Toronto Math Forum
APM346-2016F => APM346--Lectures => Chapter 5 => Topic started by: Roro Sihui Yap on November 02, 2016, 01:59:12 PM
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1. I think it should be $\hat{h}(k)$ instead of $\hat{h}(x)$ in the proof of theorem 4a
\begin{equation*}
\hat{h}(x)=\frac{\kappa}{2\pi} \int e^{-ix k }h(x)\,dx =
\frac{\kappa}{2\pi} \iint e^{-ix k }f(x-y)g(y)\,dxdy;
\end{equation*}
2. In Example 2, Since $f(x)=e^{-\frac{\alpha}{2}x^2}$, it should be $f'=- \alpha x f$ and not $f'=\alpha x f$
3. In Example 2, last paragraph, If $\alpha=\pm i\beta$ then it should be $f=e^{\mp\frac{i\beta}{2 }x^2}$ and not $f=e^{\mp\frac{i}{2\beta }x^2}$
4. In Example 2, for the last equation, there shouldn't be an "x" variable in the exponent
\begin{equation*}
\hat{f}( k )=\frac{\kappa}{2\sqrt{\pi\beta}}
(1\mp i)e^{\pm\frac{i}{2\beta} k ^2}.
\end{equation*}
Instead of
\begin{equation*}
\hat{f}( k )=\frac{\kappa}{2\sqrt{\pi\beta}}
(1\mp i)e^{\pm\frac{i}{2\beta} k ^2x}.
\end{equation*}
5. At the start of theorem 5, I think it should be $(-\infty,\infty)$ instead of $(\infty,\infty)$
http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter5/S5.2.html
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Thanks. Done