Abstract
Here we set the stage for the next chapter (on contour integration) by showing how an early (and pretty casual) use of the imaginary quantity \( i=\sqrt{-1} \) was used by such pioneers as Euler to evaluate some challenging integrals. Such integrals include the Fresnel integrals, and Euler’s log-sine integral involving the zeta function. Euler’s famous identity eix = cos (x) + i sin (x) plays a central role these calculations. The use of the classic transforms of physics, math, and engineering (Fourier, Laplace, and Hilbert) in evaluating definite integrals is illustrated with numerous examples, including applications to AM radio signal transmission and reception. Rayleigh’s energy theorem is developed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
An Imaginary Tale: the story of \( \sqrt{-1} \), and Dr. Euler’s Fabulous Formula: cures many mathematical ills, both published by Princeton University Press (both in multiple editions).
- 2.
A nice discussion of the history of the ‘Fresnel’ integrals is in R. C. Archibald, “Euler Integrals and Euler’s Spiral—Sometimes Called Fresnel Integrals and the Clothoїde or Cornu’s Spiral,” The American Mathematical Monthly, June 1918, pp. 276–282.
- 3.
You can find a proof of the lemma (it’s not difficult) in Georgi P. Tolstov’s book Fourier Series (translated from the Russian by Richard A. Silverman), Dover 1976, pp. 70–71.
- 4.
The plot (created by a MATLAB code) in Figure 7.2.1 is yet another example of the naming of mathematical things after the wrong person. The curve is often called Cornu’s spiral , after the French physicist Marie-Alfred Cornu (1841–1902). While Cornu did indeed make such a plot (in 1874), Euler had described the spiral decades earlier. Well, of course, if everything Euler did first was named after him maybe that would be a bit confusing.
- 5.
This assertion follows from the Riemann-Lebesgue lemma (see note 3). In our case here, f(t) = \( \frac{1}{\sin \left(\mathrm{t}\right)} \) which is absolutely integrable over \( 0<\mathrm{x}\le \mathrm{t}\le \frac{\uppi}{2} \) since over that interval |f(t)| < ∞.
- 6.
That is, since \( {\mathrm{e}}^{i\mathrm{y}}=\cos \left(\mathrm{y}\right)+i\ \sin \left(\mathrm{y}\right)={\sum \limits}_{\mathrm{k}=0}^{\infty}\frac{{\left(i\mathrm{y}\right)}^{\mathrm{k}}}{\mathrm{k}!}=\frac{1}{0!}+\frac{i\mathrm{y}}{1!}-\frac{{\mathrm{y}}^2}{2!}-\frac{i{\mathrm{y}}^3}{3!}+\dots \) then setting the real and imaginary parts from Euler’s identity equal, respectively, to the real and imaginary parts of the exponential expansion, we immediately get the power series expansions for cos(y) and sin(y).
- 7.
How do we know we can do this? This is a non-trivial question, and a mathematician would rightfully want to vigorously pursue it. But remember our philosophical approach—we’ll just make the assumption that all is okay, see where it takes us, and then check the answers we eventually calculate with integral.
- 8.
The angle is given by \( \upphi ={\tan}^{-1}\left\{\frac{\mathrm{a}\ \sin \left(\upphi \right)}{1+\mathrm{a}\ \cos \left(\upphi \right)}\right\} \), but we’ll never actually need to know this.
- 9.
- 10.
The reason (7.4.2) is correct is two-fold: (1) both sides are equal (to zero) at x = 0, and (2) the derivatives with respect to x of each side are equal (use Feynman’s favorite trick to show this).
- 11.
It is easy to show that \( \upzeta (3)={\sum \limits}_{\mathrm{n}=1}^{\infty}\frac{1}{{\mathrm{n}}^3}={\int}_0^1{\int}_0^1{\int}_0^1\frac{\mathrm{dxdydz}}{1-\mathrm{xyz}} \), but a direct evaluation of this triple integral over the unit cube continues to elude all who have tried. MATLAB numerically calculates integral3(@(x,y,z)1./(1-x.∗y.∗z),0,1,0,1,0,1) = 1.2020… .
- 12.
Mathematicians will want to check that the limiting operations b → ∞, a → 0, and that the reversal of the order of integration in the double integral, are valid, but again remember our guiding philosophy in this book: just do it, and check with integral at the end.
- 13.
To see this, write \( {i}^{2\mathrm{n}-1}={ii}^{2\mathrm{n}-2}=i\frac{i^{2\mathrm{n}}}{i^2}=i\frac{{\left({i}^2\right)}^{\mathrm{n}}}{\left(-1\right)}=i\frac{{\left(-1\right)}^{\mathrm{n}}}{\left(-1\right)}=i{\left(-1\right)}^{\mathrm{n}-1} \).
- 14.
I mention this only for completeness. If Ohm’s law is of no interest to you, that’s okay.
- 15.
Where these defining integrals in a Fourier pair come from is explained in any good book on Fourier series and/or transforms. Or, for an ‘engineer’s treatment’ in the same spirit as this book, see Dr. Euler (note 1), pp. 200–204.
- 16.
