Analytic functions · Definition

Complex derivative

complex analysiscomplex differentiation

Before you start you need to know

It is written like the real derivative, but requires the limit to exist when approaching from the whole plane, which is far more restrictive.

Definition

The derivative \(f'(z_0)\) of a complex function \(f\) at a point \(z_0\) is the limit of the difference quotient as \(z\) approaches \(z_0\) from any direction in the plane, if that limit exists.

\[f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}\]

Putting \(z=z_0+h\), the derivative is the limit of \(\frac{f(z_0+h)-f(z_0)}{h}\) as \(h\to0\) with \(h\) complex. Unlike the real line, there are infinitely many directions of approach (real, imaginary, spiral…), and the limit must be the same along all of them. This requirement is what makes complex theory so rigid.

The rules of real calculus are proved by the same arguments and still hold: \((z^n)'=nz^{n-1}\), \((fg)'=f'g+fg'\), \((f/g)'=(f'g-fg')/g^2\), the chain rule and \((e^{z})'=e^{z}\). Also, differentiable at \(z_0\) implies continuous at \(z_0\), but not conversely.

The key example is \(f(z)=\bar z\): the quotient at any point is \(\bar h/h\), which is \(1\) for real \(h\) and \(-1\) for purely imaginary \(h\), so there is no limit. It is continuous everywhere and differentiable nowhere. Likewise \(|z|^2\) is differentiable only at the origin. Looking at the quotient along real and imaginary \(h\) is how the Cauchy-Riemann equations appear.

A frequent mistake is to think that, because \(f(x+iy)=u+iv\) has partial derivatives, it is complex differentiable; that is not enough. Being differentiable on a whole open set is a very strong condition, and it is the starting point of holomorphic functions.

Think about it

Why is \(f(z)=\bar z\) continuous but not differentiable?

Because \(|\bar z-\bar z_0|=|z-z_0|\) gives continuity, but the quotient \(\bar h/h\) equals \(1\) and \(-1\) depending on the direction of \(h\), so it has no limit.

If \(f'(z_0)\) exists, can it depend on the direction in which \(h\to0\)?

No: by definition the limit is unique and the same along every direction; if it changed with direction, the derivative would not exist.

Exercises 10

  1. Compute \(f'(3)\) for \(f(z)=z^2\).
  2. If \(f(z)=z^5\), what is \(f'(1)\)?
  3. Compute \(f'(2)\) for \(f(z)=\dfrac1z\).
  4. What is the derivative of \(e^{z}\) at \(z=0\)?
  5. If \(f\) is differentiable at \(z_0\), what follows?
  6. Compute \(f'(1+i)\) for \(f(z)=z^3-2z^2+5z\).
  7. Compute \(g'(0)\) for \(g(z)=z\,e^{z}\).
  8. Compute \(h'(1)\) for \(h(z)=\dfrac{z^2}{z+1}\).
  9. For \(f(z)=\bar z\) at \(z_0=2\), what happens to the difference quotient?
  10. For \(f(z)=|z|^2\), at which points does \(f'(z)\) exist?

References

  • Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 1, §2.1
  • James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §19–23