See Analytic Continuation, Branch Cut, Tutorials: AppellF1[a,Subscript[b, 1],Subscript[b, 2],c,x,y] has singular lines in two-variable complex (x,y) space at Re(x)=1 and Re(y)=1, and has branch cut discontinuities along the rays from 1 to [Infinity] in x and y.
In[1]:= AppellF1[1, 1, 1, 2, x, y] // Simplify
Out[1]= -(Log[(-1 + x)/(-1 + y)]/(x - y))
