Mathematics - 2012-05-29 (page 3 of 4)

Eugene

18:10

@MarianoSuárezAlvarez thanks for clearing that up

Matt N.

Algebraists, can anyone help with this one?

Ta.

unNaturhal

Hi people!

Matt N.

Hello.

unNaturhal

Could someone tell me how is called in english the "Tangent to $f(x)$ if $x = 0$"?

Gigili

18:33

@unNaturhal I don't understand what you're asking.

What is what called in English?

The tangent of 0?

unNaturhal

Mmmmh

I have a function

This function passes throw x=1

I have to calculate the tangent in that point

@Gigili Could you figure what I mean?

t.b.

@unNaturhal The tangent line to $f$ in zero?

unNaturhal

@tb Umh?

t.b.

You mean $t(x) = f(0) + x f'(0)$, no?

I'd call that the tangent line to $f$ at $(0,f(0))$.

unNaturhal

The formula is: $y = f'(x_0)(x - x_0) + y_0$

t.b.

18:47

You mean $f'(x_0)$, there. I plugged in $x_0 = 0$ and $y_0 = f(x_0) = f(0)$.

Gigili

@unNaturhal The tangent line passes through $x=1$ which lies on the function $f(x)$?

Nimza

Hello again! Can anybody tell me what is the $\overline{\partial}$ operator?

t.b.

@Nimza $\bar{\partial} f = \frac{1}{2}\left(\frac{d}{dx} + i \frac{d}{dy}\right)f$. A function is holomorphic if and only if $\bar{\partial}f = 0$.

unNaturhal

@tb I don't thik to have undestood

Nimza

@tb thank you, it is good if $\overline{\partial} = \frac{\partial}{\partial \overline{z}}$. Thank you!

t.b.

18:52

@Nimza Yes, that's what it usually stands for.

Ilya

@Nimza привет!

t.b.

@unNaturhal You want the tangent line to pass through $(x_0,y_0) = (x_0,f(x_0))$. So I plugged in $y = f'(x_0)(x-x_0) + y_0 = f'(0)(x-0) + f(0) = f(0) + x f'(0)$. Our formulas really are the same.

Hi, Ilya

unNaturhal

@Gigili Maybe :P However, it's not important... The important thing is that I don't know how to study the tangent on this function $\left|x^2 - x\right|e^\left|x^2 - x\right|$

@tb Yeah! Exactly :D

Ilya

@tb hi, I'm away a bit, watching mr. Nobody. Didnt know it's a European movie

t.b.

@unNaturhal You should first get rid of the absolute value signs, there.

unNaturhal

18:56

@tb rid?

t.b.

@unNaturhal write the formula in such a way that they disappear, by thinking about when $x^2 - x \gt 0$ and $x^2-x \lt 0$.

@Ilya have fun, see you later!

unNaturhal

@tb I alresdy studied about all the function: Domain, Simmetry, Positivity, Derivatives (only the first), Asymptotes, and the study of the min, max, inf and sup of the function. It remains to be done this tangent, and the graphic draw...

Ilya

@tb thanks. I'm a bit fascinated by Jared. Or maybe it's all because of beers

t.b.

@unNaturhal Very good then. What's the derivative at $x = 0$?

robjohn

@tb $1=0$ :-)

unNaturhal

19:01

@tb In $x = 0$ I made the limit of the incremental***, and there is an angular point (because the limit is 1 for $0^+$ and -1 for $0^-$

Ilya

@robjohn hullo

robjohn

@Ilya howdy. How's the day treating you?

I wonder if that works into the etymology of howdy Nope, it's how do ye

Ilya

It was tough, what about yours?

robjohn

@Ilya so far, so good.

Ilya

I'm trying to reset my mind from math and biology with beer and movies

t.b.

19:04

@robjohn Isn't that short for how do you do?

(Oh, I missed your edit, sorry)

Ilya

Oh no, they are doing that!!!!

Poor kids

t.b.

@unNaturhal This sounds right. So you don't have a tangent line there.

robjohn

@Ilya doing what? :-p

Ilya

I don't know, what you're doing :)

Lost our connection somewhere in China

unNaturhal

@tb Mmmmh.. I supposed that.. Also because for $x = 1$ the first derivative doesn't exist.. So, why in the exam exercise it ask to study the tangent to the function in $x = 1$?

