# eta(a) is Dirichlet eta function, = sum((-1)^k/(k+1)^a,k,0,inf)= -sum((-1)^k/k^a,k,1,inf)= integrate(t^(a-1)/(e^t+1),t,0,inf);
# eta(a,b) is Hurwitz eta function, = sum((-1)^k/(k+b)^a,k,0,inf);
# eta(a,1,x) is incomplete Dirichlet eta function of integral from 0 to x = integrate(t^(a-1)/(e^t+1),t,0,x);


eta(a_,b_,0):= 0;
eta(a_,b_,inf):= eta(a,b);

eta(a_,0.5) := 2^a*beta(a);
eta(a_,1) := eta(a);
eta(a_,inf) := 0;

eta(0) := 1/2;
eta(1):=ln(2);
eta(2):= pi^2/12;
eta(inf) := 1;