Complex differentiability and the Cauchy–Riemann equations

Let f be defined on an open neighborhood of a complex point z₀. Its derivative is the limit of $(f(z_0+h)-f(z_0))/h$ as nonzero complex h tends to zero through arbitrary directions. The same limit must hold for horizontal, vertical, and curved approaches. For $f(z)=z^2$, expansion gives the quotient $2z_0+h$, which tends to $2z_0$ independently of direction. For conjugation, the quotient is $\overline h/h$, which equals one on real increments and minus one on purely imaginary increments.

Multiplication by a complex number α is a very special real-linear transformation. If α=a+ib, it scales all lengths by its modulus and, when α is nonzero, rotates every direction by its argument. Its real matrix has equal diagonal entries and opposite off-diagonal entries. A general Jacobian can shear or stretch two directions differently and therefore cannot represent complex multiplication. The zero multiplier is allowed for differentiability, although it does not preserve angles as a nondegenerate local map.

Write $f(x+iy)=u(x,y)+iv(x,y)$. Horizontal increments identify the derivative with $u_x+iv_x$, while vertical increments identify it with $v_y-iu_y$. Equality yields the Cauchy–Riemann equations. Complex differentiability implies real differentiability and these equations at the point. In the converse direction, real differentiability at that point together with the equations is sufficient: the real-linear approximation then has exactly the complex-multiplication form, and its remainder is small compared with the increment length.

A convenient stronger hypothesis is that u and v have continuous first partial derivatives on an open set. If the equations hold throughout that set, the map is complex differentiable there and hence holomorphic. Existence of partial derivatives and a pointwise algebraic check alone is not that theorem; partial derivatives need not assemble into a valid first-order approximation. Check regularity and the set on which the equations hold before drawing a conclusion about an open neighborhood.

The function $f(z)=|z|^2$ has derivative zero at the origin because its difference quotient there is $\overline h$, which tends to zero. Away from the origin its Cauchy–Riemann equations fail. Thus the function is differentiable at one point but holomorphic on no nonempty open set. This example separates the definition of a derivative from the neighborhood condition needed for Cauchy theory, power-series representations, and the other global consequences developed later.

Holomorphic means complex differentiable at every point of an open set. A theorem of complex analysis then shows that the function has derivatives of every order and a convergent local power series; those conclusions are not extra assumptions hidden in the definition. For the moment, a reliable workflow is to state the open domain, inspect singularities or conjugates, establish real regularity, and test the equations there. This preserves both the power and the limits of the coordinate calculation.