Editing The Road to Reality Study Notes (section)

== Chapter 7 Complex-number calculus ==
=== 7.1 Complex smoothness; holomorphic functions ===
An outline for section 7 is presented, in which calculus with complex numbers is introduced.  The material in this chapter leads up to the explanation of [https://en.wikipedia.org/wiki/Holomorphic_function holomorphic functions], which play a vital role in much of the mathematical as well as physics material later in the book.

To do so, the notion of a special type of integration along a contour in the complex plane is to be defined. This integration can then be used to solve for the coefficients of a [https://en.wikipedia.org/wiki/Taylor_series Taylor series] expression which allow for us to see that any complex function which is complex-smooth in the complex plane is necessarily analytic, or holomorphic.

As will be stated in 7.3, instead of directly providing the definition of holomorphic functions, Penrose chooses to demonstrate the argument with the ingredients in order to show a "wonderful example of the way that mathematicians can often obtain their results. Neither the premise (<math>f(z)</math> is complex-smooth) nor the conclusion (<math>f(z)</math> is analytic) contains a hint of the notion of contour integration or the multivaluedness of a complex logarithm.  Yet these ingredients provide the essential clues to the true route to finding the answer".

=== 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).

=== 7.3 Power series from complex smoothness ===
The example in section 7p2 is a particular case for the well-known [https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula Cauchy Integral Formula], which allows us to know what the function is doing at the origin (or another general point <math>p</math>) by what it is doing at a set of points surrounding the origin or the general point <math>p</math>. 

:<math>\frac{1}{2πi}\oint\frac{f(z)}{z-p}dz=f(p)</math>

A 'higher-order' version of this formula allows us to inspect <math>n</math> number of derivatives with the same relationship.

:<math>\frac{n!}{2πi}\oint\frac{f(z)}{(z-p)^{n+1}}dz=f^{(n)}(p)</math>

If we use this to provide the definition of a derivative at a point, we can then construct a Maclaurin formula (if using the origin, otherwise the more general [https://en.wikipedia.org/wiki/Taylor_series Taylor series]) for <math>f(z)</math> using the derivatives in the coefficients of the terms.

:<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(p)}{n!} (z-p)^{n} </math>

This can be shown to sum to <math>f(z)</math>, thereby showing the function has an actual <math>n</math>th derivative at the origin or general point <math>p</math>.  This concludes the argument showing that complex smoothness in a region surrounding the origin or point implies that the function is also holomorphic. Penrose notes that neither the premise (<math>f(z)</math> is complex-smooth) nor the conclusion (<math>f(z)</math> is analytic) contains contour integration or multivaluedness of a complex logarithm, yet these ingredients are essential for finding the route to the answer and that this is a ‘wonderful example of the way that mathematicians can often obtain their results’.

=== 7.4 Analytic continuation ===
We now know that complex smoothness throughout a region is equivalent to the existence of a power series expansion about any point in the region.  A region here is defined as a open region, where the boundary is not included in the domain.

For example, if there is no singularity in the function, the region can be thought of as a circle of infinite radius. Taking <math>f(z)=\frac{1}{z}</math> however forces an infinite number of circles centered at any point with boundary radii passing through the origin (noting that an open region does not contain the boundary) to construct the domain.

Now we consider the question, given a function <math>f(z)</math> holomorphic in domain <math>D</math>, can we extend the domain to a larger <math>D’</math> so that <math>f(z)</math> also extends holomorphically?  A procedure is formed in which we use a succession of power series about a sequence of points, forming a path where the circles of convergence overlap.  This then results in a function that is uniquely determined by the values in the initial region as well as the path along which it was continued.  Penrose notes this [https://en.wikipedia.org/wiki/Analytic_continuation analytic continuation] as a remarkable ‘rigidity’ about holomorphic functions.

An example of this rigidity and path dependence is ‘our old friend’ <math>logz</math>.  There is no power series expansion about the origin due to the singularity there but depending on the path chosen of points around the origin (clockwise or anti-clockwise) the function extends or subtracts in value by <math>2πi</math>.  See chapter 5 and the euler formula (<math>e^{πi}=-1</math>) for a refresher.