If $$w=f(z)=u+iv$$ and $$u-v = e^x(\cos y – \sin y)$$, find $w$ in terms of $z$, where $z$ denotes the complex variable.
What i tried :- I tried differentiating the second equation partially w.r.t $x$ and $y$ , to somehow calculate $u_x,v_x,u_y,v_y$ but somehow , i need the condition that the function $f(z)$ is analytic to apply Cauchy-Reimann's equation. Is something missing in the problem statement (function being analytic )? Is there a way out ?
Best Answer
You need another condition. As given, you have one equation in the two indeterminates $u, v$.
If the function $f$ is assumed to be differentiable, that gives another equation.