Tagged Questions

Questions related to real and complex logarithms.

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The diferential equation $y' = \frac{\ln(x^2+y^2)}{x^2 + y^2}$

In my University, the integral calculus teacher gave me this diferential equation to solve. $$y' = \frac{\ln(x^2+y^2)}{x^2 + y^2}$$ I dont have any clue of what form has the solution of this ...
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optimal hill “rank” cannot be solved?

Okay so I was thinking about the following problem today: We have a guy who is h tall stand upon a paraboloid shaped hill of the form $z=-ar^n$ How far away (in r) does his friend who is also h ...
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Checking whether sequence $x_n = \ln(n^2 + 1) - \ln(n)$ converges or diverges

I have to show whether $$x_n = \ln(n^2 + 1) - \ln(n)$$ converges or diverges. I can write $$x_n = \ln(n^2 + 1) - \ln(n) = \ln\left(\frac{n^2+1}{n}\right) = \ln\left(n + \frac{1}{n}\right).$$ ...
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An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$\tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon)$$ ...
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Bounds on twin prime counting function

I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove (...
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Solving the logarithmic rational equation

I'm wondering there exist the way to solve the equation form of: $$\log f(x) + g(x) = c$$ where $f(x)$ and $g(x)$ are rational functions, $c$ is a constant. Is there any general(in closed form) ...
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What is logarithm & how log table can be constructed by me?

I'm studying properties of logarithm but I don't understand how base e works. Base 10 looks simple while doing calculations of numbers having multiple of 10. As other numbers are not multiple of 10 ...
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Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
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Moving the branch cut of the complex logarithm

The complex logarithm is defined as $\log z:=\operatorname{Log} |z|+i\arg z$ , with the branch cut on the non-negative real axis. Determine a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$ ...
I get the expansion of $h$ to be $$h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r$$ $$\Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}}$$ ...
If $a$ and $b$ are non-negative integers and $c$ and $d$ are non-negative real numbers, for what values is the following inequality true? \$\log((a+b)!) - \log(a!b!) \ge(a+b) \log(c+d) - (a \log(c) +...