All Questions
Tagged with logarithms analysis
170
questions
4
votes
1
answer
178
views
Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.
The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as
$$
\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}
$$
where $\arg(z)$ is the standard branch of the ...
- 936
0
votes
0
answers
21
views
Inequality for log-convex functions
Let $s:\mathbb{R}^+\to\mathbb{R}^+$ be a positive, decreasing, log-convex function such that $s(0)=1$, $\lim_{x\to\infty}s(x) = 0$. In addition, $-s'$ is also assumed to be log-convex.
Are these ...
- 93
1
vote
1
answer
59
views
Convergence for a sequence using logarithmic properties?
Given the sequence $x_n=\frac{\log(3n+2)}{\log(n^2+2)}$, which has the limit $\frac{1}{2}$ when n goes to infinity, give a formal proof of the limit using the epsilon definiton.
$$| x_n - L |< \...
- 41
0
votes
0
answers
34
views
Non existence of Continuous logarithm on $\mathbb{C}^*$
This is an exercise problem, which asks to prove the non-existence of continuous logairthm on $\mathbb{C}^*$, Using the results of path integrals, In that book they already proved the non-existence ...
3
votes
1
answer
156
views
Simplify $n$th derivative of $\frac{\log^3 x}{1-x}$
I need to simplify $n$th derivative of $$f(x)=\frac{\log^3 x}{1-x}$$ where $0<x<1$
I tried writing $f(x)=u_1u_2u_3u_4$ where $u_1=u_2=u_3=\log x$ and $u_4=\frac{1}{1-x}$.
Using Leibniz rule for ...
- 936
1
vote
2
answers
72
views
Limit of $\frac{\log(R)^p}{ R^n }$ as $ R \rightarrow \infty$
I have been reading this old post:
Showing that an entire function with the following inequality is constant.
In the solution to the problem, we claim the following for $C$ constant, $n$ is a positive ...
- 19
1
vote
2
answers
70
views
Asympototic estimation of log log function
Let $N>0$ be an integer and consider the sum $\sum_{n=3}^N(\log \log N-\log \log n)$. It is not hard to see that this sum has the complexity $O(N^2)$ since $\log \log x <\log x<x$, so that ...
- 488
0
votes
0
answers
30
views
Show that for $a \neq b$ it holds: $\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$ [duplicate]
Show that for $a \neq b$ it holds:
$$\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$$
My first idea was to rearrange
$$2 \cdot (e^b-e^a) < (b-a)(e^b+e^a)$$
$$2e^b-2e^a < be^b + be^a - ae^b -e^a$$
...
0
votes
1
answer
66
views
Quick doubt on this function domain
$$f(x) = \frac{e^{1-\ln(x-x^2)}}{\ln(1 - e^{x-x^2})}$$
I was solving this, and I found the domain is the emptyset.
Yet, then I checked with Mathematica and it returned me a different domain. When I ...
- 3,482
-2
votes
1
answer
58
views
Prove using logarithms
I don't know how to prove that:
$$(2n-1)^{2n-1} \geq n^{2n} \mid \forall n\in \mathbb N$$
I have to use logarithms. I have tried to transform it into the form: $\log_n(2 - 1/n) \geq \frac{1}{2n-1}$, ...
- 25
0
votes
1
answer
32
views
Proof $\frac{ln(1+xy)}{x^2ln64}-\frac{y}{x(1+xy)ln64}>-\frac{1}{4x^2ln4}\ \forall x\in[\frac{1}{4};4]$ when $y\ge1$
Proof $$\frac{\ln(1+xy)}{x^2\ln64}-\frac{y}{x(1+xy)\ln64}>-\frac{1}{4x^2\ln4}$$ for all $x\in[\frac{1}{4};4]$ when $y\ge1$
I tried to do this: $$\frac{\ln(1+xy)}{x^2\ln64}-\frac{y}{x(1+xy) \ln64}=\...
0
votes
0
answers
78
views
Why does this show Log can't be extended to whole $\mathbb{C}^*$
Why does the following show Log can't be extended to whole $\mathbb{C}^*$?
Here's another proof which I think I understand, though I'm not sure what's the connection between the two proofs:
I ...
- 451
7
votes
4
answers
446
views
How to prove $\frac{\sqrt{5}+1}{2}>\log_23$?
I want to compare the magnitude of $2\cos36^\circ$ and $\log_23$ without using calculators.
Using $\sin72^\circ=\cos18^\circ$, I can get $$4\sin18^\circ\left(1-2\sin^218^\circ\right)=1$$
then I can ...
- 331
0
votes
0
answers
38
views
Fourier transform of the operator $\log^2(-\Delta)u$
I need to calculate the Fourier transform of the operator $\log^2(-\Delta)u $, but I don't know any property that can help me. I know that the transform of $\log(-\Delta)u $ is $\log(|\xi|^2)\hat{u}(t)...
2
votes
1
answer
71
views
A question related to a branch of complex logarithm
Define a branch of logarithm on the set $[0,\infty i) $ (the complex plane excluding the positive imaginary axis by $$\log\left(z=re^{i\theta}\right):=log(r)+i \theta,$$ where $\theta \in \left(-\frac{...
- 520