4.3 Antiderivatives
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Our main method for calculating the Riemann integral
is to find differentiable with and apply the fundamental theorem of calculus to get
easily. The difficult part is finding such a .
In the previous section we defined the contour integral
for a contour. In principle such contour integrals can be calculated as Riemann integrals, but finding antiderivatives may be too challenging in practice. Moreover, this course is not about calculating contour integrals using the tools of Riemann integration. It is about the interactions between contour integration, holomorphic functions and domains. As a first result in this direction, we introduce here antiderivatives in the complex setting and will see how the intricacies of domains in the complex plane make their existence relatively rare, in stark comparison with the real case.
Definition (Antiderivative)
Fix a domain and continuous. A holomorphic function is an antiderivative of is for all .
Theorem
Fix a domain and continuous. If is an antiderivative of then
for every smooth path in .
Proof:
Write and as real and imaginary parts. From
we obtain and . Now
as claimed.
Example
Define by and let be any contour from to . If we can find holomorphic with then
by the theorem. We can take so that the integral is .
Notice that the right-hand side only involves and and is therefore the same no matter the route from start to finish! If has an antiderivative then its contour integrals only depend on the start and end points. This is called path independence. In particular, the following is true.
Lemma
Fix a domain and continuous. If is an antiderivative of and is any closed contour in then .
Proof:
If starts at then
by the theorem.
The hypothesis that has an antiderivative is crucial, as the following example shows.
Example
Define by and by . Then
even though is a closed contour! It must therefore be the case that there is no holomorphic function with !