Heaviside’s Operator Calculus

heaviside3An operational calculus converts derivatives and integrals to operators that act on functions, and by doing so ordinary and partial linear differential equations can be reduced to purely algebraic equations that are much easier to solve. There have been a number of operator methods created as far back as Leibniz, and some operators such as the Dirac delta function created controversy at the time among mathematicians, but no one wielded operators with as much flair and abandon over the objections of mathematicians as Oliver Heaviside, the reclusive physicist and pioneer of electromagnetic theory.

The name of Oliver Heaviside (1850-1925) is not well-known to the general public today. However, it was Heaviside, for example, who developed Maxwell’s electromagnetic equations into the four vector calculus equations in two unknowns that we are familiar with today; Maxwell left them as 20 equations in 20 unknowns expressed as quaternions, a once-popular mathematical system currently experiencing a revival for fast coordinate transformations in video games. Heaviside also did important early work in long-distance telegraphy and telephony, introducing induction-loading of long cables to minimize distortion and patenting the coaxial cable. At one time the ionosphere was called the Heaviside layer after his suggestion (and that of Arthur Kennelly) that a layer of charged ions in the upper atmosphere (now just one layer of the ionosphere) would account for the puzzlingly long distances that radio waves traveled. But Heaviside was an iconoclast who saw little need to ingratiate himself with others or spend time justifying his methods to them. It took later mathematicians such as Carson and Bromwich to demonstrate that his operators are analogous to later, well-developed integral equations and contour integrals in the complex plane.

Actually, absence of rigor is less unusual historically than it might appear—much mathematical science has progressed on very shaky ground indeed, and often proofs of mathematical techniques lag by many years their application. The famous mathematician G. Hardy wrote

All physicists and a good many quite respectable mathematicians are contemptuous about proof.

and certainly mathematicians often discover things by intuition that require a great deal of time and labor to prove. Often mathematics progresses from the specific to the general, bottom to top. Carl Friedrich Gauss once wrote in his diary about one of his most important discoveries, which was based on a purely computational observation: “I have the result, but I do not yet know how to get it.” The mathematician J. Hadamard observed, “The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.”

Today “experimental mathematics”—using computers to search for number-theoretic results that can be generalized—is a hot field in mathematics. As David Berlinski writes,

The computer has in turn changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen.

And even back in 1951, Kurt Godel wrote,

If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.

Heaviside was openly dismissive of attempts to provide rigor for his operator calculus. Here’s what he had to say regarding his bold generalization of an experimental result to a general one that we will discuss shortly:

Those who may prefer a more formal and logically-arranged treatment may seek it elsewhere, and find it if they can; or else go and do it themselves.

Perhaps his refusal to validate it was a good thing because his operator calculus was not generally rigorous, particularly when initial values are non-zero. Heaviside used a great deal of intuition to guide him in the process of applying his calculus. In the end Laplace transforms, easier to use with a more rigorous structure and incorporating the powerful tool of convolution, overtook the operational calculus of Heaviside, and his methods largely fell victim to history. But the ingenuity of Heaviside appeals to me (along with his sheer indifference to the complaints of those he left in the dust) and so this essay is a short appreciation of his work in this regard.

Let’s first get a flavor of how his operators worked. Heaviside took the basic equations for voltage v and current i for a discrete resistance R, capacitance C and inductance L and rewrote them using his operator p, which performed the derivative with respect to time on the function to the right of it (effectively d/dt), as in p·f(t). He also assumed that the inverse of p, or 1/p, is the operator performing the integral of a function, so that p · 1/p = 1/p · p = 1. This is not generally true because when the integral is taken after the derivative there is a constant term, but it is true for a function where f(0)=0. So we have

v  =  iR
i  =  Cpv     from v = (1/C)  ∫ i dt
v  =  Lpi      from v = L di/dt

Then he separated the operators from their functions, giving them an independent existence, to form a general AC impedance defined as v/i that he called Z (a term we still use today):

