Substitute \(\displaystyle{z}={x}+{i}{y}\ \text{ and }\ \overline{{{z}}}={x}-{i}{y}\) in the above function and simplify as follows.

Step 2

\(\displaystyle{f{{\left({z}\right)}}}={\left({x}-{i}{y}\right)}{\left({1}-{e}^{{{i}{\left({x}+{i}{y}+{x}-{i}{y}\right)}}}\right)}\)

\(\displaystyle={\left({x}-{i}{y}\right)}{\left({1}-{e}^{{{i}{2}{x}}}\right)}\)

\(\displaystyle={x}-{i}{y}-{\left({x}-{i}{y}\right)}{e}^{{{i}{2}{x}}}\)

\(\displaystyle={x}-{i}{y}-{\left({x}-{i}{y}\right)}{\left({\cos{{\left({2}{x}\right)}}}+{i}{\sin{{\left({2}{x}\right)}}}\right)}\)

\(\displaystyle={x}-{i}{y}-{\left[{x}{\cos{{\left({2}{x}\right)}}}+{i}{x}{\sin{{\left({2}{x}\right)}}}-{i}{y}{\cos{{\left({2}{x}\right)}}}+{y}{\sin{{\left({2}{x}\right)}}}\right]}\)

\(\displaystyle={x}-{i}{y}-{x}{\cos{{\left({2}{x}\right)}}}-{i}{x}{\sin{{\left({2}{x}\right)}}}+{i}{y}{\cos{{\left({2}{x}\right)}}}-{y}{\sin{{\left({2}{x}\right)}}}\)

\(\displaystyle={\left[{x}-{x}{\cos{{\left({2}{x}\right)}}}-{y}{\sin{{\left({2}{x}\right)}}}\right]}+{i}{\left[-{y}-{x}{\sin{{\left({2}{x}\right)}}}+{y}{\cos{{\left({2}{x}\right)}}}\right]}\)

\(\displaystyle={u}{\left({x},{y}\right)}+{i}{v}{\left({x},{y}\right)}\)

Thus, \(\displaystyle{u}{\left({x},{y}\right)}={x}-{x}{\cos{{\left({2}{x}\right)}}}-{y}{\sin{{\left({2}{x}\right)}}}\ \text{ and }\ {v}{\left({x},{y}\right)}={y}{\cos{{\left({2}{x}\right)}}}-{y}-{x}{\sin{{\left({2}{x}\right)}}}\)