I would like to integrate ∫ d 3 k → ∫ d 3 q → 1 q 2 under the condition that | k → | ≤ A and | k → + q → | ≤ Awhere A is some constant.Is this executable?

Question

I would like to integrate  ∫      d    3              k      →        ∫      d    3              q      →            1          q      2      under the condition that      |              k      →            |    ≤  A           and               |              k      →        +            q      →            |    ≤  Awhere A is some constant.Is this executable?

plomet6a · Accepted Answer

This integral is in fact analytically tractable.Denote       |        k        |    =  k for any vector and first consider a change of variables       q    +    k    =  R      p  ,       R    T    R  =  1 such that       R    T        k    =  k                    z        ^             and then regular spherical coordinates. Then we obtain      ∫                  |                    q            +              k                    |            ≤      A            d    3    q      1          q      2        =      ∫          p      ≤      A            d    3    p      1          (              p            −      k                                    z            ^                                      )        2              =  2  π      ∫    0    A        q    2    d  q      ∫          −      1        1    d  (  cos  ⁡  θ  )      1                  q        2            +              k        2            −      2      q      k      cos      ⁡      θ          =            2      π        k        ∫    0    A    q  d  q  ln  ⁡            |                  q      +      k              q      −      k                  |        =            π              A        2              k              (        ln  ⁡            A      +      k              A      −      k        +  1  +            k      2              A      2        ln  ⁡                    A        2            −              k        2                    k      2                  )      This can be integrated once again without problem in spherical coordinates,since it depends only on the magnitude of the vector k:      ∫          k      ≤      A            d    3    k            π              A        2              k              (        ln  ⁡            A      +      k              A      −      k        +  1  +            k      2              A      2        ln  ⁡                    A        2            −              k        2                    k      2                  )          =  4      π    2        A    4        ∫          0        1    x  d  x            (        ln  ⁡            1      +      x              1      −      x        +  1  +      x    2    ln  ⁡            1      −              x        2                    x      2                  )        =  4      π    2        A    4    (      5    6    −      2    3    ln  ⁡  2  )

I would like to integrate int d^3 vec(k) int d^3 vec(q) (1)/(q^2) under the condition that |vec(k) |<=A and |vec(k) +vec(q)| <=A where A is some constant. Is this executable?

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