Pre-scriptum (dated 26 June 2020): the material in this post remains interesting but is, strictly speaking, not a prerequisite to understand quantum mechanics. It’s yet another example of how one can get lost in math when studying or teaching physics. 
Original post:
Roger Penrose packs a lot in his chapter on complex-number calculus (Road to Reality, Chapter 7). He summarily introduces the concept of contour integration and then proceeds immediately to discuss power series representations of complex functions as well as fairly advanced ways to deal with singularities (see the section on analytic continuation). Brown and Churchill use not less than three chapters to develop this (Integrals (chapter 4), Series (chapter 5), and Residues and Poles (chapter 6)), and that’s probably what is needed for some kind of understanding of it all. Let’s start with integrals. However, let me first note here that Wordpress does not seem to have a formula editor (so it is not like MS Word) and, hence, I have to keep the notation simple: I’ll use the symbol ∫C for a contour (or line) integral along a curve C, and the symbol ∫[a, b] for an integral on a (closed) interval [a, b].
OK. Here we go. First, it is important to note that Penrose, and Brown and Churchill, are talking about complex integrals, i.e. integrals of complex-valued functions, whose value itself is (usually) a complex number too. That is very different from the line integrals I was exposed to when reading Feynman’s Lectures on Physics. Indeed, Feynman’s Lectures (Volume II, on Electromagnetism) offer a fine introduction to contour integrals, where the function to be integrated is either a scalar field (e.g. electric potential) or, else, some vector field (e.g. magnetic field, gravitational field). Now, vector fields are two-dimensional things: they have an x- and a y-coordinate and, hence, we may think the functions involving vectors are complex too. They are but, that being said, the integrals involved all yield real-number values because the integrand is likely to be a dot product of vectors (and dot products of vectors, as opposed to cross products, yield a real number). I won’t go into the details here but, for those who’d want to have such details, the Wikipedia article offers a fine description (including some very nice animations) of what integration over a line or a curve in such fields actually means. So I won’t repeat that here. I can only note what Brown and Churchill say about them: these (real-valued) integrals can be interpreted as areas under the curve, and they would usually also have one or the other obvious physical meaning, but complex integrals do usually not have such ‘helpful geometric or physical interpretation.’
So what are they then? Let’s first start with some examples of curves.
The illustration above makes it clear that, in practice, the curves which we are dealing with are usually parametric curves. In other words, the coordinates of all points z of the curve C can be represented as some function of a real-number parameter: z(t) = x(t) + iy(t). We can then define a complex integral as the integral of a complex-valued function f(z) of a complex variable z along a curve C from point z1 to point z2 and write such integral as ∫C f(z)dz.
Moreover, if C can be parametrized as z(t), we will have some (real) number a and b such that z1= z(a) and z2= z(b) and, taking into account that dz = z'(t)dt with z'(t)=dz/dt (i.e. the derivative of the (complex-valued) function z(t) with respect to the (real) parameter t), we can write ∫C f(z)dz as:
∫C f(z)dz = ∫[a, b] f[z(t)]z'(t)dt
OK, so what? Well, there are a lot of interesting things to be said about this, but let me just summarize some of the main theorems. The first important theorem does not seem to associated with any particular mathematician (unlike Cauchy or Goursat, which I’ll introduce in a minute) but is quite central: if we have some (complex-valued) function f(z) which would happen to be continuous in some domain D, then all of the following statements will be true if one of them is true:
(I) f(z) has an antiderivative F(z) in D ; (II) the integrals of f(z) along contours lying entirely in D and extending from any fixed point z1to any fixed point z2 all have the same value; and, finally, (III) the integrals of f(z) around closed contours lying entirely in D all have value zero.
This basically means that the integration of f(z) from z1to z2 is not dependent on the path that is taken. But so when do we have such path independence? Well… You may already have guessed the answer to that question: it’s when the function is analytic or, in other words, when these Cauchy-Riemann equations ux = vy and uy = – vx are satisfied (see my other post on analytic (or holomorphic) complex-valued functions). That’s, in a nutshell, what’s stated in the so-called Cauchy-Goursat theorem, and it should be noted that it is an equivalence really, so we also have the vice versa statement: if the integrals of f(z) around closed contours in some domain D are zero, then we know that f(z) is holomorphic.
In short, we’ll always be dealing with ‘nice’ functions and then we can show that the s0-called ‘fundamental’ theorem of calculus (i.e. the one that links integrals with derivatives, or – to be somewhat more precise – with the antiderivative of the integrand) also applies to complex-valued valued functions. We have:
∫C f(z)dz = ∫[a, b] f[z(t)]z'(t)dt = F[z(b)] – F[z(a)]
or, more in general: ∫C f(z)dz = F(z2) – F(z1)
We also need to note the Cauchy integral formula: if we have a function f that is analytic inside and on a closed contour C, then the value of this function for any point z0 will be equal to:
f(z0) = (1/2πi) ∫C [f(z)/(z – z0)]dz
This may look like just another formula, but it’s quite spectacular really: it basically says that the function value of any point z0 within a region enclosed by a curve is completely determined by the values of this function on this curve. Moreover, integrating both sides of this equation repeatedly leads to similar formulas for the derivatives of the first, second, third, and higher order of f: f'(z0) = (1/2πi) ∫C [f(z)/(z – z0)2]dz, f”(z0) = (1/2πi) ∫C [f(z)/(z – z0)3]dz or, more in general:
f(n)(z0) = (n!/2πi) ∫C [f(z)/(z – z0)n+1]dz (n = 1, 2, 3,…)
This formula is also known as Cauchy’s differentiation formula. It is a central theorem in complex analysis really, as it leads to many other interesting theorems, including Gauss’s mean value theorem, Liouville’s theorem, the maximum (and miniumum) modulus principle It is also essential for the next chapter in Brown and Churchill’s course: power series representations of complex functions. However, I will stop here because I guess this ‘introduction’ to complex integrals is already confusing enough.
Post scriptum: I often wondered why one would label one theorem as ‘fundamental’, as it implies that all the other theorems may be important but, obviously, somewhat less fundamental. I checked it out and it turns out there is some randomness here. The Wikipedia article boldly states that the fundamental theorem of algebra (which states that every non-constant single-variable polynomial with complex coefficients has at least one complex roots) is not all that ‘fundamental’ for modern algebra: its title just reflects the fact that there was a time when algebra focused almost exclusively on studying polynomials. The same might be true for the fundamental theorem of arithmetic (i.e. the unique(-prime)-factorization theorem), which states that every integer greater than 1 is either a prime itself or the product of prime numbers, e.g. 1200 = (24)(31)(52).
That being said, the fundamental theorem of calculus is obviously pretty ‘fundamental’ indeed. It leads to many results that are indeed key to understanding and solving problems in physics. One of these is the Divergence Theorem (or Gauss’s Theorem), which states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Huh? Well… Yes. It pops up in any standard treatment of electromagnetism. There are others (like Stokes’ Theorem) but I’ll leave it at that for now, especially because these are theorems involving real-valued integrals.
Reading Feynman 