i should have said differentials haha i dont wnna go to hyperreal numbers
I saw this from wikipedia "The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor."
@anon I guess you're right...I suspect you can glean (by now) that I am fairly permissive about content on the site, questions etc ... So it's not usually like me to write off a question. I think I've been a little soured today...or I'm tired, or both. I stand by my downvote, but would have rather not voted to close.
we have 6 machines the lifetime of each ~ exp(-ln(.7)=0.35) if one machine fail it is reapair the next day .What is the probability that some particular day NO machine work?? ./So far I got : If the lifetime is grater than the perticular day the probability = 0 I also think to separate the even days of the not even /
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can someone help me evaluate $$\lim_{s\to 1} \frac{d}{ds}(\zeta(s,q)-\frac{1}{s-1}) $$ in terms of a special function where $\zeta(s,q)$ is the hurrwitz zeta function
I dont know asking this question in math SE is appropriate or not thats y i am writing this question here. I want to know whether limit obeys distributive law
Is there a transcendental number which does not have a trigonometric explanation? As all rationals and non-transcendental irrationals have an algebraic explanation?
@DanielMargolis Using at most countably many operations, we can still only write down countably many expressions, so at most countably many real numbers can be of the form you describe. The rest is just... let us say, intangible.
Yes. It means using a cosine function, a hyperbolic sine function, or a inverse tangent function, you can form this value by plugging in any algebraic function.
@DanielMargolis Well, no. My comment was more to the extent that if you take, say $\{\sin,\cos,\exp,\log,+,\times,\ldots\}$ (at most countably many operations), and apply them to the algebraic numbers (of which there are countably many), then you can still only express countably many real numbers.
Now, the current tools in number theory aren't very good at helping us determine if e.g. $\pi^e$ is tangible over the six operations I wrote above, but we know that there must be a large reservoir of inexpressible reals.
@DanielMargolis No. To say that there are intangible numbers between any two tangible numbers is wholly different from the existence of a number $x$ that has $0< x< \frac1n$ for all natural $n$.
(The quantifiers are interchanged.)
@DanielMargolis Indeed. And that's how mathematics has often progressed throughout the centuries. Many large research fields arose when trying to solve a certain open problem.
and presumably "trigonometric transcendentals" in this context are those numbers that cannot be expressed as a basic trig function evaluated at an argument in a certain set (e.g. rationals or rational multiples of pi)
Haha I think it's the collapse of cos(x) + isin(x) onto reals, but this requires linear algebra, complex variables of root 2, and logic functions of graphs.
if you replace x with -ix in the expression e^ix = cos(x)+i sin(x), you get e^x=cosh(x)+sinh(x). no linear algebra, complex numbers are involved yes, but complex variables of root 2 and logic functions of graphs are very strangely constructed phrases.
there are n distinct nth roots of unity in the complex plane. as corollary, there are n distinct nth roots of any nonzero complex number. for instance, there are two different square roots of 1: namely, 1 and -1.
with choices of things called "branch cuts" in the complex plane, one can construct "nth root functions," i.e. a function such that f(x)^n=x for all x. of course f() can only ever take on the value of one of the nth roots of a number, not all of them.
I gave you four different values, namely {z,iz,-z,-iz}, all of which when taken to the fourth power equal -1, i.e. four different fourth roots of -1. how does showing four things prove there is only one thing?
generally you can define nth root functions for all real numbers if n is an odd whole number, or nonnegative reals if n is an even whole number, without any issues.
generally you can define nth root functions for all real numbers if n is an odd whole number, or nonnegative reals if n is an even whole number, without any issues.
n=3 is odd; you can define a third root function on all real numbers, -1 is a real number. in the context of real numbers you can say (-1)^1/3=-1. however in the complex plane one needs to make choices in how one constructs a 3rd root function, and in the process (-1)^1/3 may or may not be -1 depending on those choices.
Here is lovely series I'm thinking of right now: if $b(k)$ is the number of zeros in base $7$ representation of $k$, then I want to find for which $x$ the series converges $$\sum_{k=1}^{\infty} \frac{x^{b(k)}}{k^7}$$
@Chris'swisesister It's radius of convergence is $1$. Calculating it for a given $x$ will be a pain.
I'm glad you encounter things you consider beautiful in mathematics. To me, such hints at where you should try to focus on, since more often than not, joy will team up with skill and ability.
$\zeta(s)/(s-1)$ has no poles. the residue of $\zeta(s)$ at the pole $s=1$ is simply $1$. you are probably thinking of $\lim\limits_{s\to1}\left[\zeta(s)-\frac{1}{s-1}\right]=\gamma$
@Lord_Farin Any idea if ZF is sufficient to prove that any compact connected Hausdorff space with at least two points is uncountable? I've been asking around and searching around and coming up empty.
In this article, I show that
$$
\zeta(s)=\frac1{1 - 2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}\tag{1}
$$
converges for $\mathrm{Re}(s)>0$ by applying the Generalized Dirichlet Test. To apply the test, we need to show that the partial sum of $(-1)^{n-1}$ is bounded, $|n^{-s}|\to0$, and
$$
\s...
