I have a reason to believe that Euler introduced the complex potential in his Continuation des recherches sur la theorie du mouvement does fluides, published in 1757. However, I am having hard time recognizing where it appears. The relevant place seems to be here:
After establishing that $(u-iv)(dx+idy)$ and $(u+iv)(dx-idy)$ are integrable, he introduces functions $\phi$ and $\psi$. The problem is I am not sure what $\phi:(x+yi)$ means. Normally, I would expect the expression to involve some kind of derivative, such as $$ (u-iv)(dx+idy) = d\phi + id\psi. $$
I am hoping somebody could explain what is going on here.
If you need, Euler continues on the following page as:
Thanks!
