Semester 1 / Mathematics / Complex Analysis / Cauchy-Riemann Equations

Cauchy-Riemann Equations

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Suppose ff is a complex-valued function of a complex variable. If the derivatives are the same for the 2 paths —real and imaginary axes— then ff is analytic.

Suppose f(z)=u(x,y)+iv(x,y)f(z)=u(x,y)+iv(x,y) for the theorems below.

The Equations

The set of equations mentioned below are the Cauchy Riemann Equations, where u,vu,v are functions of x,yx,y.

Cartesian Form

∂u∂x=ux=∂v∂y=vy      ∧      ∂u∂y=uy=−∂v∂x=−vx\frac{\partial{u}}{\partial{x}} =u_x = \frac{\partial{v}}{\partial{y}} =v_y \;\;\; \land \;\;\; \frac{\partial{u}}{\partial{y}} =u_y = - \frac{\partial{v}}{\partial{x}} =-v_x

Polar Form

Here the partial derivatives are about r,θr, \theta.

ur=1rvθ      ∧      vr=−1ruθu_r = \frac{1}{r}v_\theta \;\;\; \land \;\;\; v_r = -\frac{1}{r}u_\theta

Complex Form

fx=−ifyf_x = -if_y

Theorem

If ff is differentiable at z0z_0, then

  • All partial derivatives ux,uy,vx,vyu_x,u_y,v_x,v_y exist and
  • They satisfy the Cauchy Riemann equations
f′(z0)=ux(x0,y0)+ivx(x0,y0)f'(z_0)=u_x(x_0,y_0) + iv_x(x_0,y_0)

Theorem

If:

  • All partial derivatives ux,uy,vx,vyu_x,u_y,v_x,v_y exist and
  • They satisfy Cauchy-Riemann equations and
  • They are continuous at z0z_0

Then:

  • ff is differentiable at z0z_0 and
f′(z0)=ux(x0,y0)+ivx(x0,y0)f'(z_0)=u_x(x_0,y_0) + iv_x(x_0,y_0)

Theorem

If ff is analytic at z0z_0, then its first-order partial derivatives are continuous in a neighbourhood of z0z_0.

Written by October 19, 2026 1 min read
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