After getting to the last part of complex analysis, our teacher had mentioned that to solve harder, more intricate looking complex analysis problems, there are usually “cookbook” methods for solving these problems, for example splitting up multiple piecewise contours with different function definitions based on their shapes or using special functions that transform into another function. Most research articles and even math stack exchange is good enough for these types of problems, but i was wondering if anyone knows of a cookbook, or a book with a set of methods specifically targeting complex valued functions (similar to how physicists have mathematical methods for physicists but broader), usually solving a complex integral? If this question is a duplicate, which i couldn’t find, please specifically note the book itself, thanks.

# Tag: complex

## a torsion-free connection that preserves a complex structure

Let $ (M,I)$ be a complex manifold with a complex structure $ I$ , i.e. an endomorphism $ I$ of the tangent bundle such that $ I^2 = -Id$ and such that the subbundle $ T^{1,0}$ of eigenvectors of $ I \otimes \mathbb{C}$ with eigenvalue $ i$ in $ TM \otimes \mathbb{C}$ is involutive.

How to construct a torsion free connection such that $ \nabla(I)=0$ ?