Editing The Road to Reality Study Notes
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Continuous deformation in this sense is defined as [https://en.wikipedia.org/wiki/Homology_(mathematics)#:~:text=When%20two%20cycles%20can%20be,in%20the%20same%20homology%20class. homologous deformation], where parts of the path can cancel each other out if they are traversed in opposite directions. This contrasts with homotopic paths where you may not cancel parts. As a visualization we take the function of <math>\frac{1}{z}</math> with a homologous path: | Continuous deformation in this sense is defined as [https://en.wikipedia.org/wiki/Homology_(mathematics)#:~:text=When%20two%20cycles%20can%20be,in%20the%20same%20homology%20class. homologous deformation], where parts of the path can cancel each other out if they are traversed in opposite directions. This contrasts with homotopic paths where you may not cancel parts. As a visualization we take the function of <math>\frac{1}{z}</math> with a homologous path: | ||
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The amazing result here is that a general contour from <math>a</math> to <math>b</math> 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 <math>a</math> and <math>b</math> (or the point of non-analyticity) lie in the complex plane. Note that since <math>logz</math> 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 <math>a</math> to <math>b</math> 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 <math>a</math> and <math>b</math> (or the point of non-analyticity) lie in the complex plane. Note that since <math>logz</math> 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). |