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Mathematics : Recent Questions and Answers (Page 102)

Find answers to your questions about Mathematics or help others by answering their Mathematics questions.

Does iterating the complex function $zmapstofrac{2sqrt z}{1+z}$ always converge?

Given $z_0inmathbb Csetminus{-1}$, define the sequence $$z_{n+1}=frac{2sqrt{z_n}}{1+z_n}$$ where the square root is the one with positive real part (or, if that's not possible, non-negative imaginary part). This is...

Asked on 11/02/2021 by mr_e_man

3 answer

If an infinite set $S$ of positive integers is equidistributed, is $S+S$ also equidistributed?

By $S+S$, I mean ${x+y,$ with $x,y in S}$. By equidistributed, I mean equidistributed in residue classes, as defined here (the definition is very intuitive, and...

Asked on 11/02/2021 by Vincent Granville

1 answer

How to evaluate $int frac{dx}{sin(ln(x))}$?

I am wondering how to evaluate the indefinite integral $$int frac{dx}{sin(ln(x))} quad (1)$$Attempt 1 I tried using Weierstrass substitution.The Weierstrass substitution, (named after K.Weierstrass (1815)), is a substitution used...

Asked on 11/02/2021

6 answer

$lfloorfrac12+frac1{2^2}+frac1{2^3}+cdotsrfloor;$ vs $;lim_{ntoinfty}lfloorfrac12+frac1{2^2}+cdots+frac1{2^n}rfloor$

Is there any difference between answers of $[1]$ and $[2]$?$$Bigglfloorfrac12+frac1{2^2}+frac1{2^3}+cdotsBiggrfloor tag*{$space.....[1]$}$$$$lim _{n rightarrow infty} Bigglfloorfrac{1}{2}+frac{1}{2^{2}}+frac{1}{2^{3}}+cdots+frac{1}{2^{n}}Biggrfloor tag*{$ space.....[2] $}$$ If yes then please do...

Asked on 11/02/2021 by dRIFT sPEED

2 answer

Finding the Center of Mass of a disk when a part of it is cut out.

From a uniform disk of radius $R$ a circular disk of radius $frac{R}{2}$ is being cut out. The center...

Asked on 11/02/2021

6 answer

Do functions with the same gradient differ by a constant?

Let $f,g:mathbb{R}^ntomathbb{R}$ be such that $nabla f=nabla g$. I believe this implies that $f$ and $g$ only differ by a constant, like in the one-dimensional...

Asked on 11/02/2021

4 answer

What loops are possible when doing this function to the rationals?

What loops are possible when doing this function to the rationals? Let's define this function on a simplified fraction $frac{a}{b}$. $$fleft(frac{a}{b}right)=frac{a+b}{b+1}$$ I started this with $f(frac{2}{3})=frac{5}{4}$ then...

Asked on 11/02/2021 by user808945

2 answer

Is there an explicit construction of this bijection?

As part of my answer to another question, I needed the following fact: if $S = {1, ldots, n}$ and $k leq n/2$, then there is a...

Asked on 11/01/2021 by Gregory J. Puleo

2 answer

How can I determine the radius of 4 identical circles inside an equilateral triangle $ABC$?

How can determine the radius $r$ of 4 identical circles inside equilateral triangle $ABC$ of side $a$ ? ...

Asked on 11/01/2021 by user766881

5 answer

Prove that $tan^{-1}frac{sqrt{1+x^2}+sqrt{1-x^2}}{sqrt{1+x^2}-sqrt{1-x^2}}=frac{pi}{4}+frac 12 cos^{-1}x^2$

Let the above expression be equal to $phi$$$frac{tan phi +1}{tan phi-1}=sqrt{frac{1+x^2}{1-x^2}}$$$$frac{1+tan^2phi +2tan phi}{1+tan^2 phi-2tan phi}=frac{1+x^2}{1-x^2}$$ $$frac{1+tan^2phi}{2tan phi }=frac{1}{x^2}$$$$sin 2phi=x^2$$$$phi=frac{pi}{4}-frac 12...

Asked on 11/01/2021

4 answer

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