Indepenence of Path of Contour Integrals

Independence of Path of Contour Integrals

Many contour integrals are independent of path, i.e. any two contour integrals between the same points gives the same value.

Some theorems:

The contour integral of a continuous function $f$ is independent of path in a domain $D$ if and only if $f$ has an antiderivative in $D$.

If $f$ is continuous in a domain $D$ and has an antiderivative $F$ in $D$, then for any piecewise smooth curve $C$ in $D$ joining point $z_0$ to point $z_1$,

\[\int_C f(z) dz = F(z_1) - F(z_0).\]

The contour integral of a continuous function $f$ is independent of path in a domain $D$ if and only if the contour integral of $f$ around every closed, piecewise smooth curve $C$ in $D$ vanishes,

\[\oint_c f(z) dz = 0.\]

We have to be careful about branch cuts. Different antiderivatives of $1/z$ have different branch cuts, and we must pick one where the branch cut does not lie in $D$ in order for it to be a proper antiderivative. When $D$ contains all possible branch cuts of $\log_\phi$, $1/z$ has no antiderivative in $D$.

We also have to be careful when a path encircles a singularity - in this case, the antiderivative may not be single-valued or consistent along the entire path. In fact, when a closed path enircles a singularity, we can’t really show whether or not the function has an antiderivative defined along that path without using the residue theorem.