Question

Path-independence of Complex Integrals?

Answer 1

Let \(\gamma_1\) be the upper-half circle from \(-1\) to \(1\), and \(\gamma_2\) be the path from \(-1\) to \(-1+i\) to \(1+i\) to \(1\). Show that \[\large \int_{\gamma_1}\bar{z}\,dz\ne\int_{\gamma_2}\bar{z}\,dz\]

I realize that the LHS integral gives \(-\pi i\) and the RHS integral gives \(-4i\). But what condition is not satisfied such that path independence does NOT apply here?

Answer 2

Let \(\gamma_1\) be the upper-half circle from \(-1\) to \(1\), and \(\gamma_2\) be the path from \(-1\) to \(-1+i\) to \(1+i\) to \(1\). Show that \[\large \int_{\gamma_1}\bar{z}\,dz\ne\int_{\gamma_2}\bar{z}\,dz\]