Z  =  R
Z  =  1 / (Cp)
Z  =  Lp

Heaviside generally treated problems in which a constant voltage is applied to a circuit at time t=0, in other words an impulse (or step function) such as might be encountered as transient signals on cables. In telegraphy these transient effects limit the signaling speed, while in telephony the transient effects limit the line length. But this step function also has the property that f(0) can be treated as 0, the condition for the commutivity of the inverse operator given above. Using the superposition property from this waveform also allowed him to analyze distortion at all signal frequencies. In fact, Josephs remarks that, explicit or not, it is always assumed in the differential equations for transient currents that the applied voltage is 0 for t<0. Heaviside wrote this step function as the bold symbol 1. So if we assume we have a circuit with a resistance and inductance,

Z  =  R + Lp

Then for v = 1 here,

i  =  v/Z

   =  1 / (R+Lp)

   =  (1/R)  [1 / (1 + Lp/R)]

   =  (1/R)  [R/(Lp) / (R/(Lp) + 1)]

   =  (1/R)  [R/(Lp)]  [1 / (1 + R/(Lp))]

   =  (1/R)  [R/(Lp)]  [1 – R/(Lp) + (R/(Lp))2 – (R/(Lp))3 + · · ·]    by the Binomial Theorem

   =  (1/R) [(R/L) · 1/p – (R/L)2 · 1/p2 + (R/L)3 · 1/p3 – · · ·]

Now for the step function 1, the “integral” 1/p · 1 = t, and in general 1/pn · 1 = tn/n! , so

i  =  1/R [R/L · t – (R/L)2 · t2/2! + (R/L)3 · t3/3! – · · ·]

But the power series expansion of e-t is

e-t  =  1 + t + t2/2! + t3/3! + · · ·

so i can be rewritten as:

i  =  1/R [1 – e-(R/L)t]

yielding the correct exponential rise of current through the circuit with a time constant of 1/(R/L) = L/R.

To solve the differential equation y” – y = 0 for t>0 and y(0)= 0 and y'(0) = 0, Heaviside would rewrite this as p2y – y = 1, or y = 1 / (p2 –1). But 1/(p2 – 1) can be expanded into the series (1/p2 + 1/p4 + 1/p6 + · · · ), and since we have 1/pn · 1 = tn/n! from above, then y = t2/2! + t4/4! + t6/6! + · · ·  =  ½(et – e-t) – 1. You can see how easily one can get in the habit of dropping the 1 altogether and working solely with p’s once the knack is acquired.

As another example, Heaviside presented an operator function [p/(p+B)]1/2 acting again on a step function. Here he divides through by p to get (1 + B/p)-1/2 and again expands this into a power series, arriving at a solution in terms of modified Bessel functions. Can p’s be handled in these ways? Apparently they can, for Heaviside was able to provide correct solutions using his operator calculus.

Now in circuits with continuous, distributed impedances such as in telegraph lines, (particularly distortionless ones in which an infinite current appears instantaneously at t=0) Heaviside was faced with fractional powers of p such as p1/2. Unfazed, he found a specific problem that had been solved using Fourier series methods (the solution of the diffusion equation) and applied it to a problem for which p1/2 appeared when written in his operator form (the current from a step voltage in an infinitely long cable). Heaviside equated the form of his solution to the Fourier solution to deduce p1/2 and he declared the result to be generally true! (Later, Heaviside presented a direct derivation based on the gamma function, but it is also derivable using Carson integrals and other methods).

He then arrived at his fractional powers of p:

    p2 1  =  1 / (πt)1/2

 p3/2 1  =  p(p1/2 1) = – t-3/2 / 2π1/2

p-1/2 1  =  t1/2 / (1/2)!

p-3/2 1  =  1/p (p-1/2) 1

and so forth.

