Questions tagged [complex-analysis]

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• 143
668 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. ...
• 51
1 vote
128 views

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, ...
• 672
5k views

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?...
• 672
125 views

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 ...
• 31
361 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 ...
• 814
2k views

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 ...
• 33
935 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 ...
242 views

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 ...
529 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 (...
117 views

• 73
1k views

What was the motivation for Cauchy's Integral Theorem?

How did Cauchy go about Cauchy's integral theorem? What was his motivation?
• 345
278 views

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 ...
• 4,499
572 views

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 ...
• 255
295 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 ...
• 4,499
65 views

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 ...
87 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 ...
• 617
841 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 ...
• 4,499
227 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 ...
• 133
483 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 ...
815 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 ...
• 153
584 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?
700 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. ...
• 1,193
461 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?
• 161
14k views

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 (...
• 610
356 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 ...
• 151
1k 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 ...
83 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 ...
432 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 ...
• 1,242
339 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. ...
• 2,441
654 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\...
• 211
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 ...