Transcription of LECTURE 2: COMPLEX DIFFERENTIATION AND CAUCHY
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LECTURE 2: COMPLEX DIFFERENTIATION AND CAUCHY
LECTURE 2: COMPLEX DIFFERENTIATION AND CAUCHYRIEMANN EQUATIONSWe have seen in the first LECTURE that the COMPLEX derivative of a functionfat apointz0is defined as the limitf (z0) = limh 0f(z0+h) f(z0)h,whenever the limit exist. We have also seen two examplesi) iff(z) =z2thenf (z) = 2z, ii) the functionf(z) =zis not a differentiable function. Now we willgo for a detail differentiable atz0thenfis continuous (z0) = limz z0f(z) f(z0)z z0it follows thatlimz z0f(z) = limz z0f(z) f(z0)z z0(z z0) +f(z0) =f(z0).
LECTURE 2: COMPLEX DIFFERENTIATION AND CAUCHY RIEMANN EQUATIONS 3 (1) If f : C → C is such that f0(z) = 0 for all z ∈ C, then f is a constant function. This is because, by CR equation u x = u y = v x = v y = 0. So by MVT of two variable calculus u and v are constant function and hence so is f.
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