Differentiation Summary

Complex Analysis · Axiom Academy

SUMMARY Complex Differentiation Summary Everything from the complex-derivative unit: Cauchy–Riemann, analyticity, and harmonic functions, in one place. A complex derivative must give the same limit from every direction — a far stricter demand than a real derivative, so it carries far more information. The Cauchy–Riemann equations u_x = v_y and u_y = -v_x are the working test for differentiability: necessary always, and sufficient when the partials are continuous. A function differentiable on a whole neighborhood is analytic (holomorphic) — and analytic is enormous: it forces infinite differentiability and agreement with its own Taylor series. The real and imaginary parts of an analytic function are harmonic : each satisfies Laplace's equation . Core Concept The Complex Derivative Same difference quotient as in calculus, but along any path in the plane. The single limit must agree for all of them, which is what makes complex differentiability so demanding. When to use: the power, product, quotient, and chain rules all carry over unchanged. Watch out for: "differentiable at a point" is weaker than "analytic" — you need a whole neighborhood. Core Concept Cauchy–Riemann Equations Writing , these two equations are equivalent to the limit existing. They are necessary wherever f is differentiable, and sufficient when u,v have continuous first partials. When to use: the fastest check for "is this function differentiable here?"

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