Fourier transform of wave packet

schniefen
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Homework Statement
Consider ##h(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \Re\{a(k)e^{i(kx-\omega t)}\}\mathrm{d}k.## What is the Fourier transform ##\hat{h}(k,t)## evaluated at ##t=0##, i.e. ##\hat{h}(k,t=0)##? (##a(k)## is given, but I do not think it needs to be specified)
Relevant Equations
The Fourier transform (FT) ##\hat{f}(k,t)=\int_{-\infty}^{\infty} f(x,t)e^{-ikx}\mathrm{d}x## and the inverse Fourier transform ##f(x,t)=\frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(k,t)e^{ikx}\mathrm{d}k##.
I am unsure if ##h(x,t)## really is a wave packet, but it looks like one, hence the title. Anyway, so I'd like to determine ##\hat{h}(k,t=0)##. My attempt so far is recognizing that, without the real part in the integral, i.e.

##g(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} a(k)e^{i(kx-\omega t)} \mathrm{d}k,##
then ##a(k)## is just the Fourier transform of ##g(x,t=0)##. However, I can not remove the real part from the integral and I am unsure how to proceed.
 
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schniefen said:
(##a(k)## is given, but I do not think it needs to be specified)
I disagree. What is a(k)?
 
DrClaude said:
I disagree. What is a(k)?
##a(k)=A2\pi\delta(k)+2B/((k-C)^2+B^2)##, where ##A, B## and ##C## are real constants. ##\delta## is the Dirac delta.
 
schniefen said:
However, I can not remove the real part from the integral and I am unsure how to proceed.

For complex numbers z and w, <br /> \Re(zw) = \Re(z)\Re(w) - \Im(z)\Im(w). However for w = e^{i\theta} = \cos \theta + i\sin \theta for real \theta you may prefer to write <br /> \begin{split}<br /> \Re(e^{i\theta}) &amp;= \frac{e^{i\theta} + e^{-i\theta}}2,\\<br /> \Im(e^{i\theta}) &amp;= \frac{e^{i\theta} - e^{-i\theta}}{2i}.\end{split}
 
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