Fourier transform of wave packet

[schniefen]

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.

 

[DrClaude]

(##a(k)## is given, but I do not think it needs to be specified)

I disagree. What is a(k)?

 

[schniefen]

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.

 

[pasmith]

However, I can not remove the real part from the integral and I am unsure how to proceed.

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