I havent done an integral in years:

e^(i*w0*t + i*w*t - t^2/2) dt

with respect to t from -inf to inf

My guess is that this requires some sort of integration by parts?

7/16/2007 2:23:30 AM

That integral's a bitch. Here is what you can do:

7/16/2007 9:58:14 AM

good first post

7/16/2007 10:18:38 AM

nice.

7/16/2007 2:31:36 PM

yeah; impressive.

are you studying in budapest? (noticed the image is on a budapest server)

7/16/2007 5:18:26 PM

i'd like to be able to type something and have it come out like that

7/16/2007 5:21:32 PM

Thx.
^^Just graduated in BP, will come to NCSU this fall.

7/16/2007 5:29:38 PM

Are you coming to the NCSU math department?

7/16/2007 6:24:47 PM

i'll ask my question again

what program did you use to make that pretty derivation?

7/16/2007 9:59:26 PM

Apparently the method we were supposed to use to solve this was

1. Given integral of e^(i*w0*t + i*w*t - t^2/2) dt
2. Apply "solve the fucking problem" property
3. There's no step 3!

Lame as hell


* Property was e^(i*wo*t) * f(t) = F(w + wo) where F(w) is the fourier transform of f(t). This property can blow me.


[Edited on July 16, 2007 at 10:32 PM. Reason : wrong country]

7/16/2007 10:14:42 PM

^^ I bet it's either Latex or a Latex based addon to something.

There is a program called "ribbit" which will make pictures out of latex code real quick then you can
shove them into word docs. It's very flexible, but the downside is if you send it to someone w/o ribbit then their computer just sees garbage where all your lovely mathematics were.

Of course there are ways to fix these problems, I'm just not very good at that sort of thing.

7/16/2007 11:00:38 PM

there are lots of latex to html compilers available online. a really basic one is:

http://www.artofproblemsolving.com/LaTeX/AoPS_L_TeXer.php

(undoubtedly not what the author used)

7/16/2007 11:16:28 PM

welcome to ncsu OranjeBoom, glad to have ya, from the looks of it

7/16/2007 11:30:49 PM

^ My pleasure.

^^^^ The "solve the fucking problem" property can be quite handy
sometimes But the other rule you mentioned might get nicer results, I
just haven't heard of it...

^^^^^ It's a screenshot from a PDF made with LaTeX.

^^^^^^ No.

7/17/2007 4:37:03 AM

Wow this dude is cool... knows how to use carats and stuff!

So OranjeBoom, are you Hungarian, or an American studying in Hungary? What did you graduate in, and what will you study in grad school at NCSU?

This might be the first instance of someone not affiliated with NCSU/US and living outside the US (and possibly a non-US citizen), joining TWW!!!

7/17/2007 9:48:39 AM

I wonder what did I do...

I am Hungarian and just graduated in computer science. Will
study the same at NC State, too.

Could I be the first non-US citizen on TWW?
That's impossible...
But I am still writing from Hungary

7/17/2007 10:44:42 AM

Quote :

"I wonder what did I do... "

1) You used these correctly: "^"
2) You are using the smileys correctly/coolly!
3) Your English is better than that of most Americans!

Quote :

"Could I be the first non-US citizen on TWW?"

Oh no, I never said that! I am not one either, but I studied at NCSU. And there are a couple more. And then there are some Americans on here who never studied at NCSU.

But you are probably the first to register who is a:

Non-US citizen AND
Outside the US AND
Has no connection with NCSU/US (yet). (and maybe never been to US?!)

Rock on!

7/17/2007 11:01:24 AM

Sorry for not getting the point at first. Well, the fact is that I am already registered
at NCSU and have been to the US for short periods as a little kid. But the other
two criteria do hold

7/17/2007 12:17:43 PM

This one's better.

Let A=w0+w, and note that


The integral may now be simplified:

7/17/2007 4:41:27 PM

^The modulation property for Fourier transforms: http://mathworld.wolfram.com/ModulationTheorem.html

7/17/2007 5:57:55 PM

I'm glad in ECE most professors don't usually expect us to solve a bunch of hardcore integrals and differential equations. sucks for math guys.

