Nice proof. —Dan
On Nov 12, 2015, at 8:05 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
log(x^2 + 1) = log(x + i) + log(x - i), so it suffices to show that the (4n + 2)th derivatives of log(x + i) are all pure-imaginary when x = 1.
Let x = 1 + y, so we are interested in the Taylor series of:
log(y + (1 + i))
about the point y = 0. We want to show that the (4n + 2)th coefficients are all pure-imaginary.
But log(y + (1 + i)) is just the derivative of 1/(y + (1 + i)),
I think calculus is taught differently in the U.K. from in the U.S. (:-)>
so it suffices to show that the (4n + 1)th coefficients of the Taylor series of 1/(y + (1 + i)) about 0 are all pure-imaginary.
This follows from the claim that the coefficient in y^n of the series expansion of 1/(y + a) is [a real multiple of] a^-n, which is true by dimensional analysis.