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--- notes:cft [2020/11/24 16:08] – created cjj
+++ notes:cft [2020/11/25 19:43] (current) – [Fourier integral theorem and Dirac δ-function] cjj
@@ Line 1: / Line 1: @@
 ======Continuous Fourier transform======
+Consider the Fourier transform:
+\begin{eqnarray}
+D(x) & = & \frac{1}{4\pi^2} \int_{-\infty}^\infty d\kappa e^{-i\kappa x}\tilde{D}(\kappa) \\
+ & \approx & \frac{1}{4\pi^2} \int_{-K}^K d\kappa e^{-i\kappa x}\tilde{D}(\kappa) \\
+ & \approx & \frac{1}{4\pi^2} \sum_{k=0}^{N-1} \frac{2K}{N} \exp\left[-i\left(\frac{2K}{N}k-K\right) x\right]\tilde{D}_k \\
+D_m & \propto & e^{iKx}\sum_{k=0}^{N-1}\exp\left[-i\frac{2K}{N}k\left(\frac{2L}{N}m-L\right)\right]\tilde{D}_k \\
+ & \propto & \exp\frac{i2KLm}{N} \sum_{k=0}^{N-1}
+  \exp\frac{-i 4KLkm}{N^2}\exp\frac{i2KLk}{N} \tilde{D}_k
+\end{eqnarray}
+Let $2KL=\pi N$, we get
+\begin{equation}
+e^{-i\pi m}D_m\propto\sum_{k=0}^{N-1}\exp\left(-2\pi i\frac{km}{N}\right)e^{i\pi k}\tilde{D}_k
+\end{equation}
+=====Fourier integral theorem and Dirac δ-function=====
+{{  :notes:delta_dirichlet.svg?320|}}
+[[https://en.wikipedia.org/wiki/Dirichlet_integral|Dirichlet integral]]
+\[
+\int_{-\infty}^\infty dx \frac{\sin(xL)}{x} = \pi
+\]
+As $L$ goes to infinity, the integrand becomes more and more like a δ-function.
+  - [[http://www1.maths.leeds.ac.uk/~frank/math3383/AppA.pdf|Fourier Integrals & Dirac δ-function]] {{ :notes:appa.pdf|PDF}}

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