix}a&-b\\b&a\end{pmatrix} =|f'| \begin{pmatrix}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{pmatrix}. \] This is a positive scaling followed by a rotation.
At a zero of \(f'\), conformality fails. If \(f(z)-f(z_0)=a(z-z_0)^m+o(|z-z_0|^m)\), \(m\ge2\), angles are multiplied by \(m\) and the map is locally \(m\)-to-one away from the branch point.
Holomorphicity is stronger than smoothness
A \(C^1\) solution of \(f_{\bar z}=0\) is holomorphic; Cauchy's theorem then implies it has derivatives of every order and a convergent local power series. Thus one first-order elliptic equation forces analyticity. This rigidity is the organizing fact behind the rest of the note.