Abstract
The use of \( i=\sqrt{-1} \) to compute integrals is nicely illustrated with a quick example. Let’s use i to do
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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.
Euler used the gamma function (another of his creations that you’ll recall from Chap. 4) in his 1781 analysis, and you can see how he did the Fresnel integrals in An Imaginary Tale, pp. 175–180. As you’d expect from Euler, it’s breathtakingly clever.
- 3.
You can find the proof of this (which requires only elementary algebra) in just about any freshman calculus textbook. Look in the index under the ‘conditional convergence of alternating series,’ or something along those lines. In my old copy of Thomas’ Calculus and Analytic Geometry, for example, that I mentioned back in the Introduction (Chap. 1, Sect. 1.4), it’s on pp. 614–615.
- 4.
This assertion follows from the almost intuitively obvious Riemann-Lebesgue lemma, which says that if f(t) is absolutely integrable over the interval a to b, then limm → ∞ ∫ ba f(t)cos(mt) dt = 0. In our case, 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)| < ∞. 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.
- 5.
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 quad.
- 6.
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.
- 7.
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 quad at the end.
- 8.
To see this, write \( {i}^{2\mathrm{n}-1}= i{i}^{2\mathrm{n}-2}= i\frac{i^{2\mathrm{n}}}{i^2}= i\frac{{\left({i}^2\right)}^{\mathrm{n}}}{\left(-1\right)}= i\frac{{\left(-1\right)}^{\mathrm{n}}}{\left(-1\right)}= i{\left(-1\right)}^{\mathrm{n}-1}. \)
- 9.
I mention this only for completeness. If Ohm’s law is of no interest to you, that’s okay.
- 10.
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.
- 11.
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.’
- 12.
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?!
- 13.
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).
- 14.
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!
- 15.
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.
- 16.
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).
- 17.
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).
- 18.
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.
- 19.
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.
- 20.
Carl Bender, et al., “Observation of PT phase transitions in a simple mechanical system,” American Journal of Physics, March 2013, pp. 173–179.
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Nahin, P.J. (2015). Using \( \sqrt{-1} \) to Evaluate Integrals. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1277-3_7
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