Heaviside developed the Heaviside Expansion Theorem to convert Z into partial fractions to simplify his work. For i=1/Z and Z a polynomial in p, the roots of Z can be found and i expressed as a sum of terms consisting of constants divided by the simpler factors. A similar thing is done when using Laplace Transforms, but Heaviside developed his own method of calculating these constants, the Heaviside Cover-Up Method (See this webpage for a description).

Heaviside expressed the use of this theorem for a step function as

i  =  v/Z0 + v ∑ ept / [p(dZ/dp)]

So let’s look again at our first problem above of a series circuit of a resistance and an inductance, where Z = R + Lp and the steady state Z0 = R. Now dZ/dp = L and setting R+Lp = 0 implies a root p = – R/L. Then from the equation just above after setting the step value v to 1:

i  =  1/R + e-(R/L)t / [(–R/L) · L]


i  =  1/R [1 – e-(R/L)t]

which is the same solution we obtained earlier using power series.

Nahin describes Heaviside’s ingenious solutions when using his operators for the time-varying current in circuits with additional continuously-distributed parameters such as found in actual telegraph lines. He provides an example in which Heaviside adds a section of cable to the beginning of an infinite line, finds the current as a function of time for that configuration, and then “removes” the initial section to end up with the solution for the original cable. In fact, Heaviside used his operator calculus to design a transmission line with zero distortion (but with exponential attenuation over distance).

Now when Z is a polynomial of degree greater than 4, its roots are difficult or impossible to find directly. Also, the Expansion Theorem does not work for a Z with a root of zero or with any repeated roots, a situation not encountered in passive networks but one that can occur when an internal component such as an amplifier sources energy. In true Heaviside fashion we can in this case treat equal roots as unequal, solve for the transient current, and let the roots approach equality as a limit!

But Heaviside removed even these difficulties by instead expanding Z in inverse powers of p and then replacing p by tn/n! exactly as we saw in our first example. I’ve seen this referred to as Heaviside’s Extended Expansion Theorem, and I really have to wonder whether punchier names for his methods would have preserved more interest in them. Heaviside, who coined some terms much disliked in his day but which have stuck (such as impedance, inductance, conductance, admittance and reluctance), awkwardly referred to “algebrising” a differential equation with his operators and “logarising” when taking a logarithm, and he christened his operator e-ph the “Spotting function” because it isolates, or spots, a certain value of the function. Along with his “Cover-Up Method” these are not exactly memorable names.

Now as one more example of this method, consider a series circuit of R, L and C, where Q represents the charge in the circuit, so the current i = dQ/dt = p·Q:

Z  =  R + pL + 1/pC

i  =  1 / [R + pL + 1/pC]

[R + pL + 1/pC] i  =  1

[p2L + pR + 1/C] Q  =  1

Q  =  1 / [p2L + pR + 1/C]

1 { p2L · [1 + (R/L)p-1 + (1/LC) p-2]-1}

Expanding the term on the right using the Binomial Theorem, we have

Q  =  (1/L) · {p-2 · [1 – [(R/L)p-1 + (1/LC)p-2] + [(R/L)p-1 + (1/LC)p-2]2 – · · · }

     =  (1/L) · {p-2 – (R/L)p-3 – [(1/LC) + R2/L2]p-4 + · · · }

and as before we replace p-n · 1 by tn/n! to arrive at a power series in t without having to find the roots of Z. These power series are sometimes not expressible in terms of elementary functions, but Heaviside dismissed this with the astute observation that calling the sum of a particular power series an exponential or trigonometric function didn’t simplify finding the original solution of it.

In 1893 Heaviside published the first of a three-part series describing his operator calculus in the Proceedings of the Royal Society. Later in the year the second part appeared, but this “was the last straw for mathematicians” [Nahin], and his third part was rejected, leading to a series of written attacks best enjoyed by reading Nahin’s book.