Ilya

19:08

Oh, his dad is strange

t.b.

@unNaturhal I don't know. Maybe they want you to find out that there's none. Maybe they want you to find the one-sided tangents.

robjohn

@unNaturhal I usually use $\frac{\mathrm{d}}{\mathrm{d}x}|x|=\frac{x}{|x|}$ and then apply the chain rule

that is a nice form since it carries over to the gradient

unNaturhal

@tb Exist one sided tangents? O.o However, the tangent doesn't exist means that it is parallel to the x-axis?

@robjohn I got it

robjohn

@unNaturhal okay. Just throwing in a couple of my pennies.

t.b.

@unNaturhal well, don't worry about one-sided things if you never heard of it. I don't know. I don't think it makes sense to speak of tangents if there are none.

unNaturhal

19:12

@robjohn What? :P

robjohn

@tb I agree.

Matt N.

Vote for calls.

unNaturhal

@tb You're right, I agree :P So, I just write down: there are no tangents in $x = 1$

robjohn

@unNaturhal I was paraphrasing the idiom 'giving my two cents worth'

Matt N.

And halp please if you can. I can't do any thinking now otherwise I'd help him.

Gigili

19:14

@MattN Called.

Matt N.

@Gigili Great thanks.

unNaturhal

@robjohn Eheh, the problem is that I didn't know how to name "tangents to the function in $x = 1$", there are no possibilities for me to know the meaning of an idiom :(

Ilya

@MattN done

Matt N.

@Ilya Thanks.

unNaturhal

@tb @robjohn Thank you very much for the help! :) I go to eat, see ya :)

robjohn

19:20

@unNaturhal bon apetit!

Peter Tamaroff

Maybe someone can help this guy over here.

This question is really interesting. I favorited it because of sos440 solution.

t.b.

Votes to re-open please

Peter Tamaroff

@tb But you closed it. ??

t.b.

@PeterTamaroff Yes, I didn't look closely enough. Jyrki pointed out that it wasn't exactly the same.

Peter Tamaroff

Oh, OK.

@tb I started reading a little of Apostol's Intro to Number Theory.

I am curious about the motivation for the Möbious function. It is not zero if $n$ is squarefree right?

and if it is the product to $k$ dsictinct primes, then it is $(-1)^k$.

t.b.

19:33

@PeterTamaroff I don't remember but isn't there also something with an even/odd number of prime factors that determines the sign?

Peter Tamaroff

@tb Right.

t.b.

Yeah Wikipedia Confirms

Peter Tamaroff

But the motivation for defining it interests me.

t.b.

You ask the wrong guy :)

poor me. After pestering me with algebra they start with number theory of all things.

Peter Tamaroff

@tb Who pestered you? What do you mean?

t.b.

19:36

@PeterTamaroff there's one guy who always asks me about Atiyah-MacDonald

It's not exactly my cup of tea.

Peter Tamaroff

@tb Oh. I heard Atiyah made great discoveries, but I never heard of Mac Donald.

Ilya

who finds the duplicate first?

t.b.

@PeterTamaroff Anyway: as far as I understand the motivation simply is that the Möbius inversion formula holds.

Ilya

@Gigili what does it mean?

Nimza

@Ilya $\pi \rho i \phi \varepsilon \tau$ !

Ilya

19:46

@Nimza приФет???

use $\beta$! :)

$\omicron \lambda\omicron \lambda\omicron$

Nimza

@Ilya I think $\nu$ это н!

Ilya

@Nimza $\tau$ы $\pi\rho\alpha\beta$

t.b.

@PeterTamaroff Atiyah-Macdonald is a famous book on commutative algebra. Both authors are pretty famous. The first certainly more so.

Nimza

$ \tau \rho \omega \lambda \omega \lambda \omega $

Ilya

:)

Gigili

19:48

If nothing, we could at least make a robot to hit the MathJax bookmark when needed.

Ilya

@Gigili: we have skull already

and what does "tease in a mean while" mean?

Gigili

@Ilya Oh? don't tell me he's going to my room to do that. I don't have enough space for both of us!

Ilya

I won't

Gigili

@Ilya I'll show you, cannot explain it properly.

Ilya

future tense makes your statement hard to verify

Gigili

19:51

We first need a Matt who is AFK for a short while.

Ilya

ok

I'll be afk soon too

Gigili

So that won't work. I can't tease you while you're AFK.

