Questions tagged [complex-analysis]

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How did Hardy and Littlewood formulate the k-tuple conjecture?

Let $\mathcal{H}_k = (h_1,h_2,\cdots,h_k)$ be an admissible k-tuple. The k-tuple conjecture predicts that the number of primes $(b+h_1,b+h_2,\cdots,b+h_k)\in \mathbb{P}^k$ with $b+h_k \leq x$ is: $$\...
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$[\operatorname{Cos}(x)+i\operatorname{Sin}(x)]\cdot[\operatorname{Cos}(y)+i\operatorname{Sin}(y)]=\operatorname{Cos}(x+y)+i\operatorname{Sin}(x+y)$

Had this non-analytic theorem ever been known before Euler discovered his analytic formula: $\operatorname{Cos}(x)+i\operatorname{Sin}(x)= e^{ix}$ ? Please notice the distinction between, the theorem ...
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History of a contour integral method for summing series

A folklore result I have seen used in evaluations of infinite sums is the following clever use of the residue theorem: $$\begin{align*}\sum_{1}^\infty f(k)&=\frac1{2\pi i}\oint f(z)\pi\cot\pi z\,\...
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Cauchy's Integral Theorem Motivation

How did Cauchy go about Cauchy's integral theorem? What was his motivation?
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1answer
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Why is Riemann's dissertation (from 1851) considered a turning point in the history of the theory of conformal mappings?

The intention behind my question is to understand what are the kind of general problems of which the ideas of Riemann's dissertation (1851) lie at the heart of it's solution methods. In his ...
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162 views

An english translation of Cauchy's “Cours d'Analyse”

I am quite interested in the origins of our modern way of understanding analysis. I know that Augustin-Louis Cauchy was one of pioneers regarding a rigorous foundation towards real and complex ...
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4answers
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Mathematics development can sometimes **exceed** the practical needs, right?

I read below paragraph from the book "A Friendly Introduction to Number Theory": The use of "$i$" to denote the square root of negative $1$ dates back to the days when people viewed such numbers ...
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1answer
111 views

What is the modern interpretation of Gauss's “Summatorische Function”?

In Buhler's biography of Gauss (Gauss: A Biographical Study), at the chapter on modular forms and hypergeometric series, he mentions a function that Gauss called "Summatorische Function", which he ...
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when were dot product and cross product discovered ? Also, was quaternions discovered before or after this? [duplicate]

also, who created these ? how to rotate 3-D vectors using quaternions ?
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gauss' opinion on de moivre's theorem

What did Gauss mean when he said that you'd never be a good mathematician if you didn't think that De Moivre's theorem was obvious? I'm looking for a specifically historically correct answer with ...
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Variants in graphical presentation of real and complex numbers

It's standard nowadays for the real line to be horizontal (negative numbers on the left, positive numbers on the right) and for $i$ to be above (rather than below) 0 in the complex plane. Were these ...
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Did Gauss know the residues theorem in complex analysis in 1811?

My question refers to Gauss's 1811 letter to Friedrich Bessel, which contains a statement of Cauchy integral theorem. I have access to the letter, but I'm unable to read it. I know that Gauss gives ...
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154 views

Were complex number first considered of limited usefulness?

Were complex numbers ever considered to be of limited usefulness that is not very useful in practice (unlike modern science where we strongly need complex numbers)? Note my question is not about ...
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372 views

Who came up with the link between the spectrum of an operator and the poles of a meromorphic function?

From Dieudonné's "History of Functional Analysis" I learned that Picard in 1893 gave a characterization of an eigenvalue of the Laplacian as the simple pole of a meromorphic function. Is there an ...
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323 views

What is the history of using $i$/$\iota$ as the imaginary unit?

I'm interested in particular in knowing about when $\iota$ began to be used as the imaginary unit/who began to use it. A majority of all text books that I have seen tend to just use $i$ as the ...
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345 views

History of complex analysis

Does anyone know of a good book on the history of imaginary numbers and complex analysis and its role in physics?
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226 views

Introduction of $\imath$ and $\jmath$ notations for the imaginary unit

The imaginary unit is generally denoted $i$ or $\imath$. I have learned that the term imaginary ("imaginaires") was coined by R. Descartes in 1637, and the "i" notation was introduced by L. Euler (cf. ...
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Origin of Klein's $j$-invariant

Today Klein's $j$-invariant is used in various context's, the most famous one being maybe "Monstrous Moonshine". But what was the original motivation for the study of the $j$-invariant?
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Euler's first proof of $e^{ix}=\cos(x)+i\sin(x)$

What was Eulers first proof of his famous formula? In Euler's book on complex functions he used the following proof. But was this his first proof? Euler starts with writing down De Moivre's Formula (...
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1answer
138 views

The origins of complex differentiation/integration

What questions led to the invention of complex differentiation/integration? How were their definitions agreed upon? Real differentiation/integration has an obvious meaning. To extend calculus to the ...
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When did people know that all real polynomials of degree greater than 2 are reducible?

Admittedly, this may not be a research level question, but I am deeply curious about this. Let $f(x) \in \mathbb{R}[x]$, and write $d = \deg f$. It is well known that if $\deg f > 2$, then $f$ is ...
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Analytic and holomorphic functions, definitions and foundations?

If you search for the definition of analytic and holomorphic functions in books and online, you get crosses between two definitions (one involving taylor series and one differential) I can find no ...
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First papers on holomorphic functions

Briot and/or Cauchy are often said to have written the first papers on holomorphic functions, explicitly discussing them as such and their special properties. Which papers are these? When and where ...
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Did Dedekind show any evidence of pictorial geometric sense?

The most pictorial geometric thinking I find in Dedekind is the intuitive-geometric idea of a continuum which he criticizes as too vague before he gives his account of the continuum based on cuts. ...
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324 views

Who named the fugacity, who coined the variable name and did it already relate to complex analysis?

In Riemanns monumental paper, he expresses a prime counting function as an inverse Mellin transform of the log of the function he analytically continued into the complex plane $$\Pi(x) = \frac{1}{2\...
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What was Euler's motivation for introducing $i$ for $\sqrt{-1}$?

[Mauro Allegranza has answered the question of who introduced the notation $i$ (Euler, followed later by Gauss), so I have changed the title. I have also edited the question in other ways to make it ...