What killed the third part was Heaviside’s cavalier use of divergent series, dismissing their apparent tendencies to infinity while producing accurate results by manipulating them at whim, or at least this was how it appeared. In fact, Heaviside often produced two versions of his power series solution, a convergent one that was useful for small t but was too slow to converge for large t, and a divergent one that was useful for large t when it was taken to a small number of terms. Josephs provides an example of two such series for the Bessel function solution of a current entering a particular type of transmission line:

e-φtI0(φt)  =  1 – (φt) + (1·3/(2!)2) (φt)2 – (1·3·5/(3!)2) (φt)3 + · · ·

e-φtI0(φt)  =  (2πφt)-1/2 [ 1 + (8φt)-1 + 12·32/(2!) · (8φt)-2 + 13·33·53/(3!) · (8φt)-3 + · · · ]

The first expression is convergent but is slow to converge for larger values of φt, while the second expression is a divergent series of an asymptotic type (a series in inverse powers of the argument). The plot below shows the effect of taking terms in each series for relatively large values of φt. The first expression is shown as convergent, but slowly, while the second expression converges quickly to approximately the final value of the convergent expression as the values of the terms decrease, then takes off and rises to much larger values. The larger the value of t, the less is the error at the point of the minimum term. The trick is to stop taking terms after the one with the smallest value, which Heaviside could do empirically.


In general, Heaviside found that if an operational equation is a series in integral and fractional powers of p, he could discard the terms with integral powers of p, express the fractional powers of p as pn p1/2 or pn (πt)-1/2, and create a divergent series in pn useful for large t. Josephs provides an example of this procedure in deriving a convergent and divergent series for the voltage at the terminals of a non-inductive cable when a 1-Volt battery is applied through a terminal resistance.

Heaviside used physical intuition to guide him in handling these series, and he was unparalleled in his electromagnetic intuition. And of course he went far beyond the short flavor of his operator calculus I’ve described here—Heaviside found solutions for very complicated electromagnetic problems whose solutions were intractable by any other method at the time.

But Heaviside also checked his steps with exhausting numerical calculations to make sure he didn’t make a misstep. I really appreciate this because I have often done that in deriving approximations to elementary functions, an interest of mine apparent on my MyReckonings.com website—I was really gratified to find a professional doing something like this. Heaviside also took his solution and verified that it satisfied the original equation, although of course this doesn’t validate the uniqueness of it. Because Heaviside’s work was results-oriented, he sometimes provided ad hoc arguments to support his derivations. He often suggested in rather abrupt prose that mathematicians should provide rigorous proofs for what he did. I see a bit of his humanity in a footnote in Volume 2 of his Electromagnetic Theory (a volume almost exclusively concerned with his operator methods):

It is rather disagreeable to have to be self-assertive and dogmatic (especially when one thinks of the always possible risk of error); but there may be times when it becomes a duty—e.g., when mathematical rigourists are obstructive.

And Heaviside certainly did exercise the option, often with a Victorian flair for the erudite put-down:

I think I have given sufficient information to enable any competent person to follow up the matter in more detail if it is thought to be desirable. It is obvious that the methods of the professedly rigorous mathematicians are sadly lacking in demonstrativeness as well as in comprehensiveness.

Mathematicians were not amused, but I am.


Heaviside, Oliver. Electromagnetic Theory, Vols. 1-3. New York: Cosimo Classics (2007). These three volumes, available at Amazon, are reprints of Heaviside’s books on electromagnetic theory. Volume 2 contains a modified version of the rejected Part III of his series on operators for the Proceedings of the Royal Society. The three volumes were also printed as one large book by Dover in 1950.

Heaviside, Oliver. On Operators in Physical Mathematics, Part I. Proceedings of the Royal Society, vol. 52, Feb., 1893, pp. 504-529. Online scans of this can be found at http://gallica.bnf.fr/ark:/12148/bpt6k56145p/f512.item and its following page selections.