Ilya

I have some time before December, 21st

bbl

Gigili

Umm.

Seems my mind is infected with the virus, it doesn't interpret things immediately.

Peter Tamaroff

@Gigili If you feel the urge to eat another man's face, let us know.

28 days is usually the time it takes.

Gigili

20:01

@PeterTamaroff Ekskioozmi?

t.b.

@robjohn: didn't you have something on the Möbius function on your webpage? Maybe I misremember.

Anyway, Peter might be interested in that.

Ilya

@Gigili the virus has a name

I left math and worker for 2 years in the industry, then returned to math again. Haven't done anything interesting yet

t.b.

Hi, @leo, how are you doing?

Gigili

@Ilya What's it called?

Ilya

@Gigili you should remember

Gigili

20:13

The first time I see Tee-Bee says hello to someone before the one says hello.

2

@Ilya Rajull?

leo

@tb everything is fine. Reading about some faithful group actions. How are you?

Ilya

@Gigili of course, not

Gigili

@Ilya I can't think of a more annoying name for a virus. Perhaps you would drop me a hint?

Ilya

@Gigili perhaps

leo

by the way, I don't what term is used in spanish to "faithful action". I think must be "acción efectiva"

Matt N.

20:16

So, what have I missed since an hour ago?

Ilya

@Gigili: Matt is here, so would you care to explain now?

Gigili

You missed me.

Go AFK for a short while @MattN, I'd tease @Ilya in the mean time.

t.b.

@Gigili slander!

Gigili

Did I say in a mean time?

Ilya

@MattN: what?

Matt N.

20:18

@Gigili Ok. Where did you tease him? Or do you want me to read all the transcript? Looks as if you'd been a chatty bunch : )

@Ilya What?

Ilya

@MattN what did you remove?

Gigili

@MattN What?

Ilya

@Gigili: so...

t.b.

@leo At least it's synonymous in English and it sounds good :)

leo

@tb (-:

Gigili

20:20

The first time I see Tee-Bee talks to someone with smile.

Ilya

@leo are you from Australia, if I may ask?

@Gigili how cool. So many of new things for you today!

Matt N.

@Ilya Boring stuff.

leo

@Ilya I'm Costa Rica :-)

Gigili

The first time I see Tee-Bee.

The first time.

Matt N.

@Gigili Ok, going afk to pour another drink. In the meantime you tease Ilya. Ok?

Gigili

20:21

Okay, I'll stop now.

Ilya

@MattN what does it mean?

Gigili

teases Ilya

Matt N.

@Ilya I don't know, just trying to play along.

Gigili

@MattN I expected you more ...

Why you pretend to be male when you are female @Ilya?

Ilya

@MattN you should add some eggs, sugare and meaning to this tree

t.b.

20:22

@Ilya take your pick: make fun of, poke fun at, laugh at, make a monkey out of; taunt, bait, goad, pick on; deride, mock, ridicule;

Ilya

@Gigili I don't pretend

Matt N.

@Ilya What tree?

Ilya

@tb the first time I see Tee-Bee being a vocabulary ;)

Gigili

Dictionary, you meant.

t.b.

thesaurus.

Matt N.

20:24

You are all so funny : D

Either that, or it's the drink.

Ilya

@tb you missed "O", TheO

Matt N.

@Ilya LOL

Gigili

@MattN I knew you'd say that.

Ilya

@MattN the latter

@Gigili you lie. I'm not a female and you didn't know Matt will say that :)

Gigili

@MattN You didn't even let us enjoy the compliment for a moment.

leo

20:25

@tb when we are constructing a nonmeasurable set on $\Bbb R$, we use: 1. Action of choice. 2. Density of $\Bbb Q$. 3. (Strongly) Translation invariance of the Lebesgue measure. So given a metric space $(X,d)$ with a dense subset $D$ and a borel measure which is translation invariant, we can do the same. What about other spaces? For example in $(\mathbb{N}, \mathcal{P}(\mathbb{N}), \#)$ all the sets are measurable.

Is it interesting/important to characterize all the measure spaces where we can get nonmeasurable sets?

Ilya

@tb dear Diary, what does strip-tease mean then? Making a monkey out of somebody being naked in the meanwhile?

Matt N.

@Gigili I'm not that sort of person, ya know. : )

@Ilya At least this time it wasn't me.