Although Dirac won the 1933 Nobel Prize in physics, he was the Lucasian Professor of mathematics at Cambridge University. His physical insight into such a bizarre thing as an infinite derivative was powered (by his own admission) with his undergraduate training in electrical engineering: he graduated with first-class honors in EE from the University of Bristol in 1921. Dirac was clearly ‘a man for all seasons’! The mathematics of impulses has been placed on a firm theoretical foundation since Dirac’s intuitive use of them in quantum mechanics. The central figure in that great achievement is generally considered to be the French mathematician Laurent Schwartz (1915–2002), with the publication of his two books Theory of Distributions (1950, 1951). For that work, Schwartz received the 1950 Fields Medal, often called the ‘Nobel Prize of mathematics.’
- 17.
In an analogy with white light, in which all optical frequencies (colors) are uniformly present, such an energy distribution is also often said to be a white spectrum. To continue with this terminology, signals with energy spectrums that are not flat (not white) are said to have a pink (or colored) spectrum. Who says radio engineers aren’t romantic souls?!
- 18.
There are two frequencies in the transform because of the two exponentials in the transform integral, each of which represents a rotating vector in the complex plane. One rotates counterclockwise at frequency +ω0 (making an instantaneous angle with the real axis of ω0t) and the other rotates clockwise at frequency −ω0 (making an instantaneous angle with the real axis of −ω0t). The imaginary components of these two vectors always cancel, while the real components add along the real axis to produce the real-valued signal cos(ω0t).
- 19.
Note, carefully: the ∗ symbol denotes complex conjugation when used as a superscript as was done in Sect. 7.8 when discussing the energy spectrum, and convolution when used in-line. Equation (7.9.16) is called the frequency convolution integral, to distinguish it from its twin, the time convolution integral, which says m(t) ∗ g(t) ↔ G(ω)M(ω). We won’t use that pair in what follows, but you should now be able to derive it for yourself. Try it!
- 20.
To be honest, multiplying at radio frequencies is not easy. To learn how radio engineers accomplish multiplication without actually multiplying, see Dr. Euler, pp. 295–297, 302–305, or my book The Science of Radio, Springer 2001, pp. 233–249.
- 21.
The mathematics of all this was of course known long before AM radio was invented, but the name of the theorem is due to the American electrical engineer Reginald Fessenden (1866–1932), who patented the multiplication idea in 1901 for use in a radio circuit. The word ‘heterodyne’ comes from the Greek heteros (for external) and dynamic (for force). Fessenden thought of the cos(ω0t) signal as the ‘external force’ being generated by the radio receiver circuitry itself for the final frequency down-shift of the received signal to baseband (indeed, radio engineers call that part of an AM radio receiver the local oscillator circuit).
- 22.
They are also sometimes called the Kramers-Kronig relations, after the Dutch physicist Hendrik Kramers (we encountered him back in Sect. 6.5, when discussing the Watson/van Peype triple integrals), and the American physicist Ralph Kronig (1904–1995), who encountered (7.10.11) when studying the spectra of x-rays scattered by the atomic lattice structures of crystals. See Challenge Problem 7.9 for an alternative way to write (7.10.11).
- 23.
Combining a time signal x(t) with its Hilbert transform to form the complex signal \( \mathrm{z}\left(\mathrm{t}\right)=\mathrm{x}\left(\mathrm{t}\right)+i\ \overline{\mathrm{x}\left(\mathrm{t}\right)} \), you get what the Hungarian-born electrical engineer Dennis Gabor (1900–1979)—he won the 1971 Nobel Prize in physics—called the analytic signal , of great interest to engineers who study single-sideband (SSB) radio. To see how the analytic signal occurs in SSB radio theory, see Dr. Euler, pp. 309–323.
- 24.
Note, carefully, the dual use of the symbol “π”—once for the number, and again for the name of the gate function. There will never be any confusion, however, because the gate function will always appear with an argument while π alone is the number.
- 25.
Despite the name, the genesis of the transform can be traced back to (are you surprised and, if so, why?) Euler. See M. A. B. Deakin, “Euler’s Version of the Laplace Transform,” The American Mathematical Monthly, April 1980, pp. 264–269.
- 26.
This name was given to (7.11.18) in 1871 by the English mathematician J. W. L. Glaisher (see note 13 in Chap. 6) because of its appearance in the probabilistic theory of measurement errors. Later, electrical engineers found it invaluable in their probabilistic studies of electronic systems in the presence of noise.
- 27.
If this isn’t immediately clear, look back at the discussion just before (3.1.8).
- 28.
Carl Bender, et al., “Observation of PT phase transitions in a simple mechanical system,” American Journal of Physics, March 2013, pp. 173–179.
- 29.
See, for example, my Transients for Electrical Engineers, Springer 2018, pp. 141–145, where the “very long communication cable” is the famous mid-nineteenth century Trans-Atlantic electric telegraph cable. (The mathematical physics and the history of the cable are presented at length in my Hot Molecules, Cold Electrons, Princeton 2020).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Nahin, P.J. (2020). Using \( \sqrt{-1} \) to Evaluate Integrals. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-43788-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-43788-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-43787-9
Online ISBN: 978-3-030-43788-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)