CVD1

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Show that \frac{d}{dz}\overline{z}\, is non-analytic everywhere.

\frac{d}{dz}\overline{z} = \lim_{\Delta z \rightarrow 0}\ \frac{\overline{z+\Delta z}-\overline{z}}{\Delta z}= \lim_{\Delta x ,y \rightarrow 0}\frac{\overline{x+\Delta x+i(y+\Delta y)}-\overline{x+iy}}{\Delta x + i\Delta y}\,

\lim_{\Delta x ,y \rightarrow 0}\frac{\overline{\Delta x+i\Delta y}}{\Delta x + i\Delta y}=\lim_{\Delta x ,y \rightarrow 0}\frac{\Delta x-i\Delta y}{\Delta x + i\Delta y}\,

If \Delta x = 0\, then \lim = -1. If \Delta y = 0\, then \lim = 1.

But the limit should be the same no matter which direction the limit takes to zero. So the function is non-analytic everywhere because the derivative does not exist.

Complex Variables

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