# Origin of the notation $s = \sigma + j\omega$ in electrical engineering/control theory

In analytic number theory it is traditional to write a complex variable as $$s = \sigma + it$$, with the letter $$t$$ going back to Riemann's paper on the zeta-function (1859) and the letter $$\sigma$$ having a more complicated explanation that was asked about on Mathoverflow here.

It recently came to my attention that in electrical engineering the notation $$s = \sigma + j\omega$$ is widely used, where $$\omega$$ describes a frequency. Engineers don't want to use $$t$$ as the imaginary part since $$t$$ always means time, and $$j$$ is a commonly used alternative to $$i$$ for $$\sqrt{-1}$$ since $$i$$ in electrical engineering often means current. My question is: who first introduced the notation $$s = \sigma + j\omega$$? I am wondering in particular if this use of $$\sigma$$ can be directly tied to the use of $$\sigma$$ as the real part of $$s$$ in analytic number theory. Edit: I am not wondering about the use of $$j$$ rather than $$i$$, so please don’t focus on that or refer me to this page, which does not answer my question.

One of the settings where the notation $$s = \sigma + j\omega$$ is used is in Bode plots, and I looked at several papers of Bode from the 1930s. In none of them did I find $$s$$ or $$\sigma$$ being used to describe complex numbers and their real parts. Complex numbers were introduced to electrical engineering by Charles Steinmetz, and in his 1893 paper "Complex Quantities and Their Use in Electrical Engineering" he introduced $$j$$ as $$\sqrt{-1}$$ and $$\tilde{\omega}$$ as a phase angle, but does not use $$s$$ or $$\sigma$$ in the above sense. He writes $$s$$ and $$\sigma$$ for completely different purposes: $$s$$ denotes inductance (a real quantity) and $$\sigma$$ denotes the EMF (electromotive force) of self-induction as a fraction of total EMF.

Note: This question is 1 day old and has 4 votes to close. I am baffled and would appreciate it if someone voting to close could please explain their reason for judging this question unsuitable to be asked here.

• It has been addressed here: hsm.stackexchange.com/questions/5082/… Feb 23, 2021 at 1:07
• @M.Farooq exactly where on that page is the origin of the use of $\sigma$ (and $s$) addressed? I saw that page before posting my question and I don't see $\sigma$ there at all in the sense I am asking about here. That page is just about $j$ as $\sqrt{-1}$, which is not what I am asking about and it is how I learned the Steinmetz introduced the use of $j$, which I write about in my question here.
– KCd
Feb 23, 2021 at 1:44
• I misread your query regarding $i$ vs. $j$. I feel this $s$=$\sigma$+$j\omega$ is somehow influenced by Laplace transforms notation. Feb 23, 2021 at 1:55
• From the English translation of Gustav Doetsch's book on Laplace Transforms he explains the choice of letter $s$. "Usually, we represent the complex variable (x + iy) by the letter $z$; here it is customary to use the letter $s$ which fits well to the time variable $t$, $t$ and $s$ being neighbours in the alphabet: $s$=$x+iy$. Feb 23, 2021 at 2:04
• Does this answer your question? Introduction of $\imath$ and $\jmath$ notations for the imaginary unit Feb 23, 2021 at 12:46

I'll be answering my own question, since an anonymous person reached out to me with the following information about the likely origin of the notation $$s = \sigma + j\omega$$ in electrical engineering.

1. Deakin's paper "The Ascendancy of the Laplace Transform and how it Came About" (Archive for History of Exact Sciences 44 (1992), 265-286) says that Gustav Doetsch wrote the first textbook treatment of the Laplace transform with his 1937 book Theorie und Anwendung der Laplace-transformation (Springer-Verlag). In this book, which was written for a mathematical audience rather than engineers and was never translated into English, Doetsch writes complex numbers as $$\sigma + iy$$ (and $$x+iy$$). His use of $$\sigma$$ is not a surprise, since Doetsch's PhD advisor was Edmund Landau (see here) and Landau was responsible for making $$\sigma + it$$ the standard notation for complex numbers in analytic number theory.

2. Gardner and Barnes wrote the book Transients In Linear Systems Studied By The Laplace Transformation in 1942, aimed directly at electrical engineers. They use the notation $$s := \sigma + j\omega$$ for complex numbers starting from page 12, where $$j = \sqrt{-1}$$ and $$\omega$$ is real. This book was largely responsible for the widespread adoption of Laplace transforms in electrical engineering education in the English-speaking world (replacing Heaviside operational calculus), so this answers my question about why the notation $$\sigma + j\omega$$ (especially the $$\sigma$$ part) is used in electrical engineering.

3. Widder's 1946 book The Laplace Transform uses the notation $$s = \sigma + i\tau$$ for complex numbers, and Widder writes in the preface that he learned about the Laplace transform from Hardy, who like Landau was an analytic number theorist. This book is aimed at mathematicians, not engineers, but again we see that the choice of notation $$\sigma$$ for $${\rm Re}(s)$$ here can be related to its use in analytic number theory.

• KCd, I had mentioned in the comments that Gutav another book has been translated as Introduction to the Theory and Application of the Laplace Transformation (Einfuhrung in. Theorie und Anwendung der Laplace-Transformation). My interest in his book was looking at the history of "deconvolution" for my article. Math Overflow has that discussion. It was translated by Walter Nader of University of Alberta. I am not sure how different it is from the German versions but both of them have the same number of figures. My German is not that good that I can do one to one comparison. Feb 25, 2021 at 2:35
• @M.Farooq In Deakin's 2nd paper on the Laplace Transform ("The Development of the Laplace Transform, 1737-1937 II. Poincare to Doetsch, 1880-1937", Arch. Hist. Exact Sci. 26 (1982) pp. 351-381) he writes about Doetsch's 1937 book "It was reprinted in the United States in 1943 (under wartime abrogation of copyright agreements) but, regrettably, has never been translated into English." See MathSciNet MR0009225 . The book translated by Nader (see MR0344810) is a translation of the 2nd edition of a book by Doetsch whose first edition came out in 1958 (see MR0107136), so it was not his 1937 book.
– KCd
Feb 25, 2021 at 2:55
• Okay thanks. You are right, the number of pages are different in both. I am not a mathematician but I find its history interesting. I will check the "The Development of the Laplace Transform" for my own general knowledge. Feb 25, 2021 at 3:02