Is the following statement true?
Suppose, $ f:D\to \mathbb C $ is an analytic function where $ D $ is the unit disc of radius $ 1 $ around $0 $. Suppose, $ f $ is analytic on the boundary of $ D $ as well. Then prove that, if $ f $ is one-one on the boundary of $ D $, then $ f $ is one-one on $ ...
Is the following statement true?
Suppose, $ f:D\to \mathbb C $ is an analytic function where $ D $ is the unit disc of radius $ 1 $ around $0 $. Suppose, $ f $ is analytic on the boundary of $ D $ as well. Then prove that, if $ f $ is one-one on the boundary of $ D $, then $ f $ is one-one on $ ...
What are some good books to learn differential equations and vectors calculus? I currently have the one by Marsden for vector calculus and Zill for differencial equations. Are those good?
the (n+1)-th term is $\dfrac{(2n-1)(2n-3)\dots 3.1}{n!.3^n}= \dfrac{(n-\frac{1}{2})(n-\frac{1}{2}-1)\dots (n-\frac{1}{2}-(n-2)).(n- \frac12 - (n-1))}{n!}\bigg(\frac{2}{3}\bigg)^n$ .. so this is binomial expansion of $(1 - \frac23)^{-\frac{1}{2}}$
@robjohn, wrt. math.stackexchange.com/q/371262, I do recall a similar but not equal question which was in my filter recently but which I can't find just now. It had a nice illustration, and lots of “what I did”, but it was asking about surface area not volume iirc. It had a comment to it asking for texification, and another pointing to some PDF, some name containing the term “window”. Do you think you can identify a recently deleted post matching these criteria?
I feel I'm missing a number of posts today. I recently looked for an answer I had posted some time ago, and I can't seem to find it anymore. I really whish there were any kind of indication that an answer of mine has been deleted, so I know I can stop looking. As it stands, this is driving me crazy.
@Sush you can start with a book called Challenges and thrills in pre-college mathematics (thats where I started learning Number Theory and geometry from) .. apart from that Elementary Number Theory by Burton is a good book
@robjohn @robjohn: That's the one. I guess the question was deleted by the user, but I'd like to know whether it was in response to an answer or not. If there was an answer, this feels like cheating so undeleting the question might be warranted. Even if not, I'm kind of sad that this version got removed, instead of declared a duplicate of e.g. this one.
Good to know that I don't have to include the name if I reply to a specific message…
I'd believe it more likely that the one comment with the link somehow told him enough so that the question was no longer relevant. I'll have to work on reaching 10k…
@robjohn I only noticed that second question while searching for the question I remembered, so there is no indication that the poster of said question was aware of the previous solution before deleting his post. Might be, though. OK, in dubio pro reo I'll not assume any attempt at cheating, since there are plausible honest explanations. Thanks!
to prove that being H,K subgroups of G, H U K can't be a subgroup if H intersection K is neither H or K, ppl generally take into account an element x in H but not in K, and an element y in K but not in H. Then they argue that their product xy can't be in H and can't be in K, which i understand.
But why does the product need to be in H or in K ? Take (Z,+) and the subgroups H={...,-4,-2,0,2,4,...} and K = {...,-6,-3,0,3,6,.. } . We can take x = 2 in H, and y=3 in K. x+y is neither in H nor in K
"But why does the product need to be in H or in K ?" Because subgroups must be closed under the binary operation, and in particular, for $H\cup K$ to be a subgroup we must have $xy\in H\cup K$.
I am currently investigating the convergence of the following function,
$f(x)=\sum_{k=1}^{\infty} \frac{x^{k}+sin(k)}{k^{2}}$
for different "senses". I have shown that $f(x)$ converges uniformly and pointwisely for its domain. All I have left to do is investigate whether it converges in the $L^...
If you have an idea, please share your thoughts so I can give you rep. Also, I have to go to sleep - I have to wake up early tomorrow so I have no more time left to do study :( . Goodnight !
$f_n \rightarrow f$ in $L^p$ means that $\lim_{n \rightarrow \infty} \|f_n - f\|_p = 0$
which means $$\left(\int_{-1}^1 |f_n(x)-f(x)|^pdx\right)^{1/p}$$
@eXtremiity you might want to read about what are called "$L^p$ spaces", the vector spaces of $p$-integrable functions. these can be quite different for different $p$!
@r9m The questions he answer and his elegant geometric methods are much more delightful than his bloog, which is is not to say, that its not awesome as well!
To answer the second question I ever asked on here... 1. Mathematics - covers a mixture of both pure and applied mathematics 2. Pure Mathematics - focuses on theory and proofs 3. Applied Mathematics - focuses on the physical side of mathematics and how we can apply it to the real world
Suppose we have a function $f(x)$ that we can only observe through some noise. We can not compute $f(x)$ directly, only $f(x) + \eta$ where $\eta$ is some random noise. (In practice: I compute $f(x)$ using some Monte Carlo method.)
What methods are available for finding roots of $f$, i.e. comp...
I am currently investigating the convergence of the following function,
$f(x)=\sum_{k=1}^{\infty} \frac{x^{k}+sin(k)}{k^{2}}$
for different "senses". I have shown that $f(x)$ converges uniformly and pointwisely for its domain. All I have left to do is investigate whether it converges in the $L^...
Suppose we have a function $f(x)$ that we can only observe through some noise. We can not compute $f(x)$ directly, only $f(x) + \eta$ where $\eta$ is some random noise. (In practice: I compute $f(x)$ using some Monte Carlo method.)
What methods are available for finding roots of $f$, i.e. comp...
