Questions tagged [complex-analysis]

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How were complex analysis methods, like the Joukowsky transform, used in early aircraft design?

The Joukowsky transform is a conformal mapping of a disk to an airfoil shape. The wiki page says that "it was historically used to understand some principles of airfoil design". That's kind ...
Daniel Shapero's user avatar
8 votes
1 answer
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Who first considered signed area?

Who first suggested that the area enclosed by a closed path and the area enclosed by that same path traversed in reverse could be regarded as equal in magnitude but opposite in sign? Cauchy must have ...
James Propp's user avatar
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2 answers
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Reference request: What were the problems of accepting zero, negative numbers, and complex numbers? And how were they solved?

I asked this question on MSE and comments suggested I should ask it here I am currently reading Baby Rudin as my second analysis book (after Introduction to Real Analysis by Robert G. Bartle and ...
pie's user avatar
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Reference request: What were the problems of accepting zero, negative numbers, and complex numbers? And how were they solved? [duplicate]

I didn't know that can happen and since I already asked the question here I don't know what to do with this question should I delete it ? I am currently reading Baby Rudin as my second analysis book (...
pie's user avatar
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To what extent were Riemann surfaces a precursor to algebraic geometry?

I read that Riemann started studying the so-called Riemann's surfaces in the second half of the 19th century, introducing tools like meromorphic functions and meromorphic 1-forms. The culmination of ...
Weier's user avatar
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The history and origin of the Argument Principle ( or Cauchy's argument principle)

I am looking for a book that discusses The history and origin of the Argument Principle ( or Cauchy's argument principle) Thanks!
user17825's user avatar
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1 answer
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Origin of the term "affixe"/"affix" in the geometric treatment of complex numbers

In current French mathematical tradition, when introducing complex numbers, it is common to hear about "complex plane of Argand-Cauchy". What is particular in French treatment, it is the ...
Alexey's user avatar
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1 answer
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Who introduced the stream function?

I have found many different claimed answers to this question: Wikipedia article on the stream function claims that Lagrange introduced it in 1781. Darrigol's The Worlds of Flow says that D'Alambert ...
timur's user avatar
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Poisson integral formula

The term Poisson integral formula may refer to any of the related formulas for harmonic (or holomorphic) functions on a disk (or in a ball, half space, etc) in terms of their boundary values. This is ...
timur's user avatar
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1 answer
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History of the definition of complex derivative

Almost all of modern complex analysis (Cauchy residue theorem, analytic continuation, etc) depend on the definition of a complex derivative. That definition requires the derivative at a point $z_0$ is ...
Penelope's user avatar
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What is the significance of Gauss-Weierstrass's derivation of "Al functions"?

In a fragment entitled "inversion of the elliptic integral of the first genus" (Gauss's werke, volume 8, p. 96-97), Gauss inverts the general elliptic integral of the first kind: he writes $\...
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Reference request for Gauss's original discovery of the special property of the $j$ function

In Interchapter VII of his biography of Gauss, W.K. Buhler describes Gauss's discovery of one of the important properties that characterize the $j$ invariant (Klein's absolute invariant; Gauss called ...
user2554's user avatar
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1 answer
263 views

Original proof of the Schwarz lemma

The classical Schwarz lemma from one-variable complex analysis states that a holomorphic map $f : \Delta(r) \to \Delta(R)$ between two disks in the complex plane such that $f(0)=0$ satisfies $$|f(z)| \...
AmorFati's user avatar
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3 answers
584 views

How did Roger Cotes come up with logarithm form of Euler formula?

I have been trying to get my head around how Roger Cotes first discovered Euler Formula. I knew how Euler did it, but I wanted a new perspective, especially from someone who discovered it earlier. ...
Poon Wu's user avatar
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1 answer
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Did anyone ever propose a hypercomplex numbers system with more than one anisotropic axis?

