Holomorphic functions
Objective. Apply Cauchy-Riemann equations.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Holomorphic functions are complex functions differentiable at every point of an open set, and the Cauchy-Riemann equations give the precise test: writing f(x+iy) = u(x,y)+iv(x,y), f is holomorphic exactly when ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x, with these partials continuous. Holomorphic functions are automatically infinitely differentiable and equal their Taylor series locally, a much stronger property than real differentiability. For f(z) = z² = (x²−y²) + i(2xy), u = x²−y² and v = 2xy give ∂u/∂x = 2x = ∂v/∂y and ∂u/∂y = −2y = −∂v/∂x, confirming f is holomorphic everywhere.
Worked example
Verify whether f(z) = x² + y² + 2ixy is holomorphic by checking the Cauchy-Riemann equations.
- 1Identify u(x,y) = x² + y² and v(x,y) = 2xy.
- 2Compute ∂u/∂x = 2x and ∂v/∂y = 2x, which are equal, satisfying the first equation.
- 3Compute ∂u/∂y = 2y and −∂v/∂x = −2y, which do not match unless y = 0.
- 4Since the second Cauchy-Riemann equation fails generally, f is not holomorphic.
Answer: f is not holomorphic
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Start practisingCommon mistake. Students check the Cauchy-Riemann equations at a single point and conclude holomorphicity on a whole region; the equations must hold on an open neighbourhood, not just isolated points, for a function to be holomorphic there.
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