The loop integral can be evaluated immediately using (2.19)  which gives Now  making the change of variables v = s   w = d 2 − n$_{1}$ − n$_{2}$ − s − t   the integral becomes The contour integral over the variable v can now be evaluated using Barnes’ Lemma (B.15). Doing so  the integral becomes The Gamma function in the denominator of (2.31) can be simplified using Eq. (B.4)  which gives Note that the remaining contour integral can be evaluated in terms of the generalized hyper- 61 B ( d ; n$_{1}$   m ; n$_{2}$   m ) = i (4 π ) 2 [ m 2 4 πµ 2 ] d $_{2}$− $^{2}$( − m $^{2}$) 2 − n$_{1}$ − n$_{2}$ Γ ( n$_{1}$ ) Γ ( n$_{2}$ ) 2 1 − n$_{1}$ − $^{n$_{2}$}$π 1 2 ∫ i ∞ − i ∞ dw 2 πi $^{(}$− q 4 m 2 ) w Γ ( − w ) Γ ( n$_{1}$ + w ) Γ $^{(}$n$_{1}$ + n$_{2}$ − d 2 + w $^{)}$Γ ( n$_{2}$ + w ) Γ ( 1 2 ( n$_{1}$ + n$_{2}$ ) + w $^{)}$Γ ( 1 2 ( n$_{1}$ + n$_{2}$ + 1) + w ) . (2.32) B ( d ; n$_{1}$   m ; n$_{2}$   m ) = i (4 π ) 2 [ m 2 4 πµ 2 ] d $_{2}$− $^{2}$( − m $^{2}$) 2 − n$_{1}$ − n$_{2}$ Γ ( n$_{1}$ ) Γ ( n$_{2}$ ) ∫ i ∞ − i ∞ dw 2 πi $^{(}$− q m 2 ) w Γ ( − w ) Γ ( n$_{1}$ + w ) Γ $^{(}$n$_{1}$ + n$_{2}$ − d 2 + w $^{)}$Γ ( n$_{2}$ + w ) Γ ( n$_{1}$ + n$_{2}$ + 2 w ) . (2.31) B ( d ; n$_{1}$   m ; n$_{2}$   m ) = i (4 π ) 2 [ m 2 4 πµ 2 ] d $_{2}$− 2 ( − m $^{2}$) 2 − n$_{1}$ − n$_{2}$ Γ ( n$_{1}$ ) Γ ( n$_{2}$ ) ∫ i ∞ − i ∞ dv 2 πi ∫ i ∞ − i ∞ dw 2 πi $^{(}$− q m 2 ) w Γ ( n$_{1}$ + w + v ) Γ$^{(}$ n$_{1}$ + n$_{2}$ − d 2 + w + v$^{)}$ Γ ( − v ) Γ$^{(}$ d 2 − n$_{1}$ − v$^{)}$ Γ ( − w ) Γ ( d 2 + w ) . (2.30) B ( d ; n$_{1}$   m ; n$_{2}$   m ) = i (4 π ) 2 [ − q 2 4 πµ 2 ] d $_{2}$− 2 ( q $^{2}$) 2 − n$_{1}$ − n$_{2}$ Γ ( n$_{1}$ ) Γ ( n$_{2}$ ) ∫ i ∞ − i ∞ ds 2 πi ∫ i ∞ − i ∞ dt 2 πi [ − m 2 q 2 ] s + t Γ ( − s ) Γ ( − t ) Γ ( d 2 − n$_{1}$ − s $^{)}$Γ ( d 2 − n$_{2}$ − t $^{)}$Γ $^{(}$n$_{1}$ + n$_{2}$ + s + t − d 2 ) Γ ( d − n$_{1}$ − n$_{2}$ − s − t ) . (2.29)