Using these results, it is easy to show that Using (B.1) and (B.8), it can be shown that The Beta function is defined in terms of the Gamma function as Integral representations of the Beta function are The generalized hypergeometric function is defined as where the constants a$_{i}$ , b$_{i}$ are called indices and uniquely define each generalized hypergeo- metric function. From this definition it is clear that the ordering of the indices is irrelevant and that if any a$_{i}$ = b$_{j}$ (1 ≤ i ≤ p , 1 ≤ j ≤ q ), the generalized hypergeometric function $_{p}$F$_{q}$ reduces to p − $_{1}$F$_{q}$$_{−}$$_{1}$ . When p = q + 1, the generalized hypergeometric function has a branch cut on the interval z ∈ [1 , ∞ ). Generalized hypergeometric functions can also be 109 $_{p}$F$_{q}$ [ a$_{1}$ , a$_{2}$ , . . . a$_{p}$ ; b$_{1}$ , b$_{2}$ , . . . b$_{q}$ ; z ] = $_{p}$F$_{q}$    a$_{1}$ , a$_{2}$ , . . . a$_{p}$ b$_{1}$ , b$_{2}$ , . . . b$_{q}$ z$^{}$   = Γ ( b$_{1}$ ) Γ ( b$_{2}$ ) . . . Γ ( b$_{q}$ ) Γ ( a$_{1}$ ) Γ ( a$_{2}$ ) . . . Γ ( a$_{p}$ ) ∞ ∑ n =0 Γ ( a$_{1}$ + n ) Γ ( a$_{2}$ + n ) . . . Γ ( a$_{p}$ + n ) Γ ( b$_{1}$ + n ) Γ ( b$_{2}$ + n ) . . . Γ ( b$_{q}$ + n ) z n n ! , (B.12) B ( a , b ) = ∫ 1 0 dx x a − $^{1}$(1 − x ) b − 1 = ∫ ∞ 0 dx x a − 1 (1 + x ) a + b . (B.11) B ( a , b ) = Γ ( a ) Γ ( b ) Γ ( a + b ) . (B.10) Γ ( z ) = 1 z − γ$_{E}$ + 1 2 $^{[}$γ 2 E + ζ (2) $^{]}$z + O $^{(}$z 2 ) . (B.9) Γ (1 + z ) = 1 − γ$_{E}$z + 1 2 $^{[}$γ 2 E + ζ (2) $^{]}$z 2 + O $^{(}$z 3 ) . (B.8) B.2 Hypergeometric Functions