Proper integral from 0 to 2*pi of:

cos(sint)*e^(cost)

Any clues?

BTW, this is in a complex integration chapter which has things like Cauchy's Theorem, Cauchy's Integral Formula, Deformation Theorem, etc, so I don't see why this question is in that chapter.

There is a hint given below the question, however, I don't see how it relates to the given question. Here is the hint:

Consider Integral [e^z over z] dz around the unit circle centered at the origin. First evaluate this integral using Cauchy's Integral Formula, and then again directly using the coordinate functions of the unit circle.

?

Thanks.

7/3/2007 5:28:28 PM

is this for MA513 or 501 or what?

anyways, maple says its 2pi, but i doubt that helps you.

[Edited on July 3, 2007 at 5:36 PM. Reason : t]

7/3/2007 5:34:37 PM

502

yeah i know the answer is 2pi from the back of the book

7/3/2007 5:39:42 PM

What about just

Integral (e^cost)

or

Integral (e^sint)

?

I have looked in several tables, but not there.

7/3/2007 7:41:04 PM

Nice question, I hope I'm not too late.

1.) You can calculate from Cauchy's integral formula that Int_c(exp(z)/z)dz, where c is the unit circle in the complex plane, comes out to 2*pi*i. You just look at the Laurent expansion and pick off the coefficient of 1/z term in the integrand, that coefficient is one then the Cauchy formula supplies the 2 the pi and the i.

2.) explicitly calculate the integral. To do this we need a parametrization of c. The easy and obvious
choice is z=exp(ib) [ think b=theta but I don't want to write theta a billion times ], note
dz = iexp(ib)db and thus,

Int_c(exp(z)/z)dz = Int_0^2pi[ exp(exp(ib)) / exp(ib) ] iexp(ib)db
= Int_0^2pi[ exp(exp(ib)) ] idb
= Int_0^2pi[ exp(cos(b) + isin(b)) ] idb
= Int_0^2pi[ exp(cos(b))*exp( isin(b)) ] idb
= Int_0^2pi[ exp(cos(b))*(cos(sin(b)) + isin(sin(b)) ] idb
= Int_0^2pi[ -exp(cos(b))*sin(sin(b)) + i*exp(cos(b))*(cos(sin(b))] db
= 0 + 2*pi*i

In the last step I used part 1.). Now we can equate the real and imaginary parts of both sides to obtain two very non-obvious results,

Int_0^2pi[ -exp(cos(b))*sin(sin(b)) ] db = 0
Int_0^2pi[ exp(cos(b))*(cos(sin(b))] db = 2*pi

As usual the "i" disappears after its work is done.

How can people say there is no God after seeing stuff like this? Complex variables are neat.

7/3/2007 11:20:24 PM

Quote :

"Complex variables are neat. beautiful"

7/3/2007 11:54:00 PM

^thanks. That's better.

7/4/2007 1:05:39 AM

OMG I KEP ON MAKING THE STUPIDEST SILLY MISTAKE OVER AND OVER AGAIN IN EXPLICITLY COMPUTING THE INTEGRAL



Getting 2*pi*i from Cauchy's Integral Formula was easy, and while calculating explicitly, I kept making the same silly mistake again and again towards the end, which prevented me from reaching the last step.

DUH! It is so easy!

THANKS for that... I was ready to pull my hair out over this, esp since after doing this course, I was able to learn topics and do problems that at the beginning of the course I thought I wouldn't be able to do, and in fact, I have become quite good at it.

Amazing really, complex calculus, that is. And the way how some hard real integrals can be done using analogous complex integrals, pure beauty!

Quote :

"How can people say there is no God after seeing stuff like this?"

EXACTLY.

Quote :

"Complex variables are neat. beautiful"

Why do you think I have the username that I do!




P.S. Here is the mistake I kept on making:

e^(isint) = cost + isint

instead of cos(sint) + isin(sint)

PATTERNS, MY BOY, LEARN TO RECOGNIZE PATTERNS!


[Edited on July 4, 2007 at 7:13 AM. Reason : ]

7/4/2007 6:52:02 AM

ah ha, you were very close to the answer.

7/4/2007 4:25:56 PM

MA513 is on my list of classes to take at some point.

oohhh yeah.

7/4/2007 5:08:20 PM

I took 513 the first summer session and it didn't cover anything I hadn't already learned in high school. I was very disappointed. I was expecting a graduate level course. Anyone who has a basic understanding of calculus could take it and do fine.

7/4/2007 11:45:40 PM

^yep. Its basically no more difficult than say ma 341. You could call it calculus VI

ma 141 = calc I
ma 241 = calc II
ma 242 = calc III
ma 341 = calc IV
ma 401 = calc V
ma 513 = calc VI
...level up for courses below !
ma 425 = calc VII
ma 426 = calc VIII
ma 555 = calc IX

etc...

of course this ordering is by no means unique.
I'm to about calc. XX give or take.

7/5/2007 2:11:19 AM

I think you've got that wrong...

