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Show that each function below is non-analytic by identifying two independent directions, along which the rates of change of the function are different: (a) Re z

I don't fully understand this solution. Can anyone offer any other explanations that might make this type of question more clear to me? I have only started learning about analytic and non analytic function.

What does it mean by identify two independent directions?

original solution = (a) f(z) = Re z = x. At a given z, f(z) varies with x, i.e. along the direction y = const, but is constant along x = const. Thus it is not differentiable (in the sense of differentiability in complex analysis) at any z. Hence, it is nowhere analytic

nixluke
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1 Answers1

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Let $z_0 \in \mathbb C$.

Along the direction $y=const.$ means: consider $f$ on $\{z_0+t: t \in \mathbb R\}.$

Along the direction $x=const.$ means: consider $f$ on $\{z_0+it: t \in \mathbb R\}.$

Fred
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