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The Road to Reality Study Notes: Difference between revisions
→Chapter 7 Complex-number calculus
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The amazing result here is that a general contour from $$a$$ to $$b$$ for the function <math>\frac{1}{z}</math> has be rephrased and shown to be equal to the result for a ''closed contour'' that loops around the point of non-analyticity (the origin in this case), regardless of where the points $$a$$ and $$b$$ (or the point of non-analyticity) lie in the complex plane. Note that since $$logz$$ is multi-valued, we need to specify the actual closed contour being used (if we looped twice rather than once, then the answer is different). | The amazing result here is that a general contour from $$a$$ to $$b$$ for the function <math>\frac{1}{z}</math> has be rephrased and shown to be equal to the result for a ''closed contour'' that loops around the point of non-analyticity (the origin in this case), regardless of where the points $$a$$ and $$b$$ (or the point of non-analyticity) lie in the complex plane. Note that since $$logz$$ is multi-valued, we need to specify the actual closed contour being used (if we looped twice rather than once, then the answer is different). | ||
=== 7.3 Power series from complex smoothness === | |||
=== 7.4 Analytic continuation === | |||
== Chapter 8 Riemann surfaces and complex mappings == | == Chapter 8 Riemann surfaces and complex mappings == | ||