Questions related to real and complex logarithms.

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8
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375 views

Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
7
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0answers
89 views

On Bailey and Crandall's sum for $\sum_{n=0}^\infty \frac{1}{5^{5n}}\left(\frac{5}{5n+2}+\frac{1}{5n+3}\right)$

On page 20 of "On the Random Character of Fundamental Constant Expansions", Bailey and Crandall gave the rather unusual sum, $$u_2 = \sum_{n=0}^\infty ...
5
votes
0answers
762 views

Choosing the branch of a logarithm

The problem: I am integrating complex logarithms over an angle $\phi$ over $[0,2\pi]$. It is quite complex (pun not intended) and I called Mathematica in to aid me. I am calculating an energy of a ...
4
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0answers
60 views

Differentiating $y=x^x$ with the formal definition of a derivative

A friend and I were messing around with derivatives, and while we both know the procedure for finding the derivative of $y=x^x$ with logarithmic differentiation, i.e. $$y=x^x\\ ln(y)=x*ln(x)\\ ...
4
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0answers
64 views

Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$

I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see ...
4
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0answers
91 views

Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
4
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0answers
403 views

Logarithm of a complex number as intersections of two logarithmic spirals

In Penrose's book "The Road to Reality" page 97 figure 5.9 he shows the values of the complex logarithm in a diagram as the intersection of two logarithmic spirals. Can you please explain how these ...
3
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32 views

Exercise concerning logarithms…

I have such a problem: find all the values of real parameter "a", for which the following inequality is true for any "x" that belongs to R. I will show you my solution, and please can you verify ...
3
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0answers
81 views

Limit of $x^x$ as $x$ tends to $0$

I am trying to solve the following limit: $$\lim \limits_{x\to0} x^x$$ The only thing that comes to mind is to write $x^x$ as $e^{x\ln{x}}$ and getting the right sided limit would be easy but I ...
3
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0answers
81 views

Phrase and symbol for “geometric absolute value”$ e^{|\ln(x)|}?$

I'm calculate the median fractional difference between two vectors (to characterise the error in a quantity with a high dynamic range). If $a/b = 0.1$, the fractional difference is $10$, and if $a/b ...
2
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0answers
42 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
2
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0answers
36 views

When this inequality true?

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) ...
2
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0answers
46 views

show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$, and $0<a<b$

Show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$,and $0<a<b$ by examining the sign of the derivative of an appropriate function. This is an exercise in middle part of ...
2
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0answers
75 views

Fourier transform of a logarithm

How can one go about computing the 2d (or 1d, in either variable) Fourier transform of the function $$\ln(w^2-k^2)?$$
2
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0answers
36 views

How to prove the function is logarithmic with coefficience

I am given a set of properties for an unknown function $f(x)$. In particular, not constantly zero, not negative, additive and continues for any $x$ between 0 and 1. I am asked to show equivalence ...
2
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0answers
37 views

Can we define root extraction using Peano Arithmetic?

I've been playing with Peano Arithmetic and I've got multiplication, division, exponentiation, and logarithms. I can't figure out root extraction but I have a stab at it. Exponentiation: $a^0 = 1, ...
2
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0answers
60 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
2
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0answers
456 views

Log-concave functions whose sums are still log-concave: possible to find a subset?

Rationale: I am puzzled by a problem of log-concavity, which arises in population dynamics where the curvature of the logarithm of sums is a quantity of interest. It is well-known that sums of ...
2
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0answers
35 views

Compute $ \operatorname{Li}_{3}\left(\frac{1}{2} \right) $

Where could I find a proof of the identity $$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)$$ ?
2
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0answers
75 views

Is there a (hopefully free) graphing utility BESIDES mathematica that can graph log polar coordinates?

This is my first post to the forum. I have had a good experience with other stackexchange fora, so I have high hopes for this one. I have looked online and can not seem to find the software described ...
2
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0answers
195 views

Exponential average of logs

I'm working on a sound recognition algorithm where an "exponential moving average" is used for "adapting" to sound levels. It turns out that taking an average of logs works better than simple sums ...
2
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0answers
222 views

Can the Baker-Campbell-Hausdorff formula for $\ln(AB)$ be simplified for similar, diagonizable matrices?

