- #1

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## Main Question or Discussion Point

I'm reading some course notes for a physics class that contain the following step in a derivation:

[itex]\phi(\vec{x}) = \int \frac{d^3k}{(2\pi)^3}\frac{e^{i\vec{k}\cdot\vec{x}}}{\vec{x}^2 + m^2}[/itex]

Changing to polar coordinates, and writing [itex]\vec{k}\cdot\vec{x}=kr \cos\theta[/itex], we have:

[itex]\phi(\vec{x}) = \frac{1}{(2\pi)^2} \int_0^\infty dk \frac{k^2}{k^2 + m^2}\frac{2\sin kr}{kr}[/itex]

I'm having a bit of difficulty seeing this step. Could someone please show some of the intermediate steps between these two and explain what's happening? I understand that k

[itex]\phi(\vec{x}) = \int \frac{d^3k}{(2\pi)^3}\frac{e^{i\vec{k}\cdot\vec{x}}}{\vec{x}^2 + m^2}[/itex]

Changing to polar coordinates, and writing [itex]\vec{k}\cdot\vec{x}=kr \cos\theta[/itex], we have:

[itex]\phi(\vec{x}) = \frac{1}{(2\pi)^2} \int_0^\infty dk \frac{k^2}{k^2 + m^2}\frac{2\sin kr}{kr}[/itex]

I'm having a bit of difficulty seeing this step. Could someone please show some of the intermediate steps between these two and explain what's happening? I understand that k

^{2}in the numerator comes from converting to spherical coordinates, but that's about all I follow.