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.

Real analysis is a branch of mathematical analysis, which deals with the real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

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How do I solve $\min_x \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$ for $\lVert x \rVert_2 = 1$.

Let $f(x) = \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$. where $x, c_1, c_2, \dots, c_k \in \mathbb R^n$. What fast iterative methods are available for finding the (approximate) min of $f$ with the constraint $\lVert x \rVert_2 = 1$? Notes: $f$ is convex…
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Prove that $16(a\sin a + \cos a - 1)^2 \le 2a^4 + a^3 \sin 2a, \ \forall a\ge 0$

Problem 1: Prove that $$16(a\sin a + \cos a - 1)^2 \le 2a^4 + a^3 \sin 2a, \ \forall a\ge 0.\tag{1}$$ This is the stronger version of the following Prove that $12(a\sin a+\cos a-1)^2\le 2a^4+a^3\sin(2a)$,$\forall a\in (0,\infty)$: Problem 2:…
River Li
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Dirac Delta or Dirac delta function?

Is Dirac delta a function? What is its contribution to analysis? What I know about it: It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
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A reason for $ 64\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt =-\pi^4$ ...

Question: How to show the relation $$ J:=\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt =-\frac 1{64}\pi^4 $$ (using a "minimal industry" of relations, possibly remaining inside the real analysis)? So i have found…
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Evaluating the challenging sum $\sum _{k=1}^{\infty }\frac{H_{2k}}{k^3\:4^k}\binom{2k}{k}$.

I managed to evaluate the sum, my approach can be found $\underline{\operatorname{below as an answer}}$, I'd truly appreciate if any of you could share new methods to evaluate this series, thank you. The following are the short proofs of the other…
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Spivak's proof that every polynomial of odd degree has a root

I have the second edition of Spivak. Consider Can someone tell me why he considers $2n|a_{n-1}| \dots$? Later he shows everything is squeezed between -1/2 and 1/2 and he gets the desired result. I am perplexed as to why $2n$? He could have just…
Lemon
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Idea behind "reparameterization hiding a corner" in single variable calculus

I just solved question #2 on p. 248 from Spivak's Calculus Fourth Edition (2008). Solving it wasn't the issue. I'm trying to understand the idea behind it. This is a screenshot of the question: I'm trying to understand what is meant by…
D.C. the III
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Prove: $\int_0^{\infty} \frac{\ln{(1+x)}\arctan{(\sqrt{x})}}{4+x^2} \, \mathrm{d}x = \frac{\pi}{2} \arctan{\left(\frac{1}{2}\right)} \ln{5}$

Prove: $$\int_0^{\infty} \frac{\ln{(1+x)}\arctan{(\sqrt{x})}}{4+x^2} \, \mathrm{d}x = \frac{\pi}{2} \arctan{\left(\frac{1}{2}\right)} \ln{5}$$ This might be a repeat question (I couldnt find a question of this here). If im being honest I dont know…
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All solutions of $f(x)f(-x)=1$

What are all the solutions of the functional equation $$f(x)f(-x)=1\,?$$ This one is trivial: $$f(x)=e^{cx},$$ as it is implied (for example) by the fundamental property of exponentials, namely $e^a e^b=e^{a+b}$. But there is another…
Mr Pink
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Evaluate $\lim\limits_{n \to \infty}\frac{(-1)^n}{n!}\int_0^n (x-1)(x-2)\cdots(x-n){\rm d}x$

Evaluate $$\lim\limits_{n \to \infty}\frac{(-1)^n}{n!}\int_0^n (x-1)(x-2)\cdots(x-n){\rm d}x\,.$$ \begin{align*} I_n:&=\frac{(-1)^n}{n!}\int_0^n\prod_{k=1}^{n}(x-k){\rm d}x=\frac{(-1)^n}{n!}\int_0^n\prod_{k=1}^{n}(n-k-x){\rm…
mengdie1982
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Show that $|b-a|\geq|\cos a-\cos b|$ for all real numbers $\,a\,$ and $\,b$

$\mathbf{Question:}$ Show that $|b-a|\geq|\cos a-\cos b|$ for all real numbers a and b. $\mathbf{My\ attempt:}$ The Mean Value Theorem states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ then there exists $c \in (a,b)$ such…
Anubis
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Clausen and Riemann zeta function

This is an exercise from the American Monthly Problems from last year. I would like prove two formulas: (1) $\int_0^{2\pi}\int_0^{2\pi}\log(3+2\cos(x)+2\cos(y)+2\cos(x-y)) dxdy=8\pi Cl(\frac{\pi}{3})$ (2)…
Alexander
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Why is there no harmonic function on compact Riemannian manifold?

On a compact Riemannian manifold $(M,g)$, a function $f$ is called harmonic if $\Delta_{g} f = 0$, and it is known that the only harmonic function on a compact riemannian manifold is constant function. I wonder how one would prove this. So locally…
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The cardinality of Lebesgue sets

Suppose $A=\{S\;|\;S \subset \mathbb R^n, S\text{ is Lebesgue measurable}\}$. What is the cardinality of $A$? Is it the same as the cardinality of all of the real numbers?
Phil Wang
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