Theorem 4.2.9 gave us an antiderivative for a function that has zero integrals over closed paths in a given region. Now that we have Corollary 5.1.6, meditating just a bit more over Theorem 4.2.9 gives a converse of sorts to Corollary 4.3.8.
Theorem 4.2.9 yields an antiderivative \(F\) for \(f\) in \(G\text{.}\) Because \(F\) is holomorphic in \(G\text{,}\)Corollary 5.1.6 implies that \(f\) is also holomorphic in \(G\text{.}\)
Just like there are several variations of Theorem 4.2.9, we have variations of Corollary 5.2.1. For example, by Corollary 4.2.10, we can replace the condition for all piecewise smooth closed paths \(\gg \subset G\) in the statement of Corollary 5.2.1 by the condition for all closed polygonal paths \(\gg \subset
G\) (which, in fact, gives a stronger version of this result).
Any disk \(D[a,r]\) is simply connected, as is \(\C \setminus \R_{ \le 0 }\text{.}\) (You should draw a few closed paths in \(\C \setminus \R_{ \le 0 }\) to convince yourself that they are all contractible.) The region \(\C \setminus \{ 0 \}\) is not simply connected as, e.g., the unit circle is not \((\C \setminus \{ 0 \})\)-contractible.
If \(f\) is holomorphic in a simply-connected region then Corollary 4.3.8 implies that \(f\) satisfies the conditions of Theorem 4.2.9, whence we conclude:
Note that this corollary gives no indication of how to compute an antiderivative. For example, it says that the (entire) function \(f: \C \to \C\) given by \(f(z) = \exp(z^2)\) has an antiderivative \(F\) in \(\C\text{;}\) it is an entirely different matter to derive a formula for \(F\text{.}\)
Corollary 5.2.4 also illustrates the role played by two of the regions in Example 5.2.3, in connection with the function \(f(z) = \frac 1 z\text{.}\) This function has no antiderivative on \(\C \setminus \{ 0 \}\text{,}\) as we proved in Example 4.2.8. Consequently (as one can see much more easily), \(\C \setminus \{ 0 \}\) is not simply connected. However, the function \(f(z) = \frac 1 z\) does have an antiderivative on the simply-connected region \(\C \setminus \R_{ \le 0 }\) (namely, \(\Log(z)\)), illustrating one instance implied by Corollary 5.2.4.
Finally, Corollary 5.2.4 implies that, if we have two paths in a simply-connected region with the same endpoints, we can concatenate them—changing direction on one—to form a closed path, which proves:
If \(f\) is holomorphic in a simply-connected region \(G\) then \(\int_\gg f\) is independent of the piecewise smooth path \(\gg \subset G\) between \(\gg(a)\) and \(\gg(b)\text{.}\)
When an integral depends only on the endpoints of the path, the integral is called path independent. Example 4.1.2 shows that this situation is quite special; it also says that the function \({\overline z}^2\) does not have an antiderivative in, for example, the region \(\{ z \in \C : \, |z|\lt 2 \}\text{.}\) (Actually, the function \({\overline z}^2\) does not have an antiderivative in any nonempty region—see Exercise 5.4.7.)