Heaviside, Oliver. On Operators in Physical Mathematics, Part II. Proceedings of the Royal Society, vol. 52, Feb., 1893, pp. 504-529. Online scans of this can be found at http://gallica.bnf.fr/ark:/12148/bpt6k56147c/f112.item and its following page selections.

Josephs, H.J. Heaviside’s Electric Circuit Theory, 2nd Ed. Methuen’s Monographs on Physical Subjects, London: Methuen and New York: Wiley (1950). This tiny book (without the cover I measure it at 6-5/8″ x 4-1/8″ x ¼”) is packed with useful information on Heaviside’s operational calculus.

Lindell, I.V. Heaviside Operational Rules Applicable to Electromagnetic Problems. Progress in Electromagnetics Research, PIER 26 (2000), pp. 293-331, also found at http://ceta.mit.edu/PIER/pier26/11.9909172jp.Lindell.pdf. A comprehensive, highly mathematical collection of valid rules for Heaviside’s operator calculus along with their derivations.

Nahin, Paul J. Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age. Baltimore: Johns Hopkins University Press (1988). This is one of my favorite books of all time, a fascinating biography of Heaviside and his contemporaries and brimming with the controversies he engendered. It has a technical bent as well, as do all of Nahin’s excellent books.

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22 thoughts on “Heaviside’s Operator Calculus

  1. Jagger January 5, 2008 / 12:37 am

    “computational observation” – hmmm I am quite intrigued. Nice discussion


  2. Ed Borasky January 13, 2008 / 9:06 pm

    There’s one other piece of the story. As you note, Laplace transforms replaced the Heaviside operational calculus. But there is yet another form, discovered/invented by Mikusinski, which is purely algebraic in nature. http://en.wikipedia.org/wiki/Jan_Mikusinski. He started with a ring of functions under two operations — pointwise addition and (Laplace) convolution. Then using a well-known theorem, he showed that this ring was in fact an integral domain — there are no zero divisors. Given an integral domain, he constructed a field in the same manner one constructs the rationals from the integers or rational functions as ratios of polynomials. I think the book is out of print now, but there are a few available on line.
    Thank you for your comment, Ed—it makes a nice addendum to the essay! I had seen Mikusinski’s name when I was looking for references on Heaviside, but I didn’t pursue it. It’s also a fact that abstract algebra is not a strong suit of mine. I see that Mikusinski’s “Operational Calculus” is held by a local university library, and now I’m intrigued enough to retrieve the book and have a go at it. We’ll see.
    Again, thanks for describing Mikusinski’s achievement in “algebrising” differential equations. I’m continually amazed at how rich the mathematical experience really is, and how knowledgeable people are on sophisticated topics such as this. — Ron


  3. slon April 10, 2009 / 2:42 pm

    Heaviside was the real man.


  4. Anonymous April 19, 2009 / 9:20 pm

    Can I please ask you to convert the equations to a more readable format? Plain text formulas don’t look very good. 😦
    This was one of my early essays. In later essays I started creating the equations in LaTeX or Word Equation Editor and inserting them as images, which is infinitely better, particularly considering what the Georgia font does to numbers, etc. When I get a chance I’ll redo these. Thanks for the suggestion–I had totally forgotten that this essay had equations rendered in text. Meanwhile, if you click on the link at the bottom of the essay for the printer-friendly PDF version, you will see that the equations are rendered much better in the Times New Roman font of that file.
    ps. also what the heck is with that anti-spam thing – do you really expect people to read that they have to add 80?…
    🙂 The WordPress Askimet plug-in has caught over 7,000 spam comments submitted to this blog over the last year or so. It was only a matter of time before spam got through their filter. So I started using a WordPress anti-spam plug-in that required just the simple sum to be calculated, but it only slowed the spam mostly because, I guess, spambots worked around it. By modifying the plug-in code to require adding another value (chosen as 80) written in the text above the box, I’ve really reduced the spam that gets through to the Askimet filter. It’s also true that if an incorrect sum is submitted, a message appears reminding the commenter to add the 80. — Ron