7/17/2007 6:59:55 PM

^you think math is that way either ? Sadly not, math has gone a different direction the last century or so. Anyway this thread rocks, will comment on the integrals eventually...

7/17/2007 9:48:04 PM

^ agreed. the math education that i received as an undergrad was pretty tame, all things considered. i didn't have to whip out "real math" until i encountered problems in my research that my coursework just didn't answer.

7/17/2007 11:05:06 PM

It is a slight spinoff on the gaussian function used in wave packets. I have seen it anywhere from electrodynamics to quantum to even environmental engineering for modeling the release of some type of pollutant in the air.

7/18/2007 9:54:43 AM

I don't have LaTeX here at work to make things pretty, so I'll just sketch it:

1) Factor so that you can do Euler's formula. Let A=w0+w. You get an integral of cos(A)*exp(-t^2/2) and an integral of sin(A)*exp(-t^2/2).
2) Apply properties of even and odd functions. The integral with the sine function is thus zero.
3) Integral tables for cos(A)*exp(-t^2/2)

And you get sqrt(2*pi) * exp (-A^2/2)

[Edited on July 18, 2007 at 3:35 PM. Reason : Forumla, not identity]

7/18/2007 3:32:54 PM

OranjeBoom add some pictures of your current school's campus and other pictures of hungary to your gallery please

7/18/2007 3:38:53 PM

Have a look!

7/18/2007 6:42:58 PM

^^^ Late to the party

---------------------

Ok, new problem:

FT{ (1 + d/dt) * 1/pi * 1/(1+t^2) }

or

integral( (1 + d/dt) * 1/pi * 1/(1+t^2) * e ^(i*w*t) dt ) from -inf to inf

7/18/2007 10:36:55 PM

integral( (1 + d/dt) * 1/pi * 1/(1+t^2) * e ^(i*w*t) dt ) from -inf to inf

If we can do the part coming from 1 the d/dt part follows easily. So I'll try to do the part from the 1 that is (ignoring the 1/pi for now)

integral( 1/(1+t^2) * e ^(i*w*t) dt ) from -inf to inf

So e ^(i*w*t) = cos(wt) + i*sin(wt), consider then

1.) integral( 1/(1+t^2) * cos(wt) dt ) from -inf to inf
2.) integral( 1/(1+t^2) * sin(wt) dt ) from -inf to inf

If you look in Churchill's complex variables book the answer for 1.) is pi*exp(-w) from exercise 2 of section 61 (I've got the 6th ed.). Now what about 2.) ?

I'm sleepy, I'll leave the rest for you guys to figure, seems like there should be a way to use
partial fractions in the complex case 1/(1+z^2) = (i/2)[1/(z+i) - 1/(z-i)] plus residue theory
to derive these...

7/19/2007 3:39:01 AM

The "hint" we got was that

FT{ e^-|t| } <-----> 2/(1+w^2)

Using the symmetry property of Fourier transforms:

Quote :

"We conclude that if f(x) has the Fourier transform F(w), then F(x) has the Fourier transform 2*pi*f(-w) http://math.ut.ee/~toomas_l/harmonic_analysis/Fourier/node31.html#symmetry-property"

Soooo, FT{ 2/(1+t^2) } = 2*pi*e^-|w| = integral( 2/(1+t^2) & e^(i*w*t) dt ) from -inf to inf. I guess that takes care of the part coming from 1.


[Edited on July 19, 2007 at 4:15 AM. Reason : asd]

7/19/2007 4:11:52 AM

^Interesting, that's certainly easier . You can prove that differentiation just amounts to multiplication in w after the Fourier transform correct ? Just like in Laplace transforms when you transform the derivative it brings a multiplication in "s". This follows from integration by parts.

night.

7/19/2007 4:19:21 AM

Yep, that's it. Sweet






[Edited on July 19, 2007 at 4:35 AM. Reason : Wolfram needs to use a different math rendering program]

7/19/2007 4:27:46 AM