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Complex number are extremely useful in every branch of physics dealing with ondulatory phenomena. In electromagnetism, for example, they allow to write the solution of Maxwell's equations in a form which is particularly simple to manipulate.

When did the use of complex numbers become widespread in physics?

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It depends on the definition of "widespread". Mathematicians certainly used them in 18th century. They play prominent role in the work of Cauchy and Fourier (which belongs to both physics and mathematics). Probably they entered physics with the work of Fresnel on the wave theory of light (1823). Maxwell uses them in his Treatise on Electricity and Magnetism (1873). The standard textbook on mathematical physics of the late 19th century (Kelvin and Tait) freely uses them.

Remark. When searching the last mentioned book use the word "imaginary", not "complex". I found this word used many times without any special explanation: the use of complex numbers by that time was routine. Another English word used early in 19th century was "impossible". J. B. Airy talks about "impossible roots" meaning "complex roots".

Second remark (to address the issue raised in comments). The engineers started using complex numbers since the criteria of stability were discovered. (It is due to Maxwell (1868) that for stability of a linear system the roots of the characteristic polynomial must lie in the left half-plane.) Since then, all engineers designing steam engines use complex numbers. Maxwell was not the first to address stability in engineering from the mathematical point of view, but he gave the first general criterion. The first was J. B. Airy, an astronomer (1840). And any criterion of stability uses complex numbers. The names of 19 century engineers working on stability are still mentioned in the complex variables textbooks (Stodola, Vyshnegradski).

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    $\begingroup$ Following your last sentence about Kelvin and Tait, Charles Proteus Steinmetz (1865–1923) should probably be mentioned as being primarily responsible (1900s and 1910s) for their introduction into electrical engineering, and thus into more mainstream use. In "popular literature" (magazine articles, non-technical scientific books, etc.), I suspect they fully entered the public's eye with all the 1920s popularizations of relativity theory (and popularizations of quantum mechanics in the 1930s). $\endgroup$ – Dave L Renfro Jan 17 '18 at 18:30
  • $\begingroup$ I think you're mistaken about Kelvin and Tait. I searched pretty thoroughly through both volumes archive.org/details/treatiseonnatura01kelv_0 archive.org/details/treatiseonnatura02kelv_0 and found no mention of complex or imaginary numbers, except in one spot in Appendix G, where it appears that imaginary roots of a polynomial are only mentioned as being physically irrelevant. "Complex," "imaginary," and "quaternion" are all absent from the index. If complex numbers were going to show up, you would think they would show up in the treatment of Fourier analysis, but they don't. $\endgroup$ – Ben Crowell Jan 17 '18 at 21:09
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    $\begingroup$ @Ben Crowel: They were not called "complex" at that time. Use your computer search engine for the word "imaginary", or see formula (45), section 345 of the first volume. They use then routinely, without feeling a need for special explanation or inclusion into index. $\endgroup$ – Alexandre Eremenko Jan 18 '18 at 0:37
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    $\begingroup$ @AlexandreEremenko: I was sloppy and only searched for "imaginary" in the second volume. You're right, it appears much more frequently in the first volume. However, I think the main thrust of your answer is just plain historically wrong. Physicists and engineers in general were not commonly using vectors, complex numbers, or quaternions until about 1920-1940, as discussed in my answer. A good reference on this topic is Crowe, A history of vector analysis. $\endgroup$ – Ben Crowell Jan 18 '18 at 15:52
  • $\begingroup$ @Ben Crowel;: I do not want to argue with you, but I edited my answer, adding information about engineers. The original question was about physicists. Vectors are much later "invention" than complex numbers. $\endgroup$ – Alexandre Eremenko Jan 18 '18 at 21:43
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Roughly speaking, complex numbers seem to have become widely used in physics and engineering around 1940, after gradually gaining popularity starting around 1920.

In the late 19th and early 20th centuries, there were three algebraic systems of numbers competing for attention: complex numbers, quaternions, and vectors. Vectors were late to the party. Maxwell used quaternions (1873), and they were enthusiastically promoted by their adherents, especially Hamilton, for purposes that today we would use vectors for. Quaternions can be viewed as a generalization of both vectors and complex numbers. Most physicists considered quaternions to be very abstruse. As late as 1905, Einstein's paper on special relativity states Maxwell's equations without the use of either vector or quaternion notation. The 1895 edition of Kelvin and Tait's massive Treatise on Natural Philosophy has no index entries for "complex," "imaginary," "vector," or "quaternion," and complex numbers are used only infrequently. Tait wrote an 1890 book on quaternions, so it's not as though he didn't know about them. They just weren't the kind of thing that readers could be expected to know about.

Gibbs and Heaviside created the vector system as an easier alternative to quaternions, and it gradually gained popularity. Gibbs described it in an 1881 book, which he promoted vigorously and successfully.

Kennelly introduced the notion of a complex impedance in 1893, and this was recognized as a big selling point of the use of complex numbers. A popular 1941 textbook by Stratton, Electromagnetic Theory, used complex numbers. So I think the widespread use of complex numbers in physics and engineering must date to approximately 1900-1940. To pin this down a little more, I think it's helpful to look at the google ngrams graphs for "complex impedance" and "complex exponential," which both show nonnegligible usage starting around 1920, and an inflection point around 1940.

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