Evaluating Integrals of Exponential Functions with Dirac Delta

[Dixanadu]

Hey guys,

if I have an integral of the form $\int d^{3}x \hspace{2mm} e^{i(k\cdot x)}$, how do I evaluate this?

Thanks a bunch...

 

[NascentOxygen]

Is that d a constant, or the differential operator, or what?

 

[Dixanadu]

its the integration of measure, over 3 spatial dimensions

 

[Vagn]

its the integration of measure, over 3 spatial dimensions

How can you rewrite the exponential? Maybe try using Euler's formula if you aren't confident with the exponential.

 

[Meir Achuz]

Write ${\bf k\cdot x}=kx\cos\theta$. Then do the angular integration.

 

[jtbell]

Or in Cartesian coordinates, write out the dot product in terms of components: $\vec k \cdot \vec x = k_x x + k_y y + k_z z$.

 

[Meir Achuz]

Hey guys,

if I have an integral of the form $\int d^{3}x \hspace{2mm} e^{i(k\cdot x)}$, how do I evaluate this?
Thanks a bunch...

$\int d^{3}x \hspace{2mm} e^{i(k\cdot x)}=(2\pi)^3\delta({\bf r})$, the Dirac delta function.

 

[Dixanadu]

Thank you Meir Achuz - that's what I was looking for :D thank you! Thanks everyone else for your help, I guess I should've specified that I was looking for it in terms of the Dirac delta.