Ilya

@MattN: you are around

Dear Diary, they all are just my dreams, fragile as soap films...

Gigili

@Ilya You're mistaken, my friend. Not about your gender, of course. But about me telling lies.

Ilya

@Gigili Jasper? The-guy-who-never-lies-in-chat?

Dear Diary, today I met a woman. I also met a man. Today I went out, my dear Diary.

Matt N.

20:28

But this path-connected question is an exact duplicate.

Now it's been reopened.

Ilya

it's going crazy now

Gigili

@Ilya Oh, dear ...

Matt N.

@Ilya it = you? : )

Gigili

How can you met both a man and a woman on a single day?

Ilya

@Gigili Let me tell you a secret: I was moving

Jonas Teuwen

20:29

Or the man and woman?

Ilya

@Jonas!

Matt N.

Hey! @JonasTeuwen : )

Ilya

I have to leave

Jonas Teuwen

Bye.

Matt N.

Byee!

Ilya

20:30

and I can't leave them here, they're my dreams

Gigili

Bai.

Ilya

@Jonas: have dreams about this chat till I'll come - otherwise it disappears! Please

Gigili

@Ilya I'll keep an eye on them while you're away.

Matt N.

You guys have lost me...

Ilya

noooo. @MattN: you're still in the dream, don't be afraid

t.b.

20:31

@leo I would probably argue that in every interesting measure space we have non-measurable sets...

Ilya

@tb that's your definition of being interesting? that you cannot measure at least a part of it? otherwise you don't feel any interest? That's usual for us... I can understand you. I guess, you can't understand me

t.b.

Yes, I find $(X, P(X))$ quite uninteresting.

Ilya

@MattN oh; good luck then

Matt N.

@Ilya Thanks mate.

Ilya

which subject?

Matt N.

20:34

Functional analysis and commutative algebra.

Gigili

Which degree?

Matt N.

maths

Gigili

Which exam?

Ilya

Which virus?

Gigili

Which Matt?

Ilya

20:35

Which witch?

leo

@MattN good luck man!

Ilya

Which man? :D

Matt N.

@leo Thanks mate.

Gigili

Thanks mate.

Ilya

@MattN that was delivered to me

maybe I am leo...

Gigili

20:36

For your nice edits.

Matt N.

Welcome! : )

Ilya

@gigili: who am I

Gigili

@Ilya You could be anyone.

Ilya

that's wise

leo

or noone

Ilya

20:38

that's not :)

Matt N.

It's hot in here.

leo

what about random?

Ilya

get out of the oven

Matt N.

Wow it's supposed to be raining this week. Nice.

Ilya

@leo: probably, yes

@MattN are you surprised about the rain in your area?

t.b.

20:39

@leo It is usually hard to prove that there are non-measurable sets. For example in every uncountable complete separable metric space there are non-Borel sets that are measurable with respect to every Borel probability measure. See also universally measurable sets.

Matt N.

@Ilya Yes. Rain is a rare species here, especially in summer.

leo

be something no deterministic

@tb thanks

Ilya

Is there a picture of the void?

t.b.

@MattN that's a common misconception

Matt N.

@tb Either that. Or you shouldn't believe everything that's on the internet. : )

t.b.

20:42

I measured it myself...

Matt N.

@tb I'm not going to ask what...

t.b.

(I didn't make this statistics)

Matt N.

Anyway. June isn't my least favourite month. I think the worst are August and July.

@tb And also: isn't Bern wetter than Zurich?

t.b.

@MattN sheeeeeeeeeeeeeeeeeeeeeesh

robjohn

@tb 73°F is the annual high? We're lucky if we stay below 103°F for our high.

Matt N.

20:46

@tb Hey. Did I say anything wrong? : )

t.b.

@robjohn I think it's the maximal average over a day. It's rarely over 20 Celsius during the night.

@leo You usually need strong choice principles to get non-measurable sets. Translation invariance of a measure is not really the crucial ingredient.

Gigili

@MattN Nothing more than where Tee-Bee lives.

I'll be there by tomorrow.

Matt N.

@Gigili Heh. So you fancy him that much? Can't blame you. : )

Gigili

No.

@MattN I think that's the side effects of your drinks on me.

Matt N.

@Gigili : ) You're suggesting that Mr. Jameson can teleport?

