# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

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### On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
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### Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

This has now been cross posted to MO. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn ...
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### Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

Update 24 Nov 2015: It is solved. Please refer to this arXiv paper. For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids ...
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### Show that there are no real solutions of $1 + \sum_{n = 1}^{\infty} \frac{x^n}{\prod_{k=1}^{n} H_k} = 0$ where $H_k = \sum_{i=1}^{k} \frac{1}{i}.$

[Edited] Show that there are no real solutions of $$1 + \sum_{n = 1}^{\infty} \frac{x^n}{\prod_{k=1}^{n} H_k} = 0$$ where $$H_k = \sum_{i=1}^{k} \frac{1}{i}.$$ I managed to prove this, and I want to ...
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### Find $f$ such that $f \star f(x) = \frac{1}{1-x}$.

I'm looking for a measurable function $f$ defined on $]0,1[$ such that : $$f \star f(x) = \int_{0}^1 f(x-y) f(y) \ \mathrm{d}y = \frac{1}{1-x}$$ for (almost) any $x \in ]0,1[$. Is it possible to find ...
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Question. Let $P_d = \{ p \in \mathbb{R}[x] : \deg p = d\}$ denote the set of all degree $d$ polynomials with real coefficients. Also, for $p \in \mathbb{R}[x]$, define $$I(p) = \frac{1}{\pi} \int_{-\... 18 votes 0 answers 498 views ### Stokes' Theorem general case With the following lemma : Lemma : Let f_{+},f_{-} : U \longmapsto \mathbb{R} be C^{1} maps with f_{-} \leq 0 \leq f_{+} with U \subset \mathbb{R}^{n} open and bounded with C^{1} boundary. ... 18 votes 0 answers 911 views ### Refinement of a famous inequality I refine a famous inequality this is the following : Let x,y>0 then we have :$$x^n+y^n\leq \Big(\frac{x^n+y^n}{x^{n-1}+y^{n-1}} \Big)^n+\Big(\frac{x+y}{2}\Big)^n$$It's equivalent to :$$x^n+1\... 494 views

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### Covering a set $A \subset \Bbb R$ by two families of disjoint intervals taken from given intervals

Let $A \subset \Bbb R$ be a bounded set. Every element $a \in A$ is the center of some given open interval, let's denote it by $I_a=(a-r_a, a+r_a)$. I'm interested in knowing the following: Can we ...
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### A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...