Given two similar, diagonizable square matrices $A$ and $B$ that do not commute, can the Baker-Campbell-Hausdorff formula be simplified exploiting the similarity to obtain a nice expression for ...
1
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0answers
57 views

Is this correct: $\ln({(-1)}^{2x-1})=(2x-1)\ln(-1)$?

I would expect the answer to be positive, but it appears otherwise for some values of $x \geq 1$. Here is a simple C++ code that I have used in order to test this: ...
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0answers
32 views

Finding the $\log$ of a matrix by contour integration

My teacher presented this way of determining the logarithm of a matrix $\Omega$ in class today: $$\log \Omega = \frac1{2\pi i}\oint_{\Gamma} (\zeta I - \Omega)^{-1} \log \zeta \,d\zeta.$$ Does ...
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0answers
38 views

Is $\log^* (n+1)^{n+2} \in O(\log^* n)$?

I would like to know if $\log^* (n+1)^{n+2} \in O(\log^* n)$, where $\log^*$ is the iterated logarithm. I tried doing: $ \log^* (n+1)^{n+2} =\\ \log^{*}(\log(n+1)^{n+2})-1 =\\ \log^{*}((n+2) \cdot ...
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0answers
35 views

Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
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0answers
29 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???
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0answers
82 views

Showing that a logarithmic inequality holds

Given $0 < x_1 < x_2 < x_3 < x_4 < 1$, how can I show that the following inequality holds: $$ \frac{1}{R(x_1, x_3)}+\frac{1}{R(x_2, x_4)}<\frac{1}{R(x_1, x_2)}+\frac{1}{R(x_3, x_4)} ...
1
vote
0answers
22 views

Mapping a deleting ray to a horizontal strip

So, this is my question: D is a domain obtained by deleting the ray $x\leq 0$. And $G(z)$ is a branch of $log(z)$ on $D$. I want to show that G maps D onto a horizontal strip of width $2\pi$. Show ...
1
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0answers
51 views

When the system of equations below had a solution?

The system of equations is $$\begin{cases} \frac{c_1}{1-x_1}+\frac{c_2}{1-x_2}+\frac{c_3}{1-x_3}=0\\ \frac{c_1}{k-x_1}+\frac{c_2}{k-x_2}+\frac{c_3}{k-x_3}=0\\ ...
1
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0answers
31 views

Algebra of Exponential and Log Functions

This question may have a simple answer or a very complex one, but I am interested in what the reasons are for logarithms and exponential functions having the properties they have. To my knowledge ...
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0answers
14 views

Trying to prove a function is increasing in one variable after the second variable crosses a particular value.

Let $f(p,n)$ = $\ln\bigl(1 + \frac{ap}{b+I}\bigr) + \sum_{j = 1}^{n}\ln\bigl(1+ \frac{b_j}{b + I - b_j + ap}\bigr)$, where $b>0$ and $a>0$ are fixed constants. The positive real numbers $b_j$ ...
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0answers
49 views

Estimation of a logarithmic sum

I need to estimate the sum $$ \underset{r=2}{\overset{t}{\sum}}\left(\frac{\log\log r}{r}\right)^{2}. $$ I tried to use the Abel's partial summation, and I got $$ \frac{(\log\log ...
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0answers
24 views

Can an entire $f$ satisfy $x>k | f(x+yi)=\ln(x+yi+z)+o(1) $?

