Advanced Engineering Mathematics, 7th Edition

17.5 Cauchy–Riemann Equations

INTRODUCTION

In the preceding section we saw that a function f of a complex variable z is analytic at a point z when f is differentiable at z and differentiable at every point in some neighborhood of z. This requirement is more stringent than simply differentiability at a point because a complex function can be differentiable at a point z but yet be differentiable nowhere else. A function f is analytic in a domain D if f is differentiable at all points in D. We shall now develop a test for analyticity of a complex function f(z) = u(x, y) + iv(x, y).

A Necessary Condition for Analyticity

In the next theorem we see ...