# Question:Integration involving dot products of 3-d vectors

## Question:Integration involving dot products of 3-d vectors

Maple

I am trying to get Maple to understand, and evaluate the following integral analytically AND numerically.
The original integrand to be evaluated is given by

(w + m2)(1/4) (p2 + mu)(1/4) / ( (w2 +a2)2 (p2 +a2) (q2 + b2 2)2 ((p - q + w)2 + b12) )

which is to be integrated simultaneously with respect to dw dp dq (where w, p, q are 3-d vectors).
Limits of integration are (in cartesian co-ordinates) from -Infinity to + Infinity

All variables contained in the integrand are Real and >0.
a, m, mu, b1, b2 are constants and >0.
dp = d3p, dq = d3q, dw = d3w
p2=p2, w2=w2, q2=q2

It is of course possible to help Maple along by noting that
(p - q + w)= p2 + q2 + w2 - (2 p q Cos(\theta_pq) ) + (2 p w Cos(\theta_pw) ) - (2 q w Cos(\theta_wq) ) + b12
where \theta_pq is the angle between vectors p and q, theta_pw is the angle between vectors p and w, etc.

However, using the expanded form of (p - q + w)2 would imply that the integral would now need to be written out in tems of
dwradial, dqradial, dpradial, d\theta_w, d\theta_q, d\theta_p, and d\phi_w, d\phi_q, d\phi_p.
Now, the integral in spherical co-ordinates would be given by the pseudocode below (note that now there are no longer any vector symbols, and the limits of integration have changed)

int ( Sin(\theta_w) d\theta_w Limits {0, pi}) int ( w2 dwradial Limits:{0, Infinity} )
int ( Sin(\theta_p) d\theta_p Limits {0, pi}) int ( p2 dpradial Limits:{0, Infinity} )
int ( Sin(\theta_q) d\theta_q Limits {0, pi}) int ( q2 dqradial Limits:{0, Infinity} )

where we have omitted the integration over \phi_w, \phi_p, \phi_q, assuming that there is no dependence on the \phi angles, resulting in all the \phi angles being integrated over, with the result being a constant multiplied to the rest of the remaining integrals. No additional information on the vectors is available, e.g. the angle between p and q, p and w, q and w, are not specified, the magnitude of the vectors is not specified.

Any ideas if Maple can understand and execute the integral with p, q, w are input as vectors in the integrand directly, as shown earlier?
If it's easier for Maple to evaluate the integral in spherical co-ordinates, any suggestions on setting up the input, i.e. set up that the variables and constants are real, setup the limits of the theta and radial components etc. would also be appreciated.

﻿