  5. A.B.Leal April 20, 2009 / 6:42 am

    “The best result of mathematics is to be able to do without it.”
    — Oliver Heaviside
    Many moons ago, I had this quotation neatly penned and displayed in my study area. I must have found it in a book or magazine (yeah, those days ;-). Right now, Google only finds it in 3 pages from a couple of sources, apparently in audio electronics circles.
    It may be spurious, but it’s at least “bene trovato” … Any confirmation/source gratefully received.
    I haven’t heard this quote, but Heaviside was prolific in writing letters as well as articles, so I’m not surprised that he might have written it and not surprised that I haven’t come across it. Sounds just like him, though. If I come across the source of the quote I’ll drop you an email. Thanks for sharing it. — Ron


  6. Scott Hamilton March 22, 2010 / 12:18 pm

    Came across your site from a reference at http://newsletter.planetanalog.com/cgi-bin4/DM/y/eBXtQ0NQLGi0Frc0HeII0EF . Much of interest but I am particularly glad to see your piece on Heaviside who, as you say, is not well known these days despite his considerable contributions to electromagnetism. When I wrote my (electronics) book I found his three volumes a splendid source of appropriate references to head each section e.g.:
    Prof. Klein distinguishes three main classes of mathematicians—the intuitionists, the formalists or algorithmists, and the logicians. Now it is intuition that is most useful in physical mathematics, for that means taking a broad view of a question, apart from the narrowness of special mathematics. For what a physicist wants is a good view of the physics itself in its mathematical relations, and it is quite a secondary matter to have logical demonstrations. The mutual consistency of results is more satisfying, and exceptional peculiarities are ignored. It is more useful than exact mathematics.
    But when intuition breaks down, something more rudimentary must take its place. This is groping, and it is experimental work, with of course some induction and deduction going along with it. Now, having started on a physical foundation in the treatment of irrational operators, which was successful, in seeking for explanation of some results, I got beyond the physics altogether, and was left without any guidance save that of untrustworthy intuition in the region of pure quantity. But success may come by the study of failures. So I made a detailed study and close examination of some of the obscurities before alluded to, beginning with numerical groping. The result was to clear up most of the obscurities, correct the errors involved, and by their revision to obtain correct formulae and extend results considerably.
    Oliver Heaviside, Electromagnetic Theory, April 10, 1899. Vol.II p.460.
    Since Laplace transforms are now used in preference to Heaviside’s operators, may I suggest a topic examining the relation between these?
    ‘The Laplace Transform is one of the many contributions to mathematics and physics of the Marquis Pierre Simon de Laplace (1749-1827), who pointed out the biunique relationship between the two functions and applied the results to the solution of differential equations in a paper published in 1779 with the rather cryptic title “On what follows”. The real value of the Laplace transform seems not to have been appreciated, however, for over a century, until it was essentially rediscovered and popularized by the eccentric British engineer Oliver Heaviside (1850-1925), whose studies had a major impact on many aspects of modern electrical engineering.’
    ‘The use of impulse functions in science and engineering was popularized by the English physicist P.A.M.Dirac and by Oliver Heaviside long before impulses became “respectable” mathematically. Indeed we continue to use Dirac’s notation, δ(t), and the unit impulse is often called Dirac’s δ-function. Both Dirac and Heaviside stressed the idea that δ(t) was defined in terms of what it “did”. Thus Dirac said, “Whenever an improper function [e.g. impulse] appears it will be something which is to be used ultimately in an integrand—the use of improper functions thus does not involve any lack of rigour in the theory, but is merely a convenient notation, enabling us to express in a concise form certain relations which we could, if necessary, rewrite in a form not involving improper functions, but only in a cumbersome way which would tend to obscure the argument.’ Both quotes from Siebert W.McC. (1986): Circuits, Signals, and Systems; Cambridge, Mass.: MIT Press/ New York: McGraw-Hill. ISBN 0-07-057290-9.
    One needs, however, to beware of Stigler’s law:
    Stigler’s Law of Eponymy—“No scientific discovery is named after its original discoverer”. The law itself may be construed to contradict Stigler’s eponym. The Gaussian distribution was probably originated by de Moivre. It may also be noted that Laplace used ‘Fourier transforms’ before Fourier’s publication, and that Lagrange used ‘Laplace transforms’ before Laplace began his career. See p278. As Stigler quotes one historian’s view (from an unknown source) ‘Every scientific discovery is named after the last individual too ungenerous to give due credit to his predecessors’.
    Stigler S. (1999): Statistics on the Table: the History of Statistical Concepts and Methods; Cambridge: Harvard University Press. ISBN 0-674-83601-4.
    Thank you, Scott, I enjoyed reading this! I found your book (An Analog Electronics Companion: Basic Circuit Design for Engineers and Scientists is a link to the US Amazon site) and it looks very interesting, a physicist’s take on analog electronic design—not unlike Heaviside, really. I look forward to reading your book, as I used to be an analog/digital circuit design engineer at one point in my career. Your book looks like a refreshing read on the subject that I would particularly like.
    I appreciate you taking the time to contribute to my blog. I’m a bit proud of the quality of the comments people make here; they certainly enhance the essays and the overall value of the site. — Ron


