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Questions tagged [real-analysis]

For questions about the history of calculus and its theoretical foundations, including topics such as continuity, differentiability, and infinite series. Related topics include questions on the history of measure theory, and some aspects of general topology and classical descriptive set theory.

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Are there alternatives functions for the gamma function that was used as generalisation for the factorials?

I asked this question on MSE here $$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \ \ x>0. $$ Bohr and Mollerup showed that the gamma function is the only positive function $f$ defined on $...
pie's user avatar
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4 votes
2 answers
773 views

When and why was the concept of "having a least upper bound" dubbed "completeness", as in Axiom of Completeness?

The Axiom of Completeness states that any non-empty set with an upper bound has a least upper bound. When and why was this concept of least upper bound dubbed "completeness"? It's true, of ...
SRobertJames's user avatar
2 votes
1 answer
151 views

Height function following Borel

Borel introduces the notion of hauteur (French for 'height') in a note titled Sur l'approximation les uns par les autres des nombres formant un ensemble dénombrable in the Comptes Rendus journal in ...
Sam Sanders's user avatar
2 votes
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109 views

What were Cantor’s “real numbers of higher type”?

In the preamble to “Essays on the Theory of Numbers”, Dedekind makes passing reference to a theory (expounded in Cantor’s “Ueber die Ausdenung eines Satzes aus der Theorie der trigonometrischen Reihen”...
James Propp's user avatar
1 vote
1 answer
176 views

How good was Newton at definite integration?

On Math Stack Exchange, I am impressed by users' skill at finding closed form expressions for definite integrals. For example: Example 1: $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^...
Dan's user avatar
  • 139
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0 answers
40 views

History of functions of bounded variation

I have read that Jordan first defined the concept of variation and studied functions of bounded variation, in his 1881 publication Sur la Serie de Fourier, as referenced here: https://en.wikipedia.org/...
Addem's user avatar
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8 votes
2 answers
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When did "neighbourhood of a point" first appear in the history of Taylor series?

I am trying to track down at what point mathematicians started to use the terminology of expanding a function "around a point" or in the "neighbourhood of a point". Neither Taylor ...
StormyTeacup's user avatar
2 votes
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The habit of definition

G. H. Hardy wrote (apropos of the task of assigning values to divergent series): It is plain that the first step towards such an interpretation must be some definition, or definitions, of the 'sum' ...
James Propp's user avatar
3 votes
1 answer
2k views

What were the "weird" things people were doing in calculus at the time of Marx?

I was reading the preface of Marx's Mathematical Manuscripts. They explain the situation of calculus in the time of Marx, it seems that at the time analysis as we know today was still being forged by ...
Red Banana's user avatar
5 votes
2 answers
2k views

Source of a Poincaré quote: "Logic sometimes makes monsters..."

There's a quote by Poincare on the "new functions", such as continuous functions without derivatives, that were appearing during the second half of the 19th century. The fullest version I've ...
JMJ's user avatar
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2 votes
1 answer
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Can the so-called completeness of real numbers be understood as closure under limits in the real number system?

Someone suggested (please see the comments below) that I post this question on hsm.stackexchange. There is a connection to the history of mathematics in this, regarding the relationship between the ...
bokabokaboka's user avatar
3 votes
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197 views

Did the Maclaurin series for sine and cosine unsettle Indian mathematicians?

As many of you may know, sometime around the 14/15th centuries an Indian mathematician by the name of Madhava of Sangamagrama derived the Maclaurin series for sine and cosine for the first time in ...
voltamatron's user avatar
3 votes
1 answer
111 views

First use of ~ and ≍ (\sym and \asymp)

The relations ~ and ≍ are frequently used in math and computer science, at least within number theory and analysis of algorithms. What is their origin? Definitions Suppose $g(x)$ is an eventually-...
Charles's user avatar
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1 answer
141 views

Why the scaling rule $\delta(a x)=\frac1{|a|}\delta(x)$ was historically adopted? [closed]

It seems to me that it would be more natural to consider Dirac Delta as a piecewise-defined function, as described here, with the scaling rule $\delta (ax)=\delta(x)$. This way we keep all the ...
Anixx's user avatar
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3 votes
1 answer
388 views

Why did Clairaut's theorem take so long to prove?

I was reading the Wikipedia on Clairaut's Theorem (Symmetry of second derivatives) and the article accounts the significant amount of time and failed proofs occurred before the theorem was made fully ...
szeits's user avatar
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1 answer
83 views

When did Abel publish his test for the convergence of series?

Did Abel published of testing the convergence of series? If so, when did he published it. Also, did he offer a proof of the test? Or did he simply stated the test?
user926356's user avatar
3 votes
0 answers
231 views

How were sine and cosine functions computed before the notion of Taylor series?

