Dividing f(z) by z - i, we obtain the remainder i and dividing it by z + i, we get the remainder 1 + i, then, remainder upon the division of f(z) by z2 + 1 is

f(z)=g(z)(z−i)(z+i)+az+b; a, b∈CGiven, f(i) = i⇒ ai+b=i           (1)and f(−i)=1+i⇒ a(−i)+b=1+i      (2)From (l) and (2), we havea=i2,b=12+iHence, the required remainder is az+b=12iz+12+i