Abstract
This is a self-contained presentation of integration in the complex plane. Beginning with line integrals and the elements of complex function theory, the Cauchy-Riemann equations are derived and the concept of an analytic function is introduced. That is followed with discussions of the integral theorems of Green and Cauchy, integrand singularities, the residue theorem, and the complications caused by multi-valued integrands (which leads to the concepts of branch points and branch cuts). Numerous detailed examples are included, in each discussion, of integrating along closed curves in the complex plane. The grand conclusion is that if such curves are properly constructed (that is, they include the infinite or semi-infinite real axis and handle any singularities present correctly), then the calculation of a wide variety of real-valued, improper definite integrals can be done.
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Notes
- 1.
Jeremy Gray, The Real and the Complex: a history of analysis in the nineteenth century, Springer 2015, pp. 59–60.
- 2.
In his book Complex Analysis: Fundamentals of the Classical Theory of Functions, Birkhäuser 1998, p. 120.
- 3.
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.
- 4.
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.
- 5.
If, instead, we had started with f(z) = z3 = (reiθ)3 = r3ei3θ = r3{cos(3θ) + i sin (3θ)} = 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).
- 6.
The C-R equations had, in fact , been known before either Cauchy or Riemann had been born, as the result of studies in hydrodynamics (see Gray, note 1, p. 60).
- 7.
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.
- 8.
See, for example, Joseph Bak and Donald J. Newman, Complex Analysis (third 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.
- 9.
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.
- 10.
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.
- 11.
Can you show this? If not, go back and read Sect. 1.6 again.
- 12.
Because ∣eiTcos(θ)∣ = 1 for all T, and \( {\lim}_{\mathrm{T}\to \infty}\left|\frac{{\mathrm{T}}^2+\mathrm{aT}{\mathrm{e}}^{i\uptheta}}{{\mathrm{T}}^2+\mathrm{aT}\left({\mathrm{e}}^{i\uptheta}+{\mathrm{e}}^{-i\uptheta}\right)+{\mathrm{a}}^2}\right|=1 \).
- 13.
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.
- 14.
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.
- 15.
If you equate imaginary parts you get \( {\int}_{-\infty}^{\infty}\frac{\sin \left(\mathrm{x}\right)}{{\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}}}\mathrm{dx}=0 \), which is trivially true since the integrand is odd.
- 16.
- 17.
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.
- 18.
You should be able to show that this result immediately follows from the indefinite integral \( \int \frac{\mathrm{du}}{{\mathrm{u}}^2+{\mathrm{b}}^2}=\frac{1}{\mathrm{b}}{\tan}^{-1}\left(\frac{\mathrm{u}}{\mathrm{b}}\right) \) followed by the change of variable u = x + a.
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Nahin, P.J. (2020). Contour Integration. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-43788-6_8
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DOI: https://doi.org/10.1007/978-3-030-43788-6_8
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