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integrals. Specifically, loop integrals satisfy the relation 

<img src='content_image/327134.jpg'>

<img src='content_image/327133.jpg'>

which is proven in Ref. [39]. This identity is commonly referred to as integration by parts. The identity generalizes to multi-loop integrals, the restriction being that v$_{µ}$ must be one of the loop momenta. The technique was developed in Ref. [35] and used to calculate the renormalization group functions in scalar φ 4 theory to four-loop order. In order to illustrate the technique, we will use it to calculate the integral (2.36). Using the identity (2.37), we have 

<img src='content_image/327132.jpg'>

where the derivatives can be calculated using the result 

<img src='content_image/327131.jpg'>

Explicitly calculating the derivatives, we arrive at the following recurrence relation 

<img src='content_image/327130.jpg'>

For convenience we have defined the operators N + and N − that increase and decrease the index n$_{i}$ by one unit when acting on the integral (2.36). Essentially, equation (2.40) is a recurrence relation for integrals of the form (2.36). For instance, consider an integral of the 

