I am looking for references on Steinmetz Transform and its relation with Laplace and Fourier transforms. There is an Italian Wikipedia page about this topic but with no references.
2 Answers
(Too long for a comment.) I think the italian Wiki page is wrong. It says the transform "was conceived by the author in 1893 [probably this text] and exposed in his treatise Theory and Calculation of Alternating Current Phenomena four years later."
However, as you can check there is not a trace in these texts (nor in the others authored by Steinmetz and available on archive.org) of what Wikipedia calls "la transformata di Steinmetz"  i.e. the Fourier isomorphism $L^2(S^1)\to\ell^2(\mathbf{Z})$  which by the way, was written explicitly much before 1893.
These texts are famous for introducing complex numbers (in particular the notation $j=\sqrt{1}$) into electrical engineering, but my impression is that naming the transform in Steinmetz's honor happened much later, perhaps in Italy, with little regard to what he himself actually did (or as the case may be, never did) with it.

2$\begingroup$ "This text": Steinmetz  Complex quantities and their use in electrical engineering. $\endgroup$– LSpiceCommented Sep 13, 2020 at 18:50
The "Steinmetz transform" is most likely referring to the phasor method. For an introduction, see "Steinmetz and the Concept of Phasor: A Forgotten Story" by Araújo and and Tonidandel. (Caveat: the history of the method and characterizations of Heaviside and his and others' possible influence on Steinmetz are more accurately presented in the refs in the historical notes below.)
Historical notes:
Charles Proteus Steinmetz, the guru of Electric City, was a fascinating character and quite famous in his day"In 1922, Thomas Edison came to visit Steinmetz. By then, Edison was nearly deaf, and Steinmetz tapped out a message on Edison’s knee in Morse Code. Edison beamed, and the two continued their silent conversation in front of bewildered reporters" from Charles Proteus Steinmetz, the Wizard of Schenectady. See also the two photo ops with Einstein and luminaries in the birth of telephony and radio in Professor Einstein Looks into Radio.
For any history buff, the PBS specials "Divine Discontent: Charles Proteus Steinmetz" and "Divine Discontent: Charles Proteus Steinmetz Extended Roundtable" are a must see.
For detail on the evolution of the method (and the slow introduction of imaginary numbers in the American and European curricula and possible influence of Heaviside), see Charles P. Steinmetz and the Development of Electrical Engineering Science by Kline. Certainly Heaviside influenced Stenmetz's associates E. J. Berg and J. R. Carson. According to Kline, "Berg's conversion to the Operational Calculus came sometime between 1912 and 1918." Berg mentions Heaviside in his lecture notes in 1916. Carson published "The Heaviside Operational Calculus" in Bell System Technical Journal in 1924. Heaviside used his operational/fractional calculus in his characterization of electric circuits, particularly in telegraphy. Both Carson and Berg are with Steinmetz and Einstein in the first photo op.
For a more accurate portrayal of Heaviside's relationships with others, see The Maxwellians by Hunt.
This is my original answer posted on Sept. 13, 2020, emphasizing the lack of mutual acknowledgement of the two prominent contemporaneous 'electricians' Steinmetz and Heaviside on opposing sides of the Lake. I've decided to resurrect it after reading Kline.
Read "Steinmetz and the Concept of Phasor: A Forgotten Story" by A. E. A. Araújo & D. A. V. Tonidandel.
Berg, Carslaw, and Carson were colleagues of Steinmetz and promoted through various publications (near or after the death of Steinmetz, perhaps tactfully) the operational calculus of Heaviside, based on the Laplace (and Mellin transform), which can handle causal and transient electric circuit phenomena that the Fourier transform can't, yet Steinmetz never mentioned Heaviside nor maybe even the Laplace or Mellin transform in his publications that I have perused (he does, Fourier series and harmonics)—whether through ignorance, intentional neglect, or lack of necessity I don't know.
Steinmetz—The Forger of Thunderbolts, The Wizard of Schenectady—is a fascinating character, living and working in the Silicon Valley of the times—Electric City (Schenectady, New York, circa 1890s)—ushering in the age of electricity with other luminaries such as Edison, Westinghouse, Tesla, Marconi, and Einstein (don't forget Einstein got the Nobel for the photoelectric effect and there is a photo of him posing right next to Steinmetz at a Marconi wireless station).
I'm fairly sure Steinmetz developed his mathematical approach independently of the others I've mentioned. Proteus (Steinmetz) was regarded as a demigod at G.E. (vetting all new ideas) and was promoted as such by the astute P.R. of G.E. (they even "photoshopped" the photo I mentioned to remove everyone but Steinmetz and Einstein), but I don't recall, from my scannings of their work, him or his associates Carslaw or Carson referring to a Steinmetz transform. Perhaps the phasor concept comes closest to fitting the bill, or the Steinmetz circuit was translated as the Steinmetz transform.
A great documentary, free until Dec 2020: Divine Discontent — Charles Proteus Steinmetz.

1$\begingroup$ The Maxwellians also gives a wellrounded picture of Heaviside's contributions to mathematics, engineering, and physics, which were substantial. $\endgroup$ Commented Oct 9, 2022 at 21:55