I have this rapid procedure to construct an analytic function from a real part:
\begin{equation} u(x,y)=e^x(cosy-siny) \end{equation}
We have the partial derivatives: \begin{equation} \begin{array} fu_x=e^x(cosy-siny) \\ u_y=e^x(-siny-cosy) \\ \end{array} \end{equation}
So by the Cauchy-Riemann equations we want to find the complex conjugate function, v: \begin{equation} \begin{array} fu_x=v_y \\ u_y=-v_x \end{array} \end{equation}
By the first relation (eqn 1):
\begin{equation} v(x,y)=\int e^x(cosy-siny)dx= e^x(cosy-siny)+h(y) \end{equation}
We find h(y) by the second relation
\begin{equation} \frac{\partial}{\partial x}\big[e^x(cosy-siny)+h(y)\big]=-e^x(-cosy-siny) \end{equation}
Integrate both sides and obtain
\begin{equation} \big[e^x(cosy-siny)+h'(x,y)\big]=e^x(cosy+siny) \end{equation}
\begin{equation} h'(x,y)=e^x(cosy+siny) -e^x(cosy-siny) \end{equation}
\begin{equation} h'(x,y)=2e^xsiny \end{equation}
Then we integrate with respect to x and obtain
\begin{equation} \int h'(x,y)dx=\int 2e^xsiny dx= 2e^xsiny \end{equation}
Insert in the equation nr 1 above for $h(y)$ and multiply it by i
\begin{equation} v(x,y)=e^x(cosy-siny)+2ie^xsiny \end{equation}
Would this be correct, and accurate?
Thanks