Analytic functions · Definition

Holomorphic function

complex analysiscomplex differentiation

Before you start you need to know

It is the class of functions where most of complex analysis lives: differentiable on an open set, and therefore far more rigid than differentiable real functions.

Definition

A complex function \(f\) is holomorphic on an open set \(U\) of the plane if it is complex differentiable at every point of \(U\).

Differentiability is required on a whole open set, not at a single point. That is why \(|z|^2\), differentiable only at the origin, is holomorphic nowhere, while polynomials in \(z\), \(e^z\), \(\sin z\) and \(\cos z\) are holomorphic on all of \(\mathbb{C}\). The function \(1/z\) is holomorphic on \(\mathbb{C}\setminus\{0\}\). A function holomorphic on all of \(\mathbb{C}\) is called entire.

Sums, differences, products and compositions of holomorphic functions are holomorphic, and the quotient \(f/g\) is holomorphic where \(g\neq0\). The differentiation rules are those of real calculus, so \(((z^2+1)^3)'=6z(z^2+1)^2\).

In terms of \(u\) and \(v\), \(f=u+iv\) is holomorphic on \(U\) when it satisfies the Cauchy-Riemann equations at every point and \(u\) and \(v\) have continuous partial derivatives. An immediate consequence: if \(U\) is connected and \(\operatorname{Re}f\) is constant, so is \(f\).

It will be shown later that this seemingly first-order condition implies that \(f\) has derivatives of all orders and equals its Taylor series. A common mistake is to call holomorphic a function that is differentiable at a single point or only along a line.

Think about it

Is \(f(z)=\operatorname{Re} z\) holomorphic?

No: with \(u=x\) and \(v=0\) we have \(u_x=1\neq0=v_y\), so it satisfies Cauchy-Riemann at no point.

Why is \(\tan z=\sin z/\cos z\) not entire?

Because it is a quotient and \(\cos z\) vanishes at the points \(\pi/2+k\pi\); away from them it is holomorphic.

Exercises 10

  1. Which of these functions is holomorphic on all of \(\mathbb{C}\)?
  2. On which set is \(f(z)=1/z\) holomorphic?
  3. Saying that \(f\) is holomorphic on an open set \(U\) means that:
  4. The function \(f(z)=|z|^2\) is differentiable at \(z=0\). Is it holomorphic on an open set containing \(0\)?
  5. Compute \(f'(i)\) for the holomorphic function \(f(z)=z^2+3z\).
  6. If \(f\) and \(g\) are holomorphic on \(U\), which of these functions is not necessarily holomorphic on all of \(U\)?
  7. Let \(f=u+iv\) with \(u=x^2-y^2\) and \(v=axy\). For which value of \(a\) is \(f\) holomorphic?
  8. Compute \(f'(1)\) for \(f(z)=(z^2+1)^3\).
  9. If \(f\) is holomorphic on a connected open set \(U\) and \(u=\operatorname{Re}f\) is constant, what follows?
  10. For which pair \((u,v)\) is \(f=u+iv\) holomorphic on all of \(\mathbb{C}\)?

References

  • Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 1, §2.2
  • James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §24–§25
  • Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 2, §1.2