2
votes
1answer
66 views

Functions which satisfy $\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w)$

Let $\mathrm{f}$ be a complex-valued function with the following property: $$\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w) $$ for all $w,z \in \mathbb C$. Necessary conditions are that ...
2
votes
2answers
54 views

A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?

I did not know what to search to see if this existed elsewhere. But, I could not find it. Here's the question, does there exist a continuously differentiable function, $f: [0,1] \rightarrow ...
0
votes
2answers
90 views

function with restrictions in finding solutions

Please help... How to prove the following functional equation, which has no solution. $A + B + C + (A + B)C - 2 = 0$ has no solution, where $A = f(x)$, $B = g(x)$ and $C = h(x)$. Here $A, B$ and $C$ ...
6
votes
0answers
254 views

Can $ 262537412640768743.99999999999925 $ be beaten with simple expressions? [closed]

We know: $$\begin{align}e^{\pi \sqrt{163}} &= 262537412640768743.9999999999992500726\dots\\ x^{24} - 24&=262537412640768743.9999999999992511239\dots\end{align}$$ where $x$ is the real solution ...
3
votes
2answers
83 views

Number theory Exercise

for positive integer $n$, how can we show $$ \sum_{d | n} \mu(d) d(d) = (-1)^{\omega(n)} $$ where $d(n)$ is number of positive divisors of $n$ and $mu(n)$ is $(-1)^{\omega(n)} $ if $n$ is square ...
0
votes
0answers
113 views

a follow up question to the birthday-paradox question.

The previous question. My goal is to find a function of the difference (error) between F and G generally. F = $\Pi_{0}^{n-1}\frac{365-b}{365}$ G = $ \frac {364}{365}^{\frac {n^2-n}{2}}$ Now F ...
-1
votes
2answers
760 views

write text using an equation

Well like the batman equations and equations for heart, I once saw a site that draws equation for whatever text you type....but now I can't find it. Does anybody know such a site? Also a general ...
1
vote
1answer
81 views

Measuring how monotonically “staircase-like” a set of values is

A bit of a bizarre question here -- I'm looking for assistance in generating a robust metric to measure how monotonically "step-wise" a series of values is. The set must not start or end at a specific ...