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@SimplyBeautifulArt I already voted to close that problem, and went on to vote to close two other problems posted by same user, all three posted inside of one hour's time.
In Stein's Complex Variables I'm having trouble verifying the conjecture in $(2.)$
$(1.)$
If $f$ is a complex-valued, on $\partial D$ and if it's absolute value $|f(z)|$ is bounded by a constant $M$ for all z on $\partial D$.
$$\left|\int_{\partial D} f(z)dz \right| \leq \max:=|f(z)|\int_{a}^{...
@Simply I think the intial mistake was the result in $Lemma(3)$ since I dropped $\frac{1}{2i}$ the result was changed
I think that's where the mistake was
Looking specifically: $\oint_{\partial D} \frac{e^{rei\theta}-1}{re^{\theta}}=\frac{e^{rei\theta}-1}{re^{\theta}}\bigg\vert_{\pi}^{0}+\int_{\pi}^{0}{e^{rei\theta}-1} \frac{e^{rei \theta}-1}{(re^{i \theta})^2}$ If the coefficnet wasn't dropped the result would change
@SimplyBeautifulArt: I didn't see any comment from you directed at me except when you invited me to participate in a large number contest. I can't though because I'm very busy as I said earlier.
@SBM: Open sets are specified by the topological space and simply satisfy the axioms. As to why the axioms are interesting, I so happen to have a post on that:
Here is an interesting intuitive definition for open set that actually gives rise to the topological axioms!
An open set is a collection of objects for which we can easily verify membership of any given member.
Note that it may not be easy to verify non-membership of non-members!
...
But it occurs to me that perhaps you encounter the term "open set" in the context of some metric space, in which case it's typically not defined via the topological axioms but via open balls.
Either way, the intuitive notion I describe in the linked post for the topological axioms carry over nicely to the open-ball definition.
@SimplyBeautifulArt You're too kind! I haven't been around due to lots of more urgent things to do, and it's not my PhD stuff.
@SBM: As it is, I'm sure @shredalert here can explain more to you if you wish since he has more time to spare than me now, and likes this metric-space stuff, and needs practice. =)
If you would like some structure/motivation for your learning, I'm sure you've got plenty of us who, depending on the topic(s) you want to cover of summer, You've got us to help keep you on track!! After all, you have entered and frequented this Realm...
I'm more worried about how do I get into a good college and not let anything stop me from my motives, all the time, taking care that I continue becoming better both in the academics as well as or rather more importantly become a better person
Thank you
You've all helped me learn so much
From about supremum and partial derivatives to Real and Complex analysis
I don't mean to interrupt, but I'm wondering if I can ask about the Marshal Problem on this site? The Marshal Problem is a problem (I think I've come up with but who knows) where I'd like to know how many days it would take me to reach 100 flags, if I start at 10, can only use all my flags once a day and get a new flag every 10 flags I use. would that be ok to ask?
I asked in the other place, but mr 'dir sent me here
@SBM You've got a vast resource at your disposal. Literally just start reading and working through stuff you find interesting. Start now while you have lots of free time.
Total_flags_available_day(1) = 10 Total_flags_used_day(1) = 11 (I used 10 so gained 1, so had one more to use) Total_flags_available_day(2) = 11 Total_flags_used_day(2) = 12
Total_flags_available_day(3) = 12 (I gained another from the 1 on day one and 9 on day 2. I then had 3 added to the 10 required for my next gained flag) Total_flags_used_day(3) = 13 (I used 7 to gain a flag, I use 6 more before the end Total_flags_available_day(4) = 13 (I use 4 to gain a flag. I then use 10 and gain a second Total_flags_used_day(4) = 15
Total_flags_available_day(5) = 15 Total_flags_used_day(5) = 16 (I gain 1 flag for 10 used and get 6 closer to the next)
Total_flags_available_day(6) = 16 Total_flags_used_day(6) = 18 (I use 5 and get a new flag, I then use 10 and get a second new flag, I use 3 more and I'm now three closer to my new flag)
And on day n I reach 100 and can't continue gaining flags
@TheGreatDuck That was partly my projection (calling it a title... and not necessarily a title of a book, but the title of a song, as in the spirit of Peter, Paul, and Mary's "Where have all the flowers gone?"
I think, now that I've been listening to classics, like "Blowin' in the Wind", I might take the liberty of answering a poorly asked question: "The answer, my friend, is blowin' in the wind. The answer is blowin' in the wind"
@Mithrandir the sky is always falling. It's just so light that we don't notice. It is constantly putting weight on us. So when chicken little screamed "the sky is falling", the proper response should've been: "Yeah, so? Tell us something we didn't already know."
"Oops...I may need to call on @Mithrandir to delete the obnoxious image above!! Apologies!! Very annoying, very distracting, and it highlights only finitely many phone numbers, not including mine!!"
I came up with the attribution to "ty" as meaning "teen yodeling" off the top of my head...and interestingly, found a "pop artist" yodeling, and an on-the-verge-of-teenhood yodeler who is pretty amazing. Never listened to this much yodeling as I have this past hour.!
I'm having trouble with the following differential equation. Not sure how to proceed.
$$\frac{dy}{dx}=y(\ln x )\left(\frac{x}{y+1}\right)^2$$
In Stein's Complex Variables I'm having trouble verifying the conjecture in $(2.)$
$(1.)$
If $f$ is a complex-valued, on $\partial D$ and if it's absolute value $|f(z)|$ is bounded by a constant $M$ for all z on $\partial D$.
$$\left|\int_{\partial D} f(z)dz \right| \leq \max:=|f(z)|\int_{a}^{...
Here is an interesting intuitive definition for open set that actually gives rise to the topological axioms!
An open set is a collection of objects for which we can easily verify membership of any given member.
Note that it may not be easy to verify non-membership of non-members!
...
This is the Realm of Simply Beautiful…
This is the Realm of Simply Beautiful…