content_image/322935.jpg content_image/322934.jpg
This integral can also be evaluated using Mellin-Barnes techniques. The result is 

<img src='content_image/322935.jpg'>

Note that this result agrees with results given for this integral in Ref. [29]. This is the most complicated integral that we will evaluate in this chapter. In the next section we will consider a technique that can be used to construct recurrence relations among loop integrals. Using this approach, all integrals encountered in Chapters 3, 4, 5 and 6 can be evaluated in terms of the integrals given so far in this chapter. 

## 2.4 Integration By Parts 

Consider the two-loop integral 

<img src='content_image/322934.jpg'>

This integral is represented by the Feynman diagram in Fig. 2.6. Note that this is the most complex loop integral that can occur when calculating two-point functions at two-loop level, because at most five independent propagators can be constructed from the two loop momenta k$_{1}$ , k$_{2}$ , and the external momentum q . For this reason, the integral (2.36) is occasionally referred to as the “master” two-loop integral for two-point functions [29]. 

This integral cannot be evaluated using the methods discussed so far. However, it can be evaluated by appealing to one of the properties of dimensionally regularized momentum 

