Homework 10 collaboration area
From Jake Eppehimer:
I must be doing something wrong on problem 2 of Lesson 31. There doesn't seem to be a way to solve the integral if you use formula (1b) with the answer from problem 1. Any tips?
Also, I don't understand how to do number 5 on Lesson 29. Any help would be appreciated.
Response from Mickey Rhoades Mrhoade
I get 2/pi times a bunch of sinc function integrals which have to be evaluated with the Dirichlet Integral. I can use Table 11.10 relationship #10 and get the answer without chugging out all the integrals. ---
From Jake:
What page is this table on? I'm not seeing anything. Thanks.
From Andrew:
Table 1 in 11.10 on Page 534 is a table of Fourier Cosine Transforms.
From Eun Young:
From the lesson 30 lecture note, we know that $ \int_0^{\infty} \frac{2 \sin w }{\pi w} \cos wv \ \ dw = 1 $ if -1<v<1 and 0 otherwise. Consider only positive v. Then, $ \int_0^{\infty} \frac{2 \sin w }{\pi w} \cos wv \ \ dw = 1 $ if 0 <v<1 and 0 if v>1.
From this, we can compute $ \int_0^{\infty} \frac{2 \sin 2w }{\pi w} \cos wv \ \ dw . $
Let 2w = t. Then, $ \int_0^{\infty} \frac{2 \sin 2w }{\pi w} \cos wv \ \ dw = \int_0^{\infty} \frac{ 2 \sin t }{ \pi \frac t 2} cos (t \frac v 2 ) \frac{dt}{2} = \int_0^{\infty} \frac{2 \sin t}{ \pi t} cos(\frac v 2 t) \ \ dt = 1 \ \ \text{if} \ \ 0 < \frac v 2 < 1 \ \ \ \text{and } 0 \ \ \ \text{if} \ \ \frac v 2 >1 $.
Thus, $ \int_0^{\infty} \frac{2 \sin 2w }{\pi w} \cos wv \ \ dw =1 \ \ \text{if} \ \ 0 <v < 2 \ \ \ \text{and } 0 \ \ \ \text{if} \ v >2 $.
Use the second and the last equations. Then, you will get the answer.
From Farhan:
In response to Jake, regarding 11.6.5, I am thinking of a simple proof only with words, like a summation of only even functions will give an even function (I am not 100% sure if I can make this claim). But would like to hear from others if they are thinking of coming up with a mathematical rigorous proof.
From Craig:
In response to Jake and Farhan, for 11.6 #5 I described a proof with words and then gave examples, as the book asked. I'm not sure if this is exactly what was desired, but it gets the point across.
On a separate note, is anyone else getting a different answer than the book for 11.8 #5? When working it out, I'm getting an extra -2w/w^3 term. My math looks correct, but I still have a bit of doubt.
From Hzillmer I'm having trouble getting started on 11.5, I must have missed something critical in the notes and the book isn't much help. Can someone point me in the right direction. I wrote down an example but I don't understand how an integral with P(x) in it can be evaluated without knowing P(x).
From Craig:
Hzillmer, in Lecture 29, in the blue text near the middle of the page, Dr. Bell worked out the first few Pn(x) functions. Plug those Pn(x) functions into your integral to get your coefficients. Hope this helps.
From djkees:
Craig: I think you forgot to make x = 0 in that term, it should be 2x*cos(wx)/(w^2), taken when x = 0, = 0, which will kill that term for you and give you the answer in the back of the book.
