# Tag Info

## Hot answers tagged complex-analysis

41 votes

### Why are quaternions more popular than tessarines despite being non-commutative?

Commutativity is over-rated: in fact, it holds back bicomplex numbers: It prevents your number system characterising non-commuting operations, e.g. rotations in $3$-dimensional space, Hamilton's ...
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23 votes

### Why are quaternions more popular than tessarines despite being non-commutative?

Your description "total uselessness of quaternions" in a comment above is poorly chosen, and reflects more on your interests than on the real state of knowledge of mathematics. The Hamilton ...
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22 votes
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### What was Euler's motivation for introducing $i$ for $\sqrt{-1}$?

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 ...
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11 votes
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### What was the motivation for Cauchy's Integral Theorem?

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: ...
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10 votes
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### Who was the first person to notice logarithms of negatives numbers and for what reason?

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 ...
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10 votes
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### Mathematics development can sometimes **exceed** the practical needs, right?

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 ...
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10 votes
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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 $\... • 6,134 10 votes ### Why are quaternions more popular than tessarines despite being non-commutative? Hamilton expected that the quaternions would be of physical interest. In this, he was right. But he was too early. He had discovered them in 1843, it was almost a century later, in 1928, when Dirac ... • 3,018 7 votes ### History of complex analysis 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 ... • 42.9k 6 votes ### Who pioneered the study of the sedenions? 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. ... • 67.1k 6 votes Accepted ### History of a contour integral method for summing series Whittaker and Watson credit this method to Mittag-Leffler, Acta Soc. Sci. Fenn., XI 1880 273-293, and Acta Math., IV 1884 1-79. • 42.9k 6 votes Accepted ### What is the history of using$i$/$\iota$as the imaginary unit? 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. ... • 13.9k 6 votes Accepted ### Were complex number first considered of limited usefulness? 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 ... • 42.9k 5 votes ### The origins of complex differentiation/integration 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. ... • 42.9k 5 votes ### First papers on holomorphic functions 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" ... • 42.9k 5 votes Accepted ### First papers on holomorphic functions 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 ... 5 votes Accepted ### Original Proof of the Schwarz lemma The following paragraph in "Julius and Julia: mastering the art of the Schwarz lemma" by Professor Harold P. Boas (Amer. Math. Monthly, 117 (2010), no. 9, pp. 770-785) might be relevant: ... • 1,563 5 votes Accepted ### History of the definition of complex derivative In general, definitions in mathematics are not arbitrary. Useful definitions are those which lead to interesting and important objects. This particular definition is good because it leads to an ... • 42.9k 4 votes Accepted ### Why are complex numbers called 'complex'? Complex numbers were used long before Gauss. They appeared for the first time in 16th century when people found a formula for solving cubic equations. One problem with this formula is that even for ... • 42.9k 4 votes ### How much ground was prepared for Riemann so that he could conjecture Riemann hypothesis? Riemann wrote his paper on the zeta-function in 1859. He was the first person to consider the zeta-function in the complex plane, first for${\rm Re}(s) > 1$and then he worked out an analytic ... • 4,084 4 votes Accepted ### Who pioneered the study of the sedenions? 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 ... 4 votes ### Mathematics development can sometimes **exceed** the practical needs, right? 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 ... • 42.9k 4 votes ### Who named the fugacity, who coined the variable name and did it already relate to complex analysis? 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" ... • 1,134 4 votes Accepted ### Introduction of$\imath$and$\jmath$notations for the imaginary unit 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. ... • 1,069 4 votes ### Who introduced the stream function? d'Alembert did originally introduce the stream function in "Remarques sur les lois du mouvement des fluides" in 1761. Page 149 says that $$p = \frac{\partial \omega}{\partial x} \,,$$$$q = ... 3 votes Accepted ### Who came up with the link between the spectrum of an operator and the poles of a meromorphic function? 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 ... • 7,539 3 votes ### Origin of Klein's$j\$-invariant

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 ...
3 votes

### Did Gauss know the residues theorem in complex analysis in 1811?

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 (...
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3 votes
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### Poisson integral formula

I think that this is what you are asking for: Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques, et sur l’usage de cette transformation dans la résolution de ...
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