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According to Florian Cajori, A History of Mathematical Notations (1928 - Dover reprint), Vol II, page 128 : 498. It was Euler who first used the letter $i$ for $\sqrt -1$. He gave it in a memoir presented in 1777 to the Academy at St. Petersburg, and entitled "De formulis differentialibus etc.," but it was not published until 1794 after the death of Euler....

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The original motivation, and an inkling of the integral formula, came from Cauchy's attempts to compute improper real integrals. Here is from A Brief History of Complex Analysis in the 19th Century: "Cauchy’s first work on complex integration appeared in an 1814 paper on definite integrals (improper real integrals) that was presented to the Institute but ...

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As far as I known, logarithms of negative numbers first appeared (in the modern sense) in the 1751 paper of Euler De la controverse entre Mrs. Leibniz et Bernoulli sur les logarithmes des nombres negatifs et imaginaires, you can find here the original and the English translation. As you will see, Euler discuss the different positions of Bernoulli and Leibniz ...

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According to Boyer's A History of Mathematics, the identity first appeared in Euler's Introductio in analysin infinitorum of 1748. Euler started from the infinite series for $e^x$, $\sin x$, and $\cos x$, from which the identity quickly follows. Thus from: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$ $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{... 7 I would say that there is no good book which satisfies your description. The Book of Bottazzini and Grey mentioned in the comments is OK, but it certainly does not cover the role of complex analysis in physics. A better book is J. Dieudonne, Abrégé d'histoire des mathématiques 1700–1900, Hermann, Paris, 1978, in 2 volumes, both volumes have chapters on ... 7 Yes, this happens many times. Mathematics developed purely for internal mathematical reasons, later turns out to have applications elsewhere. See the essay by Wigner, "The unreasonable effectiveness of mathematics in the natural sciences" I heard many years ago a lecture by Heisenberg. He told how, in finding the first things about quantum mechanics, he ... 6 There are (at least) two different types of numbers called "sedenions". The first ones were introduced by Muses in 1980, who called them 16-ions, and renamed into "sedenions" by Carmody in 1988. However, Sorgsepp and Lohmus beat him to the name in Binary and Ternary Sedenions (1981), which they used more in line with "quaternions" and "octonions" (sedecim ... 6 Whittaker and Watson credit this method to Mittag-Leffler, Acta Soc. Sci. Fenn., XI 1880 273-293, and Acta Math., IV 1884 1-79. 6 Usefulness was never seriously questioned. Complex numbers are useful, for example, if you want to make Cardano's formula work. And this is how they were invented. They are useful for other mathematical problems, and since the beginning of 19th century, in physics too. What was questioned sometimes is their justification, existence. These questions ... 5 For an early occurrence of "iota", see : Alfred Cardew Dixon (English mathematician, 1865-1936), The elementary properties of the elliptic functions. With examples (London, 1894), p. 5 : §10. writing \iota for \sqrt {-1}, [...]. 5 There is no difference in differentiation. Derivative of a complex function of a complex variable is defined by the same formula$$f'(z)=\lim_{h\to 0} (f(z+h)-f(z))/h$$as for the real variable. Concerning integration, one reason was the desire to investigate integrals of real functions. One of the consequences (and motivations) of Cauchy theory is that it ... 5 The theory of analytic (holomorphic) functions was indeed created by Cauchy. Briot-Bouquet book was also very influential, in particular they introduced the modern terminology ("meromorpic functions" for sure, perhaps also the term "holomorphic", and such things as poles, removable and essential singularities etc.) Their book is Theorie des fonctions ... 5 According to F.Klein's "Lectures on the development of mathematics in the 19th century" Cauchy created this theory (by studying the convergence of series in the complex plane) in his Cours d'Analyse from 1821 ( Series 2, Volume 3 of his collected works). A direct predecessor (but with a lack of rigour seems to be Lagrange) and also Klein declares that these ... 4 On Bibliography of Quaternions and Allied Mathematics by Alexander Macfarlane I found this: On page 72; James Byrnie Shaw 1896 Sedenions (title). American Assoc. Proc., 45, 26. I couldn't find this reference, but the same author wrote this book: Synopsis of Linear Associative Algebra: A Report on its Natural Development and Results Reached up to the ... 4 It is well known when and why complex numbers were introduced. When you solve a cubic equation which has 3 real roots, using Cardano's formula, you obtain square roots of negative numbers in your formula, so to obtain the correct result you have to do arithmetic with complex numbers. Of course, when Cardano's formula was discovered, it had no practical ... 3 Yes, it is a coincidence. The concept of fugacity (from the Latin for "fleetness, tendency to flee") was originally introduced by Gilbert Lewis in his 1901 paper "The Law of Physico-Chemical Change" for the pressure of an ideal gas which has the same chemical potential as a real gas. Lewis notated this with ψ, though these days the letter f is used. The ... 3 In answer to your second part of the question regarding j for \sqrt{-1}, this was introduced into text books describing Power System Analysis of AC power circuits in the early 1900s by Charles P. Steinmetz. I am not sure of the earliest date but my guess is between late 1890s and 1920s but certainly no later than 1923 as Steinmetz died in 1923. ... 3 A paper of Kneser (1904) strongly suggests that the idea does (indeed) go back to Cauchy, in connection with Sturm-Liouville problems (i.e. ordinary differential operators, as opposed to the Laplacian in the body of your question). Given functions g,k,l and writing \smash{L=\frac d{dx}\left(k\frac{d}{dx}\,\cdot\right) - l}, Kneser considers the [for us: “... 3 For the completeness of the discussion in this post i must add a link to another relevant question - Meaning of passages by Gauss on the "convergence of expansions (in infinite series) of the (elliptical) equation of the center"? . The answer to this question, if interpreted appropriately, indicates that already in 1805 Gauss had deep insights on ... 2 According to Klein (Lectures on history of mathematics in 19 century), he absolute invariant J was introduced by Gauss in his manuscript "On summatory function", which is reproduced on p. 386 of volume III of Gauss collected works. 2 This is an example of absolute invariant in the context of classical invariant theory. Of course it is of paramount importance in the theory of elliptic curves but it is not just about them. Such a curve can be written as y^2=f(x) where f is a nonhomogenous polynomial of degree four (or three). By homogenization this corresponds to a binary quartic$$ F(...

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This is a reply to some of the points you have made rather than an answer to the very broad question you have posed, which is essentially "yes, it is a very convoluted and tortured history". A reliable source on the subject of the history of complex numbers is the mathematical historian Leo Corry. According to Corry and contra your source, ...

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A great example is binary arithmetic. When Boole first presented his work on the subject he introduced it saying, "I give you a fascinating system that has absolutely no application."* Eventually the Navy used it to replace base 10 arithmetic in computers they were building, getting us to the world we know today. Source: Dr. Goldbeck's Calc 1 class.

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A homeomorphism between two surfaces is called conformal if it preserves angles. Examples of conformal maps were known since antiquity: stereographic projection maps conformally the sphere without one point onto the plane (this map was known to the ancient Greeks, but conformality was stated and proved only in 17th century). Another example is Mercator's ...

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For a skeptical balance I recommend "Solving Wigners Mystery: The Reasonable (Though Perhaps Limited) Effectiveness of Mathematics in the Natural Sciences" which argues that: ...the connections between mathematics and the natural sciences are, and always have been, rationally although fallibly forged links, not a collection of mysterious parallelisms.

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I doubt it that the $i$ notation would have been used for $\sqrt{-1}$ before Euler because even Euler himself did not start using it until a rather late date, and moreover used $i$ in a different sense namely for an infinite integer, in his Introductio and Institutiones.

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