>>764


１．Let ａ be an arbitrary real number and ｂ a positive real number. Evaluate the integral
　　∫[0,∞） cos(ax)/cosh(bx) dx.
　　（Recall that cosh(x) = (1/2)(e^x + e^-x) is the hyperbolic cosine.)
　　{Wed., 2010/Jan/20 (Day 2)}


２．Let ｆ be a holomorphic function on a domain containing the closed disc {z : |z|≦3}, and suppose that
　　f(1) = f(i) = f(-1) = f(-i) = 0.
Show that
　　|f(0)| ≦ (1/80)･max{|f(z)| : |z|=3},
and find all such functions for which equality holds in this inequality.
　　{Thu., 2010/Jan/21 (Day 3)}