MA 511 (Adv. Calc. I) - Calc IV
MA 512 (Adv. Calc. II)- Calc V

MA's 341, 401, 425, 513, et al. are all parallel to the calculus courses. IMHO the material in 513, along with that in 141 and 241 should be covered in high school. (and, by the way, NONE of the material in 242, 341, or 401 is necessary for 513). Basically all that was covered in 513 was:

+ What is a complex number?
+ Algebraic expressions with complex numbers
+ Functions of complex numbers
+ Limits, derivatives, and analytic functions
+ Contours and contour integrals
+ Cauchy's theorem
+ Residues, poles, and zeroes (definitions but NOTHING about how useful they can be)
+ Taylor and Laurent series expansions

I learned all of this and much, much, much, much more in a 300-level course at UNCW I took while in high school.

7/5/2007 2:53:30 AM

Quote :

"How can people say there is no God after seeing stuff like this?"

Because some of us know that affirming the consequent is a logical fallacy.

7/5/2007 3:38:09 AM

^^ frankly I have even higher expectations for my kids in highschool, but I hear you. The course could demand a lot more, it should really, it is too easy for a 500-level course. I do think that ma 425 is about ten times as hard as ma 141/241, I mean sure the material is the same but there are no proofs in ma 141/241, like I said the ordering is by no means unique.

If the physics department had a proper math methods in physics course that would be good, that course when done right has problems that could stump the majority of grad students in math. Truth is math majors by in large avoid really hard problems, we leave those to the applied folks. In fact, until most grad students in math teach a course in calculus you'd be surprised how little we know. The focus of math is not really calculus anymore, its much broader than that.

Of course since I teach calculus I want to know everything and then some because I feel like that
is my duty as the teacher.

[Edited on July 5, 2007 at 3:47 AM. Reason : ^ comment injected at unfortunate time.]

7/5/2007 3:46:18 AM

HAY GUYZ

I just looked at course descriptions here: http://www2.acs.ncsu.edu/reg_records/crs_cat/MA.html

And I can tell you honestly after taking MA 502 that:

MA 502

=

MA 242 + MA 405 + MA 513

ALL IN ONE SEMESTER!!!

We seem to have covered 90% of the stuff in those 3 courses, all in one course!

As LimpyNuts said, the stuff in MA 513 is too basic and simple, and it should be covered in parallel with 141, 241, etc. It is the first course in complex calculus, so I don't know why it is a graduate course. It should be called MA 313 or 413, and then the real advanced courses in complex calculus:

MA 711
MA 712
MA 713

Should be numbered in the 500's (at least 711 and 712), as are the advanced courses in real calculus.

Quote :

"In fact, until most grad students in math teach a course in calculus you'd be surprised how little we know."

I can fully agree with that. Last year I taught a course at college entitled "College Algebra", and I learnt more from teaching it than I did while taking/doing such stuff in HS/college myself. It is when you start teaching you start realizing things and every 2 days of the course you hear yourself saying "oh so that's why it is like this..." That's when you get the insight which you didn't get while learning it yourself. You get the insight because when you are teaching it to others, you look at it deeply, not on purpose, but only because you want to teach it nicely, and then suddenly while you are looking at the stuff, you are hit with the insights.


NICE COMMENTZ GUYZ... KEEP IT GOING!

7/5/2007 7:54:49 AM

Learning a given concept = See one, Do one, Teach one

7/5/2007 8:38:54 AM

alas, as a math major who took 401, that means i can't take 501, and by extension, 502.

7/5/2007 9:25:45 AM

^^ that's great!

^ nope, wrong! click on the link above. it says you can't get credit for 401 and 501 both. 502 is completely different from 501 topic-wise, and you don't have to know one to do the other. in fact, 501 is magnitudes of order MORE difficult than 502. and even though it says 341 (diff eq 1) is a prereq for 502, you don't have to know diff eq at all do 502. 501 and 502 are grouped together only because they are both "Advanced Mathematics for Engineers and Scientists" (I and II), but as far as content goes, the difference is like heaven and earth.

as i said, 502 has 3 topics, and EACH topic covers as much as you would cover in a whole course:

matrices and determinants
vector calculus
complex variables

p.s. bad news for you it says this for BOTH 501 and 502:

Not for credit by mathematics majors

but so what? you can still take it, it just won't count towards your major. but i am sure you can make it count towards the degree requirements.


[Edited on July 5, 2007 at 9:49 AM. Reason : ]

7/5/2007 9:48:17 AM

ahahahahahahahahahaha

0EPII1 pisses the bed

7/6/2007 11:28:51 AM

^why is this clown not suspended yet?

7/7/2007 10:13:37 AM

lol

7/7/2007 7:09:02 PM

Quote :

"as i said, 502 has 3 topics, and EACH topic covers as much as you would cover in a whole course"

I know the description made it seem like that, but if it were true, I'd be shocked at how little they actually teach in an entire semester of 405 or 513. I thought 502 was ridiculously easy and cake walked through the course, and I wasn't really fulfilled with the amount taught in any of the 3 subject areas. To be fair, the professor was Charlton, and that may or may not explain it.

7/7/2007 9:16:56 PM

i had charlton too... and he is

7/7/2007 9:23:47 PM

well, probably the course is not harder because if they made it harder it would kill it. In my experience
making a course harder or more advanced than advertised is not appreciated by the majority of students.
It is a sad but relevant fact to this discussion. I doubt it is any particular professor's fault.

I still try to talk about stuff beyond the course I'm teaching but if anything I only get complaints about it. I hope
there are a few students who go on and profit from my comment here or there about linear algebra or analysis, but those future students don't write my evaluations.

7/7/2007 9:57:59 PM