thus an+1 = (2n + 1)an/2(n + 1). We know that a0 = π, and the remaining part follows by induction. Thus ∞ ∞ ∞ 2 1 tanh2n v 1 2n π 2 g(x, y) dx dy = dv = . (5) 2 cosh v 2 n 22n −∞ −∞ −∞
Equations (4) and (5) give the desired result.
Remarks. a) It is easy to see that ∞ ∞ /( m − ) ζ( ) = ··· 1 2 1 ··· , m 2 2 2 2 dx1dx2 dxm cosh x1 ···cosh xm − sinh x1 ···sinh xm −∞ −∞
for m = 2, 3, 4,.... b) If n = 0 then Example 2 reduces to Example 1. c) For n = 1Example2reducesto