I have that f(z) = 1/(z+i) and I need to write f(z+i) as a(x,y) + i*b(x,y) where a and b are real functions.

I see that f(z+i) = 1/(z+2i) and I multiplied by its conjugate z-2i, but since z is itself a complex number with x + iy, I don't see how I can separate f(z+i) into its real and complex part. Can someone help me here?

Thanks in advance.