Holomorphe Ableitung/z und z konjugiert/Beispiel

Es ist

${}{\frac {\partial z}{\partial z}}={\frac {\partial (x+{\mathrm {i} }y)}{\partial z}}={\frac {1}{2}}\cdot {\frac {\partial (x+{\mathrm {i} }y)}{\partial x}}-{\frac {\mathrm {i} }{2}}\cdot {\frac {\partial (x+{\mathrm {i} }y)}{\partial y}}={\frac {1}{2}}-{\frac {\mathrm {i} }{2}}{\mathrm {i} }=1\,$

und

${}{\frac {\partial {\overline {z}}}{\partial z}}={\frac {\partial (x-{\mathrm {i} }y)}{\partial z}}={\frac {1}{2}}\cdot {\frac {\partial (x-{\mathrm {i} }y)}{\partial x}}-{\frac {\mathrm {i} }{2}}\cdot {\frac {\partial (x-{\mathrm {i} }y)}{\partial y}}={\frac {1}{2}}+{\frac {\mathrm {i} }{2}}{\mathrm {i} }=0\,.$