You got here via a link on <a href='http://www.few.vu.nl/~jhulshof/'>my home page</a>. This page is under construction. It is not yet about analytic semigroups, but about that part of complex analysis needed for analytic semigroups, namely Chapter 18 of
 <a href='http://www.few.vu.nl/~jhulshof/2024.pdf#page556'>Fundamentals Of Analysis</a>. Chapter 17 until 17.5 will prepare you for any complex analysis course, based on a first analysis course which properly prepares for functions defined on and taking values in (preferably complete) real normed spaces.
Here I take the viewpoint that the familiar xy-plane is just a special example of such a normed space, one that can be identified with the set of complex numbers via z=x+iy.  

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<br>Chapter 18 does away with the usual approach of line integrals over arbitrary sufficiently smooth curves. All the results are formulated and proved using polygononal curves only, beginning with line segments.