# beta(s) is Dirichlet beta function of integral from 0 to inf;
# beta(s,x) is incomplete Dirichlet beta function of integral from x to oo;
# beta(s,0,x) is incomplete Dirichlet beta function of integral from 0 to x;
# beta(s,a,x) is incomplete Dirichlet beta function of integral from a to x;


#beta(s_,a_,b_):=beta(s,a)-beta(s,b);
beta(s_,a_,a_) :=0;
beta(s_,0,b_):=beta(s)-beta(s,b);
beta(s_,0,inf) := beta(s);

beta(s_,0) :=beta(s);
beta(s_,oo) :=0;

beta(0):=1/2;
beta(1):=pi/4;