diff --git a/_posts/2019-02-15-ch2-notes.md b/_posts/2019-02-15-ch2-notes.md index 8578eb08e163ddd5c53676ea0de37f54f65f0520..df038b4b27e299a7d7d89258ae6f805018d875ed 100644 --- a/_posts/2019-02-15-ch2-notes.md +++ b/_posts/2019-02-15-ch2-notes.md @@ -114,11 +114,13 @@ real $$x$$. Let's see what the Fourier transform of the conjugate of a function looks like. $$ +\begin{align*} (\mathcal{F} \overline{f})(x) -= \int_\mathbb{R} \overline{f(x')} e^{-2 \pi i x' x} \mathrm{d} x' -= \int_\mathbb{R} \overline{f(x') e^{2 \pi i x' x}} \mathrm{d} x' -= \overline{\int_\mathbb{R} f(x') e^{2 \pi i x' x} \mathrm{d} x'} -= \overline{(\mathcal{F}^{-1} f)(x)} +&= \int_\mathbb{R} \overline{f(x')} e^{-2 \pi i x' x} \mathrm{d} x' \\ +&= \int_\mathbb{R} \overline{f(x') e^{2 \pi i x' x}} \mathrm{d} x' \\ +&= \overline{\int_\mathbb{R} f(x') e^{2 \pi i x' x} \mathrm{d} x'} \\ +&= \overline{(\mathcal{F}^{-1} f)(x)} +\end{align*} $$ This implies that $$ \mathcal{F}^{-1} f$$ is the complex conjugate of $$\mathcal{F} \overline{f}$$.