We know using modern analysis techniques that $\sin x$ and $\cos x$ can be computed by their Taylor series (in fact the Taylor series are given as the definitions of these functions in today's real ...
Maximal Ideal's user avatar
2 votes
2 answers
150 views

Definition and name change of the oscillation function

I have two related questions: Who first defined the oscillation function (perhaps under a different name)? When did the switch from the phrase "saltus function"(*) to "oscillation ...
Alp Uzman's user avatar
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5 votes
0 answers
128 views

Seeking quotes rejecting early forms of the Dirac delta

Question Why is Dirac delta named after Dirac when the concept was already over two centuries old? notes the Dirac delta function effectively appeared centuries before Dirac. This was long before the ...
Colin McLarty's user avatar
1 vote
0 answers
64 views

What is the grounding of commensurability?

I understand that before Hippasus of Metapontum proved that the square root of 2 is an irrational number, it was commonly assumed that, given two line segments, it would be possible to find a third ...
Underdog's user avatar
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1 vote
2 answers
427 views

Did Berkeley's criticism of infinitesimals hobble calculus pedagogy?

I recently read an article that discussed--rather briefly--the issues of infinitesimals and the criticism of them by Berkeley. The author of the article (which, of course, I cannot find, as I read it ...
Eric Snyder's user avatar
2 votes
1 answer
134 views

History of the Darboux-Froda theorem

I am curious about the history of the so-called Darboux-Froda theorem, which is the following theorem: a monotone function $f:[0,1]\rightarrow \mathbb{R}$ has at most countably many points of ...
Sam Sanders's user avatar
0 votes
0 answers
95 views

Did anyone ever propose an analytic definition of zero divisors, including nilpotents, as opposed to algebraic definition?

I wonder, can we meaningfully define zero divisors based on analytic rather than algebraic approach? For instance, if we extend the real numbers with divergent integrals and series, and evaluate the ...
Anixx's user avatar
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3 votes
1 answer
236 views

First time the real numbers were axiomatized as the "unique complete ordered field"

(originally asked at M.SE: https://math.stackexchange.com/questions/4094361/first-time-the-reals-were-axiomatized-as-the-unique-complete-ordered-field) I'm looking for historical references on the ...
Luiz Cordeiro's user avatar
10 votes
4 answers
246 views

Historically, what led to the question of the validity of interchange of limit operations?

It seems G. H. Hardy once wrote that "The problem of deciding whether two given limit operations are commutative is one of the most important in mathematics". I was wondering what led to the ...
user avatar
5 votes
1 answer
100 views

Where could I find the reference for the statement of Euler finding out the coefficients of Fourier series?

I am reading Carlslaw's "Introduction to the Theory of Fourier's Series and Integrals", first chapter on the history of Fourier series, page 3. The author asserts that Clairaut and Euler did ...
James Warthington's user avatar
1 vote
0 answers
117 views

How did Fourier determine the coefficients of Fourier series?

I was reading a chapter of Fourier's seminal work "Analytic Theory of Heat". The third chapter of this book was translated by the famous Stephen Hawking in his book "God created the ...
James Warthington's user avatar
5 votes
2 answers
264 views

Has the heyday of mathematical formulae ended?

I have a strong emotional reaction when I read the works of Euler. I have seen many extremely beautiful and intriguing identities in the notebook of Ramanujan, so much so that I think he is indeed a ...
James Warthington's user avatar
2 votes
1 answer
275 views

How to derive the power series of $\sin(x)$ and $\cos(x)$ followed the footstep of Euler

I am reading Euler's "Introduction to analysis of the infinite", chapter 8, page at the end of page 208, beginning of page 209 and came across his derivation of the power series for $\sin(x)$...
James Warthington's user avatar
3 votes
0 answers
186 views

How did Euler obtain this formula from a paper/work in 1748?

I am reading this book on trigonometric series, "Тригонометрические ряды от Эйлера до Лебега" (Trigonometric series from Euler to Lebesgue) , it is in Russian, and my Russian is abysmal. But ...
James Warthington's user avatar
3 votes
0 answers
98 views

Why does Rolle get its own theorem?

The importance of Rolle's theorem lies in the fact that it's used to prove the mean value theorem, which is a central result of analysis, eventually leading to the fundamental theorem of calculus. But ...
user avatar
1 vote
1 answer
447 views

Who first used the Completeness Axiom for real numbers?

I was studying calculus and the following question came to my mind: Who was the first person to use or suggest the use of the Completeness Axiom of the Real Numbers?
user926356's user avatar
5 votes
1 answer
214 views

Was Lebesgue differentiation theorem the motivation for Vitali's, Riesz's and Hardy-Littlewood's results used to prove it?

I have been reading about the Lebesgue differentiation theorem from Terence Tao's book and came across a bunch of things. In his book, Tao uses the Vitali Covering lemma (finite), Hardy-Littlewood ...
red whisker's user avatar
3 votes
1 answer
731 views

When did the Notion of "Limit" Arise and for What Purpose?