Gigili

20:51

He sure can.

Matt N.

brb

Gigili

Let's say "what haven't you tried" , "what will you try", "Why don't you try" instead of "what have you tried".

I'm off to bed.

Good night.

N3buchadnezzar

21:14

How is it going guys ? =)

Matt N.

21:26

@Gigili Good night!

Hm. Did I kill the conversation?

Jonas Teuwen

@MattN Moo.

Matt N.

Hi Jonas.

It's bed time here.

Good night! x

robjohn

@tb I only have a proof of the basic Möbius inversion formula. However, it looks as if Peter is just starting in on the Möbius function, so it might be useful.

I will mention it next time I see him.

t.b.

@MattN 'night!

robjohn

@tb Was MattN leaving as well as Gigili?

@MattN Good night!

I thought he was saying goodnight to Gigili :-)

t.b.

21:39

@robjohn Thanks a lot! I guess it will be in Apostol but young guys should learn to appreciate ASCII art :)

robjohn

@tb I think so, too! Dying Arts should be a breadth requirement.

t.b.

@robjohn I thought "It's bed time here" together with seeing the red hat jumping out of the (Gravatar) window is enough indication that Matt N.'s left for a better place.

robjohn

@tb Yes, you are correct, but all I saw was him saying 'good night' to Gigili. I missed him before he left.

I wonder if I could teach Dying Arts 101 at UCLA

David Wheeler

is there an extension of F11 that has the klein 4-group as its galois group?

t.b.

@robjohn That'd be a nice course to teach. Although with such courses I'd fear of running out of good ideas after the sixth or seventh week.

David Wheeler

21:44

artfiles.com has a lovely pron collection

t.b.

I like this guy's approach to maths: non-trivial questions with the technical skills to dig through the mess and solve them.

Peter Tamaroff

@tb I was thinking of asking "What have you tried" in a "sire"ly manner.

To make it a little fun.

t.b.

@PeterTamaroff heh :) By the way: robjohn linked to a proof of the Möbius inversion formula on his webpage.

Dylan Moreland

Look at that ASCII.

Peter Tamaroff

@tb See this

hhh

22:00

I am trying to find how to determine complex fourier factors $c_n$, is that some consintent terminology?

Dylan Moreland

@PeterTamaroff Pretty subtle.

Peter Tamaroff

@DylanMoreland We have to think of a more tortuous way of saying it.

@tb AAAA that ASCII kills me.

hhh

-

Peter Tamaroff

robjohn is the amster of mathJax and he uses ASCII?

anon

Hey, my swag arrived today. Cool.

hhh

22:02

Question like this: "Suppose $f$ with $2\pi$ periodic and $f(x)=e^{-ix/2}$ when $-\pi\leq x\leq \pi$. Determine the complex fourier factors of $f$."

Peter Tamaroff

@anon What is it?

Dylan Moreland

@hhh Fourier coefficients, maybe? I'd be the wrong person to ask.

hhh

(was about to ask the same q)

@DylanMoreland Yes that is the term.

...trying to find something in English...

Jonas Teuwen

What are the complex Fourier coefficients? The bloody things are always complex.

t.b.

@PeterTamaroff what's wrong with it? It's like reading books in $\mathfrak{Fraktur}$ (or older texts written with typewriter and handwritten formulas). Takes a while to get used to it but after a while you can enjoy its own charm...

anon

22:05

Hell yes, I got bubble wrap!

4

Jonas Teuwen

Hah, I'd love to be able to write in mathfrak. Then I'd do this on the blackboard.

hhh

en.wikipedia.org/wiki/Fourier_series#Exponential_Fourier_series <--- I can find here $c_n$, that is probably the coefficient.

Fourier series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. The Fourier series is named in honour of Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation in a me...

Jonas Teuwen

Holy cow, I do know this. But thanks.

I breathe Fourier... (and eat it too).

Dylan Moreland

@hhh "coefficients" isn't English? Or am I misreading you?

Peter Tamaroff

@anon What else did you get? I think I'll get a shirt for the elections

hhh

22:08

That is the stuff, read them earlier -- but have to just somehow remember how to deduce the $a_n$, $b_n$ and $c_n$...not wanting remember.

anon

It comes with shirt, pen, marker, mug, a couple stickers and a brief thank-you letter.

t.b.

