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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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