It is my understanding that Cauchy was the first to incorporate the notion of a $\delta$-$\epsilon$ limit in his proofs, although a definition was not formulated until Weierstrass did so. How far ...
DDS's user avatar
  • 287
5 votes
2 answers
222 views

Question about the significance of "Gauss-Legendre quadrature"

I want to understand why, according to several sources, Gauss's discovery of Gaussian quadrature in his 1814 article was "the most significant event of the 19th century in the field of numerical ...
user2554's user avatar
  • 4,489
2 votes
1 answer
260 views

Why is a time series not called a time sequence?

In pure mathematics, a sequence is a list of terms, for instance $1, \frac12, \frac14, \dots, \frac{1}{2^k},\dots$, and a series is the sum of an infinite sequence, for instance $\sum_{k=1}^\infty \...
Federico Poloni's user avatar
4 votes
1 answer
121 views

Etymology of certain terms in the theory of elliptic integrals

In the theory of elliptic integrals, one encounters the terms "amplitude" and "modular angle" in relation to incomplete integrals of the first kind, which are two variables that denote the upper limit ...
user2554's user avatar
  • 4,489
6 votes
2 answers
2k views

When was the first recorded occurence of irrational and imaginary number usage in number theory?

I saw a letter of Euler to Lagrange congratulating him on his usage of imaginary numbers in the "analysis devoted to rational numbers alone", was that the first known such usage? What was the likely ...
GEP's user avatar
  • 1,515
6 votes
1 answer
348 views

Did Cauchy ever deal with double or triple integrals?

Did Cauchy ever deal with double or triple integrals? Did he give rigorous proofs of multivariable integral calculus like what came to be called Stokes's theorem, the divergence theorem, etc.?
Geremia's user avatar
  • 5,339
0 votes
1 answer
317 views

Was there a more intuitive early proof of the generalized mean value theorem?

I am interested in the early proofs of the theorem. It is often called Cauchy mean value theorem, so perhaps Cauchy proved it first. In all the proofs that I have seen we construct a contrived ...
Milan's user avatar
  • 119
5 votes
1 answer
266 views

How did Peano prove his existence theorem without Ascoli's theorem?

In modern proofs of the Peano Existence Theorem for ordinary differential equations, Ascoli's theorem is used. Ascoli's theorem came after Peano's proof. Did Peano prove a form of Ascoli's theorem in ...
LinearGuy's user avatar
  • 345
3 votes
0 answers
367 views

The Integral as a Uniform Limit of Step Functions

Who first realized that it is possible to define the integral of a function as the limit of the integrals of a sequence of step functions that converge uniformly to the given function? This is ...
user109871's user avatar
5 votes
2 answers
847 views

Riemann's Contribution to Integration

What did Riemann do for the theory of integration? I am asking because I hear his name a lot in relation to integration and it is often implied that he made large contributions, but I do not know ...
user109871's user avatar
6 votes
2 answers
661 views

How did Newton and Leibniz interpret the integral?

How did Newton and Leibniz think about the integral? Did they only see it as an anti-derivative or did they also think of it as the area under a curve?
user109871's user avatar
0 votes
2 answers
181 views

What geometric results were first proven by assuming all real numbers are rational?

Pythagoras and his followers believed that all magnitudes are commensurable; that is, the ratio of two magnitudes of the same kind, like two lengths or two areas, is equal to the ratio of natural ...
Keshav Srinivasan's user avatar
1 vote
1 answer
340 views

Who first proved Fubini's theorem for abstract measure spaces?

Fubini's theorem relates the double integral of a function $f(x,y)$ to an iterated integral with respect to $x$ and $y$. The basic idea of this theorem for Riemann integrals of continuous functions ...
Keshav Srinivasan's user avatar
5 votes
3 answers
781 views

What brought about the need for real analysis and formal logic in recent years?

I can't seem to find a clear, definitive, non-circular answer on this. For centuries and centuries, we've been doing mathematics in one form or another, be it geometry and pictures, or inventing ...
user525966's user avatar
5 votes
3 answers
773 views

Did Eudoxus really set out to present irrationals as Dedekind cuts?

I've been intrigued by the similarities between what Eudoxus' Theory of Proportions and Dedekind cuts. However, I wish to question this "perceived similarity" and would like to where the flaws are, ...
PhD's user avatar
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6 votes
2 answers
323 views

Does anybody know the history of how Peter Gustav Lejeune Dirichlet came up with the “nowhere continuous” Dirichlet function?

So I am writing a research paper on the properties of the Dirichlet function (the function with 1 if x is rational and 0 if x is irrational), and I wanted to include some historical background on how ...
serendipity456's user avatar
5 votes
1 answer
367 views

Were there proofs of the Lebesgue Differentiation Theorem without using maximal functions?

Is there a proof of the Lebesgue Differentiation Theorem that does not involve the Hardy-Littlewood Maximal Function? For example, did Lebesgue prove it? If there is such a proof, where can I find it?...
user109871's user avatar