Let $z$ be a complex number. Let $i$ be the imaginary unit. Let $x,y,k$ be positive real numbers. Consider $$x>k | f(x+yi)=\ln(x+yi+z)+o(1) $$ true for all $x>k,y$ and some $k,z$. Is there ...
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0answers
43 views

A problem on logarithm

Can this expression (see below) be written in the form $g_kw^{h_k}$. Where $g_k$ and $h_k$ are functions of only $k$?: $(1- k)^{\lceil\log_kw\rceil - 1}$. Here $k$ and $w$ are positive integers. I ...
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0answers
60 views

Matrix logarithm of sum

Let $D$ be a diagonal, $n\times n$ complex matrix, and let $X$ be an $n\times n$ complex matrix. Is there is simple formula for \begin{align} \log(D + \epsilon X) \end{align} as a power series in ...
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0answers
59 views

Closed form solution for ${d \operatorname{Tr}(A\log(X))\over dX}$

I need a closed form solution for ${d \operatorname{Tr}(A\log(X))\over dX}$. Here $A, X$ are $n\times n$ matrices, $\log$ is the matrix logarithm. ${d \operatorname{Tr}(\log(X))\over dX}=X^{-1}$, ...
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0answers
77 views

Expectation of functions of binomial random variable involving logarithms

Let $X\sim\text{Binomial}(n,p)$ where $n$ is the number of trials and $p$ the probability of success of each trial. I am trying to evaluate the expected value of the following functions of $X$: ...
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0answers
77 views

simplification of a natural log of a trigonometric function

hope you are all well. I am having a bit of a mental block, I am wondering if it is possible to simplify the following expression: $$k\cos X \cdot 4\ln(\cos X)$$ where $k$ is a constant and $X$ ...
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0answers
36 views

What's the most straight forward way to show that a function is increasing?

I am trying to show that: $$\frac{2}{n}\log\Gamma\left(\frac{x}{2}\right) - \log\Gamma\left(\frac{x+n-1}{n}\right)$$ is an increasing function for $x \ge 5$ and $n > 2$ One way to do this would ...
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0answers
38 views

transforming a straight band into a logarithmic spiral

I want to plot the labels and the graduations of an historical timeline onto a logarithmic spiral. If this timeline is on the $x$-axis, $-\infty$ would project to the center of the spiral, $+\infty$ ...
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0answers
33 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
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0answers
136 views

Using the gamma function as an upper and lower bound to the logarithm of a factorial function.

I am trying to find an upper and lower bound for the following function: $$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$ where ...
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0answers
136 views

Logarithmic simplification of a sum of power terms

It is entirely possible that there is no solution to this problem, but here goes... I have a number of equation of states for fluids that have terms that are of the form ${\phi _r} = ...
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0answers
101 views

Is this “Elegant” algorithm for logarithm by Zeckendorf representation, the same as an 'efficient' algorithm?

The algorithm here which computes the exponent $b$ given a base $a$, and given $n$ = $a$^$b$, appears no better to me than simply counting the number of times we divide $n$ through by the base $a$ ...
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0answers
111 views

Complex Logarithm

For what values of $p$ is the following valid? $$\log(z^p) = p\log(z)$$ where $$\log(z) = \ln(|z|) + i[\arg(z)+2\pi n]$$ where $n$ is an integer. I heard the expression above should not be valid for ...
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0answers
45 views

$f(n) = n^2 \lceil \log n \rceil$ is time constructible

I have a question, I want to show, that: $$f(n) = n^2 \lceil \log n \rceil $$ is time-constructible. I have no idea how to do this. I know that $n^2$ is time-constructible and I know that $\log n$ ...
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0answers
46 views

Calculating the number of states in power of 10's

This is part of a problem that was assigned to me. It might seem elementary, but I need a hint on how to start this problem. I have 10 billion bits that can either be on or off at any time. ...
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0answers
31 views

Log return of two different timeseries

Lets say I have a single timeseries, the simple return would be T/T-1-1 the log return would be ln(T/T-1) But let's say I have two different time series, T and R The values are close, but still ...
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0answers
69 views

Use of natural logarithm transformation on weighted index series

I have a value computed as sum of powers, e.g. $x^5+y^8+z^2$. The exponent represents the weight for variables, $x, y$ and $z$ in the example above. Applying natural logarithm on $x^5+y^8+z^2$, I get ...