  7. TheHeadlessSourceMan December 8, 2010 / 3:08 pm

    Okay, so the Laplace Transform replaced Heaviside’s operator calculus, but even so the full Bilateral Laplace Transform was still derived using Heaviside’s step function!!
    To quote slon, above, Heaviside was THE MAN!
    Very interesting! I didn’t know that before. Thanks for the pointer. — Ron


  8. Tom Copeland November 27, 2011 / 9:20 pm

    Just a small aside: Needham in his book Visual Complex Analysis discusses an amplitwist interpretation of complex differentiation. I remember first coming across this idea several years ago in one of Heaviside’s publications.
    Hi Tom. Thanks for your comment–I had never heard of this concept. I found that chapter of Needham’s book on one of his Visual Complex Analysis pages on the University of San Francisco website at http://usf.usfca.edu/vca//PDF/amplitwist.pdf
    Really interesting stuff! I was just looking up the Jacobian last night for its use in nomography theory, and there it was in this article. The author writes very clearly and has a really engaging style. I’m going to set out some time to read through his book.


  9. TX November 30, 2011 / 4:10 pm

    Great article. However, I believe that there’s a typo in your example regarding the equation y” – y = 0 for t>0 . I think it should be y” – y = 1 for t>0 because you go on to use p2y – y = 1.

    Just a heads-up because I’m working on a presentation about Heaviside and looking for examples of his operator method, and your site comes up pretty high in the search rankings.


  10. Tom Copeland April 27, 2012 / 7:57 am

    This is a follow-up on my earlier comment on Heaviside’s use of an amplitwist interpretation of complex differentiation/antidifferentiation, or at least a precursor to that idea. See page 436 eqn. 5 of Electromagnetic Theory Vol. II by Heaviside at http://archive.org/details/electromagnetict02heavrich. You have to mull over his idiosyncratic presentation with antiderivatives, but you can see his clever, intuitive angle on this idea.


  11. Tom Copeland April 27, 2012 / 6:39 pm

    BTW, the book that motivated me to delve a little more deeply into Heaviside’s style and ingenuity and his legacy to modern math is Friedman’s Lectures on Applications-Oriented Mathematics.
    Tom, I just saw your two comments here! Sometimes I get no notification when a comment is posted, and this is one example of that. Let me look through your reference in your previous comment and I’ll get back to you as soon as I can. Thanks! — Ron


