Toronto Math Forum
APM3462018S => APM346Tests => Quiz5 => Topic started by: Jilong Bi on March 08, 2018, 01:23:53 PM

http://www.math.toronto.edu/courses/apm346h1/20181/PDEtextbook/Chapter5/S5.2.P.html problem4 1.
given $$f(x) =e^{\frac{ax^2}{2}}$$
$$\implies\widehat {f}(k) = (\frac{1}{\sqrt{2{\pi}a}})e^{\frac{k^2}{2a}}$$
By theorem $$g(x) = f(x)e^{i{\beta}x} \implies \widehat {g}(k) = \widehat {f}(k{\beta})$$
$$cos{\beta}x =\frac{ e^{i{\beta}x}+ e^{i{\beta}x}}{2}$$
$$\implies \widehat {g}(k) = \frac{1}{2}\widehat {f}(k{\beta})+\frac{1}{2}\widehat {f}(k+{\beta})$$
$$\implies \widehat {g}(k) = \frac{1}{2\sqrt{2{\pi}a}}[e^{\frac{(k{\beta})^2}{2a}}+e^{\frac{(k+{\beta})^2}{2a}}]$$
same reson for $$sin{\beta}x =\frac{ e^{i{\beta}x} e^{i{\beta}x}}{2i}$$
$$\implies \widehat {g}(k) = \frac{1}{2i\sqrt{2{\pi}a}}[e^{\frac{(k{\beta})^2}{2a}}e^{\frac{(k+{\beta})^2}{2a}}]$$

I will learn how to code as soon as possible. :)

I just want to remind everyone who sees this so that he won't make the same mistake as I did:
$$F(\Re f)\neq\Re F(f)\text{ and } F(\Im f )\neq\Im F(f)$$

Good job (Yuxin, Jilong). Jilong: please note that it is T0101, not T0201 (messing up tutorial sections not a big deal for us, but was a real monkey wrench thrown into in TT1 for MAT244)
$\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}$
Please escape sin, cos by \ : \sin, \cos (and so on) it will make them upright
Jingxuan is correct. Also if $f$ is real, then $\hat{f}$ is real iff $f$ is even, $\hat{f}$ is purely imaginary iff $f$ is odd.

Considering the property that if $ h(x) $ is of the form $ f(x) \cdot g(x) $ that the Fourier transformation is the convolution, $ \hat h(x) = \hat f(x) \ast \hat g(x) $ that gives us the same answer? So these properties of paired functions that we're memorizing are all special cases of this general fact about convolutions?

Andrew, indeed it would give us the same answer, if we were able to calculate F.T. of $\cos(\beta x)$. The trouble is that at this moment we cannot do it since integrals do not converge.
Later we will be able to do this in the sense of distributions $1\mapsto \delta(k)$, $e^{i\beta x}\mapsto \delta(k\beta)$, $\cos(\beta x)\mapsto \frac{1}{2}(\delta(k\beta)+\delta(k\beta))$ etc and then indeed it would work.
PS
I noticed that Yuxin goes this way, but I cannot accept his arguments until he explains what does it mean. Probably he learned it in the Physics class, but these guys are sometimes reckless