Toronto Math Forum

My question for Quiz5 was to decompose a function into full Fourier Series [0, $\pi$]. I wonder is it equivalent as decompose into full Fourier Series on [$-\pi$, $\pi$]? I understand it is equivalent when the function is even, but I'm wondering what should I do when the function is odd.

In property 3 of Theorem 3 ($g(x) =𝑥𝑓(𝑥)⟹𝑔̂(𝑘)=i\hat{f}'(k)$), for $\hat{f}'(k)$, is the Fourier transform being computed before the derivative, or the other way around? Intuitively, Leibnitz rule says that the order shouldn't matter here, but I just want to get a confirmation. Thank you

For this problem we have the following equations: $u_{xx} + u_{yy} = 0 \space \space (-\infty < x < \infty , y > 0)$ and $u|_{y=0} = f(x)$.
After doing the Fourier Transformation, the equations become $\hat{u}_{yy} - k^2\hat{u} = 0$ and $\hat{u}(k,0) = \hat{f}(k)$.
In my understanding, we should get a general form as $\hat{u}(k,y) = A(k)e^{ky} + B(k)e^{-ky}$ and drop the first term if $k>0$ since as $y \rightarrow \infty$ the term $e^{ky} \rightarrow \infty$, and if $y<0$ we drop the second one.
On the answer provided by Prof. Kennedy it said $\hat{u}(k,y) = \hat{f}(k)e^{-k|y|}$, so I am a little confused since $\hat{u}(k,y) = \hat{f}(k)e^{-|k|y}$ makes more sense to me.

Hi Prof. Ivrii!

I have seen that the announcement on Quercus test 1 page said we would have four question in the test, for main sitting, 1 for first order, 2 for wave, 3&4 for heat(one whole, one half). But the practice past test that you posted is not the same coverage as the announcement, the practice contains:1 for first order, 2 for whole wave, 3 for half wave, and 4 for whole heat. I am so confused and I have checked with Prof. Kennedy, he said the test format he received is the same as the practice, and I make sure with him I am asking about the main sitting. Since I mentioned the difference coverage between practice and announcement, he took a look at something, I think that was internal announcement for instructor maybe, and he told me the format is the same as practice.

So I just want to make sure that tomorrow main sitting, we will have Q1 for first order, Q2 for whole wave, Q3 for half wave and Q4 for whole heat, right?

Hi Prof. Ivrii!

I wonder if we need to compute all the integral out for the heat and wave equation during the test?

Like for heat equation question, during the quiz, I found the final graded point is for simplifying the heat fomula, so I wonder if we can get full marks during test if we just simplify the fomula and leave the inside integral calculations? (I know I have had a quiz that the question cannot be integrated by it is an error function, maybe this is why we only need to simplify the fomula and leave it there)

The same question to the wave equation questions, do we need to do all detail calculations or just plug in (maybe simplify) the fomula and annotate the interval clear?

Thx!