Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 18 / b / ii

Let $w=x+iy$. Then $y-ix=-iw$, and the complex trigonometric identity $\sin (-iw)=-i\sinh w$ givesThus the form is exact, with complex-valued scalar potential $F(x,y)=\cosh (x+iy)$. Differentiating that potential with respect to the two real variables checks both coefficients individually.