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Analytic Functions

Source lecture(s): PHY622 Lec3

Intuition

A function is analytic if it is differentiable everywhere in a neighborhood—this forces it to be smooth and conformal.

Formal Definition

\(f(z)\) is analytic at \(z_0\) if \(\lim_{z\to z_0}(f(z)-f(z_0))/(z-z_0)\) exists and is unique.

Mathematical Formulation

Cauchy-Riemann: \(u_x=v_y\) and \(u_y=-v_x\) for \(f=u+iv\). Analytic \(\implies\) \(u_{xx}+u_{yy}=0\) (harmonic).

Derivation

Write \(f(z+\Delta z)-f(z)=\Delta z f'(z)+o(\Delta z)\). Separate real and imaginary axes; equating the two limits gives the CR equations.

Worked Example

\(f(z)=z^2\) has \(u=x^2-y^2\), \(v=2xy\), so \(u_x=v_y=2x\) and \(u_y=-v_x=-2y\).

Common Mistakes

  • Checking differentiability only along one path.
  • Confusing 'analytic' with 'entire'.

Quiz

Q1. If \(f\) is analytic and \(|f|\) is constant, what is \(f\)?

Answer

\(f\) is constant.