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I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments that can be found in many excellent standard textbooks.
Analysis is about differentiation. Hence, complex differentiation will be my starting point. It is simultaneously my finish line because its inverse, the complex integration, is closely interwoven with complex differentiation. By the lack of details, I mean that I will sometimes assume a disc if a star-shaped region or a simply connected open set would be sufficient; or assume a differentiable function if differentiability up to finitely many points would already be sufficient. Also, the sometimes necessary techniques of gluing triangles for an integration path, or the epsilontic within a region will be omitted.
The statements listed as theorems, however, will be precise. Some of them might sometimes allow a wider range of validity, i.e. more generality. Nevertheless, the reader will find the basic ideas, definitions, tricks, and theorems of the residue calculus; and if nothing else, see where all the ##\pi##'s in integral formulas come from.
Continue reading...
Analysis is about differentiation. Hence, complex differentiation will be my starting point. It is simultaneously my finish line because its inverse, the complex integration, is closely interwoven with complex differentiation. By the lack of details, I mean that I will sometimes assume a disc if a star-shaped region or a simply connected open set would be sufficient; or assume a differentiable function if differentiability up to finitely many points would already be sufficient. Also, the sometimes necessary techniques of gluing triangles for an integration path, or the epsilontic within a region will be omitted.
The statements listed as theorems, however, will be precise. Some of them might sometimes allow a wider range of validity, i.e. more generality. Nevertheless, the reader will find the basic ideas, definitions, tricks, and theorems of the residue calculus; and if nothing else, see where all the ##\pi##'s in integral formulas come from.
Continue reading...
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