How to get the first variation of a complex lagrangian?

Question

How do I get the first variation for this: $$ \int_C L\left(z,\phi,\frac{\mathrm{d}\phi}{\mathrm{d}z}\right)\mathrm{d}z$$ where: $$z=x+iy,$$ $$\mathrm{d}z=\mathrm{d}x+i\mathrm{d}y,$$ $$\phi=f(x,y)+ig(x,y).$$ The integral is a complex line integral and $\phi$ is an analytical function. I'm not sure whether the process is identical to that of real analysis. Is it even possible to get such a variation? Does it even make any sense?