Editing The Road to Reality Study Notes (section)

=== 7.2 Contour integration ===
In the real number sense, integrals are taken from a single point <math>a</math> to another point <math>b</math> along the real number line.  Usually the horizontal axis, and there is only one way to travel along this line (moving positive and negative along the axis).  However, in the complex plane points involve two dimensions, and therefore have many such routes that allow us to get from a complex point <math>a</math> to <math>b</math>.

The [https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations Cauchy-Riemann equations] (to be formally introduced later in chapter 10) allow us to narrow our focus to find a path-specific answer, where the value of the integral on this path is the same for any other path that can be formed from the first by continuous deformation in its domain.  Note that the function <math>\frac{1}{z}</math> has a hole in the domain at the origin, which can prevent a continuous deformation thereby allowing for different answers for the value of the integral depending on the path taken.

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:

[[File:Fig 7p3 png.png|thumb|center]]

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).