The real number axis is asymmetric against zero: for instance, multiplication of two negative or two positive numbers will produce a positive number, a square root of a negative number is not real, ...
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Why are quaternions more popular than tessarines despite being non-commutative?

Is this simply because of marketing, hype, etc? The bicomplex numbers (especially tessarines) look just great being commutative and all. Images source:https://citeseerx.ist.psu.edu/viewdoc/download?...
Anixx's user avatar
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History of complex trigonometric ratios

I have just started learning about trigonometric ratios of complex arguments but I couldn't find any justification or derivation for extending trigonometric ratios to complex field. Also the Euler's ...
Guji2203's user avatar
3 votes
2 answers
330 views

On the history of development of the concept of complex numbers [closed]

The history of how the concept of complex numbers developed is convoluted. On physics.stackexchange questions about complex numbers keep recurring. It seems to me this indicates that when authors of ...
Cleonis's user avatar
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Why are complex numbers called 'complex'?

I'm a high school teacher, and I was just wondering why complex numbers are called 'complex'. I have read that Gauss coined the term. But I couldn't find any reference where it was explained. I also ...
Ann's user avatar
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1 answer
924 views

Who was the first person to notice logarithms of negatives numbers and for what reason?

Who was the first person to notice logarithms of negatives numbers and why? When did they first arise naturally? I thought I saw somewhere that it had to do with integration but I can't find it ...
Pineapple Fish's user avatar
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2 answers
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How much ground was prepared for Riemann so that he could conjecture Riemann hypothesis?

Although I do not doubt in Riemann˙s originality, I would like to know how much complex analysis was developed up to the day when Riemann conjectured what is today called Riemann hypothesis and how ...
user avatar
5 votes
2 answers
507 views

Who pioneered the study of the sedenions?

I found lots of background information about the discovery of both imaginary and complex numbers, and enough information about the first two types of hypercomplex numbers; quaternions and octonions (...
Mr. J. Larios's user avatar
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0 answers
111 views

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: $$\...
LAGRIDA's user avatar
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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 ...
Eli's user avatar
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7 votes
1 answer
254 views

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\,\...
kolobok's user avatar
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7 votes
1 answer
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What was the motivation for Cauchy's Integral Theorem?

How did Cauchy go about Cauchy's integral theorem? What was his motivation?
LinearGuy's user avatar
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4 votes
1 answer
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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 ...
user2554's user avatar
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2 votes
4 answers
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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 ...
smwikipedia's user avatar
2 votes
1 answer
279 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 ...
user2554's user avatar
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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 ...
stuartboehmer's user avatar
4 votes
0 answers
84 views

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 ...
James Propp's user avatar
4 votes
2 answers
814 views

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 ...
user2554's user avatar
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3 votes
1 answer
218 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 ...
porton's user avatar
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6 votes
1 answer
471 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 ...
Jan Peter Schäfermeyer's user avatar
5 votes
1 answer
758 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 ...
SarthakC's user avatar
  • 153
8 votes
1 answer
575 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?
user3880994's user avatar
3 votes
2 answers
648 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. ...
Laurent Duval's user avatar
6 votes
1 answer
433 views

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?
Steven's user avatar
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12 votes
1 answer
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Euler's first proof of $e^{ix}=\cos(x)+i\sin(x)$

What was Euler's 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 (...
MrYouMath's user avatar
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2 votes
1 answer
354 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 ...
vafylec's user avatar
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17 votes
0 answers
957 views

When did people know that all real polynomials of degree greater than 2 were 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 ...
Stanley Yao Xiao's user avatar
3 votes
0 answers
81 views

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 ...
Quantum spaghettification's user avatar
8 votes
2 answers
421 views

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 ...
Gottfried William's user avatar
16 votes
1 answer
330 views

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. ...
Colin McLarty's user avatar
7 votes
1 answer
626 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\...
Nikolaj-K's user avatar
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41 votes
1 answer
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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 ...
Michael Weiss's user avatar