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In QFT today I learnt some insane tricks in calculating impossible integrals....I figure it'd be a good idea to see if others have similar tricks so we can all learn from each other.

here goes:

to integrate (bounds are assumed to be from negative inf to inf, k^2 means the vector dot product)

edit: missing factors of 2pi

[tex]\int \frac{d^d k}{(2\pi)^d}\frac{1}{(k^2 + m^2)^n}[/tex]

substitute

[tex]\frac{1}{(k^2 + m^2)^n}=\frac{1}{\Gamma(n)}\int_0^\infty t^n e^{-t(k^2 + m^2)} dt[/tex]

and get a gaussian in the integral. At the end of the day,

[tex]\int \frac{d^d k}{(2\pi)^d}\frac{1}{(k^2 + m^2)^n}=\frac{\Gamma(n-d/2)(m^2)^{d/2-n}}{\Gamma(n)(4\pi)^{d/2}}[/tex]

[tex]\Gamma(n)=(n-1)![/tex]

Quite crazy eh? I would've never thought of it myself... what about your favorite crazy tricks?

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# Insane integral tricks

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