I Learning Complex Integration: Endpoints & Paths

[Silviu]

Hello! I started learning about complex analysis and I am a bit confused about integration. I understand that if we take different paths for the same function, the value on the integral is different, depending on the path. But if we use the antiderivative: $\int_{\gamma}f=F(\gamma(b))-F(\gamma(a))$, where $\gamma$ is the path and a and b are the endpoints. So based on this formula, the value of the integral doesn't depends on the path but just on the endpoints. I am not sure I understand the meaning of this and when can we use this formula, as it gives just a value for all the path, so what do the other paths means? Thank you!

 

[BvU]

The antiderivative expression holds if $F$ is complex differentiable -- which is a severe constraint. See holomorphic functions

 

[Silviu]

The antiderivative expression holds if $F$ is complex differentiable -- which is a severe constraint. See holomorphic functions

But if I take $f(z)=z^2$, this has the antiderivative $F=(z)=\frac{z^3}{3}$, which is entire. So if I take the path from $0$ to $1+i$, using this formula I obtain $\int_{\gamma}f=F(i+1)-F(0)=\frac{2i-2}{3}$. However, if I use the formula $\int_{\gamma}f=\int_a^b f(\gamma(t))\gamma'(t)dt$ and I parametrize $\gamma$ using a straight line or a parabola between the same 2 points ($0$ and $1+i$), I obtain 2 different results. So in the first case only the end points matter, in the second case, the path matters, too. So how am I suppose to use the antiderivative formula?