Just began the study of complex analysis. Let $$ f(x,y) = x^2 – y^2 + 2 i xy – x – iy. $$ I need to determine if this function is analytic. This means I have to show the partials satisfy the Cauchy-Riemann equations, and that the partials are continuous. So in this case we have $u(x,y) = x^2 – y^2 – x$ and $v(x,y) = 2xy – y$. Now \begin{align*} \frac{\partial v}{\partial y} = 2x – 1 = \frac{\partial u}{\partial x} \end{align*} and \begin{align*} – \frac{\partial u}{\partial y} = – (-2y) = 2 y = \frac{\partial v}{\partial x} \end{align*} Hence the Cauchy-Riemann equations are satisfied, which is a necessary condition for being analytic, but not sufficient. Now I have to show the partials are continuous? How do I do that?
[Math] How to show if a complex function is analytic
complex-analysis
Best Answer
Let $g$ define by $g(z)=z^2-z$. $g$ is clearly analytic on $\mathbb C$. You have that $f(x,y)=g(z(x,y))$ which is a composition of two analytic function.