Abstract
In this, the penultimate chapter of the book, I’ll give you a really fast, stripped-down, ‘crash-course’ presentation of the very beginnings of complex function theory, and the application of that theory to one of the gems of mathematics: contour integration and its use in doing definite integrals. As we start this chapter I’ll assume only that you are familiar with complex numbers and their manipulation. I’ve really already done that, of course, in Chap. 7, and so I think I am on safe ground here with that assumption. The first several sections will lay the theoretical groundwork and then, quite suddenly, you’ll see how they all come together to give us the beautiful and powerful technique of contour integration. None of these preliminary sections is very difficult, but each is absolutely essential for understanding. Don’t skip them!
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Notes
- 1.
In his book Complex Analysis: Fundamentals of the Classical Theory of Functions, Birkhäuser 1998, p. 120.
- 2.
In keeping with the casual approach I’m taking in this book, I’ll just assume that these two limits exist and then we’ll see where that assumption takes us. Eventually we’ll arrive at a new way to do definite integrals (contour integration) and then we’ll check our assumption by seeing if our theoretical calculations agree with MATLAB’s direct numerical evaluations.
- 3.
There are, of course, two distinct ways we can have A = B. The trivial way is if C simply has zero length, which immediately says Ix = Iy = 0. The non-trivial way is if C goes from A out into the plane, wanders around for a while, and then returns to A (which we re-label as B). It is this second way that gives us a closed loop.
- 4.
If, instead, we had started with f(z) = z3 = (reiθ)3 = r3ei3θ = r3{cos(θ) + i sin(θ)}3, then we could have just as easily have derived the triple-angle formulas that are not so easy to get by other means (just take a look at any high school trigonometry text).
- 5.
The word finite is important: f(z) = z blows-up as |z| → ∞ and so f(z) is not said to be analytic at infinity. In fact, there is a theorem in complex function theory that says the only functions that are analytic over the entire complex plane, even at infinity, are constants. In those cases all four partial derivatives in the C-R equations are identically zero.
- 6.
See, for example, Joseph Bak and Donald J. Newman, Complex Analysis (3rd edition), Springer 2010, pp. 35-40. While the C-R equations alone are not sufficient for analyticity, if the partial derivatives in them are continuous then we do have sufficiency.
- 7.
If the function f(z) is analytic everywhere in some region except for a finite number of singularities, mathematicians say f(z) is meromorphic in that region and I tell you this simply so you won’t be paralyzed by fear if you should ever come across that term.
- 8.
For the interesting history of this theorem, named after the English mathematician George Green (1793-1841), see my An Imaginary Tale, Princeton 2010, pp. 204-205.
- 9.
Because |eiTcos(θ)| simply oscillates forever between 0 and 1, and \( \underset{\mathrm{T}\to \infty }{ \lim}\left|\frac{{\mathrm{T}}^2+{\mathrm{a}\mathrm{Te}}^{i\uptheta}}{{\mathrm{T}}^2+\mathrm{aT}\left({\mathrm{e}}^{i\uptheta}+{\mathrm{e}}^{- i\uptheta}\right)+{\mathrm{a}}^2}\right|=1. \)
- 10.
The singularity in (8.7.1) is called first-order because it appears to the first power. By extension, \( \frac{\mathrm{f}\left(\mathrm{z}\right)}{{\left(\mathrm{z}-{\mathrm{z}}_0\right)}^2} \) has a second-order singularity, and so on. I’ll say much more about high-order singularities in the next section.
- 11.
The limits on a are because, first, since n − m ≥ 2 it follows that m + 1 ≤ n − 1 and so a < 1. Also, for x ≪ 1 the integrand in (8.7.9) behaves as xa − 1 which integrates to \( \frac{{\mathrm{x}}^{\mathrm{a}\ }}{\mathrm{a}} \) and this blows-up at the lower limit of integration if a < 0. So, 0 < a.
- 12.
- 13.
Each of these branches exists for each new interval of θ of width 2π, with each branch lying on what is called a Riemann surface. The logarithmic function has an infinite number of branches, and so an infinite number of Riemann surfaces. The surface for 0 ≤ θ < 2π is what we observe as the usual complex plane (the entry level of our parking garage). The concept of the Riemann surface is a very deep one, and my comments here are meant only to give you an ‘elementary geometric feel’ for it.
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Nahin, P.J. (2015). Contour Integration. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1277-3_8
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