A coherent derivation of Stirling's formula and the Stirling series (Q1423728)

The paper provides a new derivation for Stirling's formula for \(n!\), and also for the asymptotic expansion of \(n!\). The main term is \(\sqrt{2\pi} (1/2+n)^{1/2+n}e^{-(1/2+n)}\), which gives slightly better approximation for \(n!\) than the usual \(\sqrt{2\pi} n^ne^{-n}\). (This form of Stirling's formula was found by Nörlund in 1924.) The new derivation is based on a forgotten summation formula of Lindelöf, which asserts that \[ \sum_{k=1}^n f(k) ={\pi\over 2} \int_R {F(n+{1\over 2} +it)- F({1\over 2} +it) \over \cosh^2 \pi t } \,dt, \] where \(F'(z)=f(z)\), \(f(z) \) is holomorphic on a domain containing the strip \({1\over 2} \leq \text{Re} z \leq n+{1\over 2}\), and \(F(z)\) does not go faster in this strip than the denominator in the integral. The derivation obtains the term \(\sqrt{2\pi}\) from an integration, like other terms, applying the displayed formula for \(f(z)=\log z\), and does not require the usual use of Wallis' formula.