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Cauchy–Riemann: satisfied at a point, holomorphic nowhere

Analytic FunctionsComplex Analysis, Algebra & Topologyfree

Why the equations alone are not enough, and the extra hypothesis that makes them work.

Cauchy–Riemann: satisfied at a point, holomorphic nowhereinteractivefree

Step 1 / 4The equations

Writing f=u+ivf = u+iv, complex differentiability at a point requires

ux=vy,uy=−vx.u_x = v_y, \qquad u_y = -v_x.

These are necessary. They are not, on their own, sufficient.

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