  12. Frank Firk March 16, 2014 / 8:48 pm

    March 16, 2014
    Before I became a card-carrying Nuclear Physicist, I was an electronics technician, not only fixing complicated equipment but also designing new systems for detecting nuclear particles, and for nuclear spectroscopy. At the time, (the early 1950’s) we were in the golden age of do-it-yourself physics. It was before the rise of giant electronics companies with their own research facilities. If we needed an ultra low-noise pre-amplifier for observing minute signals from some rare nuclear event, we had to design and build the equipment. We were in the “millimicrosecond” era, before transistors, and before printed and integrated circuits. Most of our work involved dealing with very short, transient pulses. We were all day-to-day users of Heaviside’s Operational Calculus. I recall giving a series of lectures at the Atomic Energy Research Establishment at Harwell in England (where I worked) to the Nuclear Physicists who were using our locally designed electronics. They were quite ignorant of his great contributions to Applied Mathematics, Physics and Engineering.
    The two most important books that we used at the time were:
    1. “Operational Methods in Applied Mathematics” by H. S. Carslaw and J. C. Jaeger,
    2nd edition, Oxford University Press (Oxford) 1953. (This is an outstanding
    2. “Millimicrocecond Pulse Techniques” by A. I. D. Lewis and F. H. Wells,
    3rd impression, Pergamon Press Ltd., (London) 1956.
    These books are not listed in your references.
    It is interesting to note that Dirac, one of the greatest theoretical physicists of the 20th century was, in fact, an undergraduate in Electrical Engineering when at Bristol University in the early 1900’s. I am certain that he would have been familiar with Heavisides’s Operational Calculus and, in particular, with his step function. The Dirac delta function then follows in a natural way.
    Frank Firk,
    Professor Emeritus of Physics, former Chairman of the Department of Physics, and Director of the Electron Accelerator Laboratory, Yale University.
    Thank you very much, Prof. Firk! I appreciate you taking the time to write this. I will find the two references you cite and read through them. I was in experimental nuclear physics in graduate school, and that was a field that, unlike high-energy particle physics, also involved designing our own circuitry as well as mechanical hardware. Today I’m a systems engineer and I find that electrical engineers I work with are not familiar at all with Heaviside. Also, thanks for pointing out the quite feasible connection between Dirac and Heaviside’s step function, which is really intriguing. — Ron


  13. Tom Copeland February 3, 2015 / 7:14 pm

    A great reference on Heaviside’s influence on Dirac is
    Masters of Theory: Cambridge and the Rise of Mathematical Physics by Andrew Warwick  (e.g., pg. 508).
    Tom, thank you very much for your contributions here! I have been remiss lately on replying to quite a few comments by various people, but I do appreciate the value that people add to the blog by their participation. I am quite proud of the quality of the comments posted here by knowledgeable people like yourself who share these interests, and I learn a lot from them. — Regards, Ron


  14. Ken Abramson February 5, 2015 / 2:24 pm

    I just found this article on Heaviside’s Operator Calculus on the internet. I have no idea who wrote this, or when or where it was published.
    I am a retired Electrical Engineer looking for some field of mathematics to study in order to stay busy.
    To the author, whoever he is: Thanks for a great article. I know know what I will be doing in my spare time.
    Thanks, Ken! I wrote this article for this blog after researching Heaviside as a matter of personal interest, and it has never been published anywhere except other websites that picked it up. Prof. Firk and Tom Copeland have provided some really great references in the comments here on the blog in addition to those listed at the end of my article.
    Paul Nahin’s book on Heaviside is a joy to read, but you may also be interested in Nahin’s other books–he is a retired professor of Electrical Engineering at the University of New Hampshire and he writes fascinating books with the kind of mathematical detail you see here in this article. I have enjoyed his older book on “The Science of Radio” as well as his newer books on methods of extrema titled “When Least is Best”, his book on pursuit theory titled “Chases and Escapes”, his book on imaginary numbers and how they apply to electrical engineering and other fields in “An Imaginary Tale”, and several others. These are aimed at people like you and me, and I think you would really like some of these. — Ron


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