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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Cauchy addressing "metaphysical difficulties" of Calculus
I am reading Everything and more: A brief history of infinity by David Foster Wallace and came across this quote:
Broadly stated, Cauchy’s project involves trying to rescue calculus from its ...
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Meaning of “expand by algebra” in an old Tripos question about $a^x$ and $\sin(x/m)$?
I came across an old Cambridge Tripos question (The 1861 Smith's Prize Exam) which asks:
Indicate methods of expanding $a^x$ and $\sin(\frac{x}{m})$. Show that the former function may be expanded ...
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Practical and historical role of Jordan measure
In my earlier questions, the proofs given by Asigan and D.R. showed that the Jordan outer/inner measure of the subgraph $[0,f]$ and the Darboux upper/lower integrals of $f$ are essentially the same ...
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Several questions about a problem of asymptotic nature dealt in entry 83 of Gauss diary
In pp. 525–529 of volume 10–1 of Gauss’s works, one can find entry 83 of his diary, which reads:
If one sets:
$$l(1+x)=\varphi'x;\quad l(1+\varphi'x) = \varphi''x; \quad l(1+\varphi''x)=\varphi'''x, \...
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Where did the real number axioms 'come from'?
Asked this on the Math Stack exchange and was redirected here:
Currently trying to self-learn Real Analysis, with the first few chapters introducing the 'Axioms for the Real Numbers'. The Field Axioms ...
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Can you suggest textbook for history of modern mathematics like topology and abstract algebra?
I want books in real analysis, topology and abstract algebra that provide the intuition and history behind each definition, theorem and proof. I want to know about the historical development of modern ...
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History of the "typewriter" sequence
Can someone please give us some history on the so-called "typewriter" sequence of functions that is often used to show that $L^p$-norm convergence does not imply the almost-everywhere (or ...
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Are there alternatives functions for the gamma function that was used as generalisation for the factorials?
I asked this question on MSE here
$$\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 $...
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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 ...
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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 ...
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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”...
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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^...
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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/...
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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 ...
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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' ...
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