Complex Analysis 8 | Wirtinger Derivatives [dark version]

TL;DR

Wirtinger derivatives define two operators: ∂/∂z=(1/2)(∂/∂x−i∂/∂y) and ∂/∂z̄=(1/2)(∂/∂x+i∂/∂y).

Briefing Cornell Notes

Briefing

Wirtinger derivatives turn complex differentiation into two independent partial-derivative operators—one with respect to z and one with respect to z̄—making holomorphicity a clean algebraic condition. For a complex function f(x+iy)=U(x,y)+iV(x,y), the complex derivative can be recovered from these operators, while holomorphic functions are exactly those for which the “conjugate” derivative vanishes. This matters because it converts the Cauchy–Riemann equations into a single test: check whether ∂f/∂z̄ equals 0.

Starting from a holomorphic function f on an open set U, the discussion introduces two differential operators built from partial derivatives in the real variables x and y. The first operator is defined as

∂/∂z := (1/2) (∂/∂x − i ∂/∂y),

and the second as

∂/∂z̄ := (1/2) (∂/∂x + i ∂/∂y).

The key payoff is that these definitions are engineered so that, for holomorphic f, the conjugate derivative ∂f/∂z̄ becomes 0—this is essentially the Cauchy–Riemann equations packaged into operator form. The usual complex derivative f′(z) can then be computed using ∂f/∂z.

To connect the operators to the real and imaginary parts, the transcript rewrites the complex derivative at a point z=x+iy as a+ib, where a and b are expressed using partial derivatives of U and V. Using the Cauchy–Riemann relations (∂U/∂x=∂V/∂y and ∂V/∂x=−∂U/∂y), the calculation shows how the x- and y-derivatives combine into the compact forms above. A helpful viewpoint is to treat f as a map from (x,y) to U+iV, so that differentiating with respect to x or y directly feeds into the z and z̄ operators.

An explicit example seals the idea. For f(z)=z^2, expanding (x+iy)^2 gives U=x^2−y^2 and V=2xy. Computing the derived operators yields ∂f/∂z = 2z and ∂f/∂z̄ = 0. The vanishing of ∂f/∂z̄ matches the expectation that polynomials in z alone are holomorphic.

The final takeaway is an equivalence that’s easy to remember and use: a complex function f on an open domain U is holomorphic if and only if ∂f/∂z̄=0 at every point. When that condition holds, the complex derivative f′(z) is given by ∂f/∂z. This reframes holomorphicity as a single operator equation rather than a pair of real PDEs.

Cornell Notes

Wirtinger derivatives split complex differentiation into two operators built from ∂/∂x and ∂/∂y: ∂/∂z=(1/2)(∂/∂x−i∂/∂y) and ∂/∂z̄=(1/2)(∂/∂x+i∂/∂y). For a function f(z)=U(x,y)+iV(x,y), these operators encode the Cauchy–Riemann equations. In particular, holomorphicity is equivalent to the single condition ∂f/∂z̄=0 everywhere on the domain. When that holds, the complex derivative is obtained from ∂f/∂z. The example f(z)=z^2 confirms the pattern: ∂f/∂z=2z while ∂f/∂z̄=0.

How are Wirtinger derivatives defined in terms of x and y partial derivatives?

They are defined by combining real partial derivatives into two complex-direction operators: ∂/∂z=(1/2)(∂/∂x−i∂/∂y) and ∂/∂z̄=(1/2)(∂/∂x+i∂/∂y). These definitions are chosen so that the Cauchy–Riemann conditions appear automatically when applying them to f=U+iV.

Why does ∂f/∂z̄ vanish for holomorphic functions?

Writing f=U+iV and applying ∂/∂z̄=(1/2)(∂/∂x+i∂/∂y) produces a combination of U_x, U_y, V_x, and V_y. The Cauchy–Riemann equations link these derivatives (V_x=−U_y and V_y=U_x), so the combined terms cancel. The result is ∂f/∂z̄=0 exactly when f is holomorphic.

What is the relationship between ∂f/∂z and the usual complex derivative f′(z)?

For holomorphic f, the operator ∂/∂z=(1/2)(∂/∂x−i∂/∂y) yields the complex derivative at the point: ∂f/∂z = f′(z). In the transcript’s example, f(z)=z^2 gives ∂f/∂z=2z, matching the standard derivative.

How does the example f(z)=z^2 illustrate both Wirtinger derivatives?

Expanding (x+iy)^2 gives U=x^2−y^2 and V=2xy. Computing the operator results gives ∂f/∂z=2z and ∂f/∂z̄=0. The zero conjugate derivative confirms holomorphicity, while the nonzero ∂f/∂z reproduces the expected derivative.

What is the practical holomorphicity test using Wirtinger derivatives?

A function f is holomorphic on an open domain U if and only if ∂f/∂z̄=0 at every point in U. This replaces the pair of Cauchy–Riemann equations with a single operator equation.

Review Questions

State the formulas for ∂/∂z and ∂/∂z̄ in terms of ∂/∂x and ∂/∂y.
What condition on ∂f/∂z̄ is equivalent to holomorphicity?
For f(z)=z^2, what are ∂f/∂z and ∂f/∂z̄?

Key Points

1
Wirtinger derivatives define two operators: ∂/∂z=(1/2)(∂/∂x−i∂/∂y) and ∂/∂z̄=(1/2)(∂/∂x+i∂/∂y).
2
For f=U+iV, applying these operators packages the Cauchy–Riemann equations into operator form.
3
A function f is holomorphic on an open domain U if and only if ∂f/∂z̄=0 everywhere on U.
4
When f is holomorphic, the complex derivative satisfies f′(z)=∂f/∂z.
5
The example f(z)=z^2 yields ∂f/∂z=2z and ∂f/∂z̄=0, matching holomorphic expectations.
6
Wirtinger derivatives let complex differentiation be computed using real partial derivatives without separately solving two real PDEs.

Highlights

Holomorphicity becomes a single check: ∂f/∂z̄=0.

The operator ∂/∂z recovers the usual complex derivative f′(z) for holomorphic functions.

For f(z)=z^2, the conjugate derivative vanishes while ∂f/∂z equals 2z.

Wirtinger derivatives are engineered combinations of ∂/∂x and ∂/∂y that encode the Cauchy–Riemann equations. 

Topics

Wirtinger Derivatives
Holomorphicity Test
Cauchy Riemann Equations
Complex Differentiation
Example z Squared