One challenge in a course like this is to maintain the connection with actual applications of the techniques one is covering. Since I was originally taught this material in a set of courses as a mathematics undergraduate, my own take on the material can be rather formal, and I have worked over the years to balance this out. However, as you might guess, my own examples are predominantly drawn from those areas of physics with which I am most familiar - for example, supersymmetry, and the restrictions that holomorphy places on superpotentials, is a nice illustration of the power of complex analysis.
See this useful link.
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