Analytic functions · Definition

Cauchy-Riemann equations

complex analysiscomplex differentiation

Before you start you need to know

They translate complex differentiability into the language of real functions of two variables, and are used to prove or rule out that a function is differentiable.

Definition

The two equalities between the partial derivatives of the real part \(u\) and the imaginary part \(v\) of a function \(f=u+iv\) that is differentiable at a point.

\[u_x=v_y,\qquad u_y=-v_x\]

If \(f=u+iv\) is differentiable at \(z_0=x_0+iy_0\), the limit of the difference quotient is the same when \(h\) is real and when \(h=ik\) is purely imaginary. With real \(h\) one gets \(f'(z_0)=u_x+iv_x\); with \(h=ik\) one gets \(f'(z_0)=v_y-iu_y\). Equating real and imaginary parts gives \(u_x=v_y\) and \(u_y=-v_x\).

As a consequence, the derivative can be computed with partial derivatives: \(f'=u_x+iv_x\). For instance, for \(f(z)=z^2\) we have \(u=x^2-y^2\) and \(v=2xy\), so \(u_x=2x=v_y\) and \(u_y=-2y=-v_x\); and \(f'=2x+2iy=2z\).

The equations are necessary but not always sufficient: if they hold at \(z_0\) and \(u\) and \(v\) are also (real) differentiable at \(z_0\), for example when they have continuous partial derivatives, then \(f\) is differentiable at \(z_0\). Without that extra hypothesis the implication can fail.

They are used mostly to rule things out: \(f(z)=\bar z\) has \(u_x=1\) and \(v_y=-1\), so it is nowhere differentiable, and \(|z|^2\) satisfies them only at the origin. A typical mistake is forgetting the minus sign in the second equation.

Think about it

If \(u_x=v_y\) and \(u_y=-v_x\) at a point, why is that not enough for \(f\) to be differentiable there?

Because the equations only compare two directions (horizontal and vertical); the complex derivative needs the same limit in all of them, which is guaranteed if \(u\) and \(v\) are also differentiable at the point.

What does \(f'=u_x+iv_x\) say about \(|f'|^2\)?

That \(|f'|^2=u_x^2+v_x^2=u_xv_y-u_yv_x\), the Jacobian determinant of \((u,v)\) with respect to \((x,y)\).

Exercises 10

  1. For \(f(z)=z^2\) we have \(u=x^2-y^2\) and \(v=2xy\). What is \(u_x\)?
  2. Which is the correct pair of Cauchy-Riemann equations?
  3. For \(f(z)=e^z\), with \(u=e^x\cos y\) and \(v=e^x\sin y\), what is \(v_y\)?
  4. For \(f(z)=\bar z\), with \(u=x\) and \(v=-y\), at which points does \(u_x=v_y\) hold?
  5. For \(f(z)=|z|^2\), with \(u=x^2+y^2\) and \(v=0\), at which points do the Cauchy-Riemann equations hold?
  6. If \(f=u+iv\) is differentiable at \(z\), then \(f'(z)=u_x+iv_x\). Compute \(f'(1+i)\) for \(f(z)=z^2\).
  7. Let \(u=x^2-y^2\). Which \(v\), with \(v(0,0)=0\), makes \(u\) and \(v\) satisfy Cauchy-Riemann?
  8. Let \(u=e^x\cos y\). Which \(v\) satisfies Cauchy-Riemann together with \(u\)?
  9. If \(f=u+iv\) satisfies the Cauchy-Riemann equations at a point, what can be asserted?
  10. For \(u=x^2\) and \(v=y^2\), where do the Cauchy-Riemann equations hold?

What comes next

References

  • Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 1, §2.3
  • James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §21–§23
  • Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 2, §1.3