@PeterTamaroff see here

anon

I don't drink coffee. Is it "normal" to drink anything in mugs or are they primarily for coffee? I guess I could make a very small root beer float.

Dylan Moreland

I use mugs for everything.

anon

Cool.

Dylan Moreland

22:09

Because, in a pinch, I want to be able to use any container for coffee.

Peter Tamaroff

@anon You can drink tea

@tb I bought the one of $X+Y+Z=A$ from Einsten a while ago, when I went to visit Princeton.

It is my favorite one so far.

@anon Why don't you drink coffee?

anon

I don't like it.

t.b.

@PeterTamaroff the success formula?

Peter Tamaroff

@tb Yeah.

"Sheafification of a presheaf through the etale space" this is why math is awesome.

anon

lulz, Diddy Pow changed his name

Peter Tamaroff

22:12

Yes, it is the third time I think.

anon

Indeed.

Peter Tamaroff

He will eventually leave the D

only

anon

Go figure.

Peter Tamaroff

@MarianoSuárezAlvarez Buenas noches =)

Jonas Teuwen

@hhh EULER'S FORMULA.

Bill Dubuque

22:19

@anon Perhaps "did" was in response to "doit"

anon

that would be an order of magnitude more hilarious

Bill Dubuque

@anon Even more so if did answered doit. Did did doit?

Peter Tamaroff

@BillDubuque LOL

David Wheeler

@hhh $c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)e^{-inx}dx$ (i think)

t.b.

did did it

anon

22:24

also here

hhh

@JonasTeuwen Yes, I think better show my understanding with example.

$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx} dx=\frac{1}{2\pi}\int_{-\pi}^{\pi} \left(e^{-ix/2}\right) e^{-inx} dx=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{-ix/2-inx} dx$

...then...

$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{-(x/2+nx)} d x$

(I wish I could find some easy way to proof-read things with Vim...)

Bill Dubuque

@tb Beat me by a few secs

Jonas Teuwen

@hhh Emacs with preview-latex + Vim mode.

You can just integrate that sucker... What the bloody monkey is going on?

hhh

$ emacs
The program 'emacs' can be found in the following packages:
* emacs23
* emacs23-nox
* e3
* emacs23-lucid
* jove

err which one?

Jonas Teuwen

Srsly?

hhh

22:26

(I think that is a good suggestion, helps visually)

Jonas Teuwen

It is called AucTeX. Check it out.

Peter Tamaroff

@tb That was me =)

t.b.

@PeterTamaroff before adopting your "sire"ly ways? :)

Peter Tamaroff

@tb Right.

t.b.

@BillDubuque yes, I'm trying to improve on my FGITW skills here in chat. I'm a bit rusty but it's coming back...

Why are people so keen on closing this question? Doesn't look all bad to me.

hhh

22:31

@JonasTeuwen no command: M-x preview-latex, did I do it right?

Peter Tamaroff

@MarianoSuárezAlvarez Sorry for being a little insistent, but have you read my mail? If you're busy it's fine, but just to know if I should be expecting any feedback.

hhh

Complex integration for $\int_{-\pi}^{\pi} e^{-ix(0.5+n)} dx$?

Some noise with WA [here](http://www.wolframalpha.com/input/?i=Integrate%28e%5E%7B-ix%280.5%2Bn%29%2Cx%2C-%5Cpi%2C%5Cpi%29), some q about this?

...wait I can change it with Euler formula, thinking...

t.b.

@hhh try this but you should just do it by hand.

anon

If you can integrate $e^x$, try integrating $e^{ax}$, then plug in $a=-i(1/2+n)$ and apply the fundamental thm of calc

t.b.

what anon said.

anon

22:43

What in the world is $\risingdotseq$?

"This becomes that"?

Image of, okay.

Peter Tamaroff

@anon I've seen some videos where the professor used $\sim$ to denote Laplace Transformation

$$e^{at} \sim \frac{1}{s-a}$$

Ie

I arcwise connected the same as pathway connected?

t.b.

@PeterTamaroff Yes, but the definitions are not the same.

The equivalence is highly nontrivial

Let me look

anon

According to MW, not necessarily the same.

Peter Tamaroff

I see...

t.b.

22:58

@PeterTamaroff For Hausdorff spaces the definitions are equivalent by the Hahn-Mazurkiewicz Theorem. See also here and Peano space

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$Mathematics$ Mathematics