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 Incoming links: Inhomogeneous Wirtinger equation with holomorphic forcing

Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2012 / 5 / b / ii / Solution 2026-10-31

The continuously differentiable function

z \overset{z}{ˉ} = x^{2} + y^{2}

is a particular solution, since

\frac{\partial}{\partial z ˉ} (x^{2} + y^{2}) = \frac{1}{2} (2 x + i, 2 y) = z .

If

f

is any solution, set

h (z) = f (z) - z \overset{z}{ˉ}

. Then

h

is continuously differentiable and its Wirtinger derivative vanishes. By the preceding part and the sufficient Cauchy-Riemann equations criterion,

h

is a holomorphic function on all of

C

, hence an entire function.

Conversely, every entire function

h

has vanishing Wirtinger derivative, so addition of

h

preserves the prescribed derivative. Therefore all solutions are

f (z) = ∣ z ∣^{2} + h (z), h any entire function .

This is an inhomogeneous Wirtinger equation with holomorphic forcing: a particular solution plus an arbitrary element of the homogeneous solution space.

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