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.
81
questions
-1
votes
1
answer
108
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 ...
- 614
1
vote
1
answer
195
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 ...
- 11
1
vote
1
answer
68
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?
- 125
2
votes
0
answers
144
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 ...
- 339
0
votes
0
answers
46
views
The role of monotonicity in integrating term-by-term preceding Lebesgue's monotone convergence theorem
Given a measure space $(\Omega,\Sigma,\mu)$ and sequence of pointwise non-decreasing, non-negative, measurable functions $\{f_n\}_{n=1}^{\infty}$ on it, Lebesgue's monotone convergence theorem says ...
- 1
1
vote
1
answer
97
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 ...
- 275
5
votes
0
answers
109
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 ...
- 2,306
1
vote
0
answers
59
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 ...
- 111
1
vote
2
answers
289
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 ...
- 121
1
vote
1
answer
94
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 ...
- 141
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 ...
- 614
2
votes
1
answer
177
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 ...
- 131
10
votes
4
answers
180
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 ...
A.D.
5
votes
1
answer
96
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 ...
1
vote
0
answers
98
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 ...
5
votes
2
answers
231
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 ...
2
votes
1
answer
245
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 $\sin(x)$...
3
votes
0
answers
166
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 ...
3
votes
0
answers
80
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 ...
Vercassivelaunos
1
vote
1
answer
311
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?
- 125
4
votes
1
answer
173
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 ...
- 143
4
votes
1
answer
456
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 ...
- 215
4
votes
0
answers
127
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 ...
- 4,087
2
votes
1
answer
191
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 \...
- 332
4
votes
1
answer
115
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 ...
- 4,087
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 ...
- 1,476
6
votes
1
answer
292
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.?
- 5,211
0
votes
1
answer
203
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 ...
- 119
5
votes
1
answer
244
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 ...
- 335
3
votes
0
answers
303
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 ...
- 486
3
votes
2
answers
641
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 ...
- 486
6
votes
2
answers
412
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?
- 486
0
votes
1
answer
132
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 ...
- 69
1
vote
1
answer
320
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 ...
- 69
5
votes
3
answers
692
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 ...
- 201
5
votes
3
answers
583
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, ...
- 619
6
votes
2
answers
258
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 ...
5
votes
1
answer
296
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?...
- 486
1
vote
2
answers
1k
views
Dirichlet's Proof of the Convergence of Fourier Series
Where can I find Dirichlet's proof of the convergence of Fourier series?
- 486
4
votes
1
answer
206
views
Reference - Schwarz's Proof of Clairaut's Theorem
Where can I find a copy (online) of Schwarz's paper that proved Clairaut's theorem for mixed partial derivatives?
His paper is: Schwarz, H. A., "Communication", Archives des Sciences ...
- 486
6
votes
2
answers
362
views
What is history behind Smith-Volterra-Cantor sets?
Looking at Wikipedia, I see that fat Cantor sets are also called Smith-Volterra-Cantor sets. Another name which is sometimes associated with these sets is Hermann Hankel.
I suppose that Cantor's name ...
- 203
3
votes
1
answer
235
views
Who proved the monotone convergence theorem for the Lebesgue integral?
The theorem often be called Lebesgue's MCT or Levi's theorem. Who did originally prove it or what is the contribution of Lebesgue and Levi respectively?
- 33
8
votes
2
answers
796
views
What was Lebesgue's original definition of a measurable set?
I found an interesting question on Math SE asked by @Dilemian that seems more on topic here, and since it lacks answers there I thought to post it here so that it can receive good answers here.
There ...
user3224
1
vote
1
answer
229
views
What was Lipschitz's original motivation for the introduction of Lipschitz continuity?
The concept of Lipschitz continuous mappings is probably at the present time the most important mathematical concept associated with Lipschitz's name. These mappings play an important role in the ...
- 421
6
votes
2
answers
1k
views
What is the correct statement of Cauchy’s erroneous theorem on continuity?
I read recently that Cours includes a famous, or perhaps infamous, error in that Cauchy states and proves a false result concerning sequences of continuous functions. (Here, obviously, continuous ...
- 6,337
0
votes
1
answer
230
views
The Origin of the Jacobian
In what work did Jacobi introduce the jacobian, and what was his motivation for doing so?
- 486
4
votes
0
answers
83
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 ...
- 261
1
vote
1
answer
106
views
Motivation of Infinite Series
What is the historical motivation of infinite series? According to Wikipedia, they are arose separately by Newton, Leibniz and Somayaji.
- 486
4
votes
1
answer
202
views
Motivation of Continuous Functions
What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?
- 486
7
votes
1
answer
366
views
Historical occurrences of mathematicians substituting terms for $x$ in the denominator of $\mathrm{d}y/\mathrm{d}x$?
This answer, to a question on teaching the chain rule, suggests writing something like this
$$
\frac{\mathrm{d}\, \mathrm{e}^\sqrt{s}}{\mathrm{d}\,s}=\frac{\mathrm{d} \,\mathrm{e}^\sqrt{s}}{\mathrm{d}\...
- 1,580