K-function

Remarks

Raabe's formula $\int_a^{a+1} \ln \Gamma(u)\, du = \frac12 \ln 2\pi + a \ln a - a$ for $a > 0$ gives $$K(x + 1) = x^x K(x) \qquad (x > 0).$$

Example: $\int_0^1 \ln \Gamma(t+1)\, dt = \int_0^1 (\ln t + \ln \Gamma(t))\, dt = -1 + \frac12 \ln 2\pi$. Hence $K(2) = (2\pi)^{-1/2} \exp(1 - 1 + \frac12 \ln 2\pi) = 1$. The recurrence then gives $K(3) = 2^2 = 4$, $K(4) = 4 \cdot 3^3 = 108$, and in general $K(n + 1) = \prod_{k=1}^n k^k$.

Non-example: the function $x \mapsto x^x$ does not satisfy the recurrence; for instance $3^3 = 27 \ne 2^2 \cdot 2^2 = 16$.

The formula above is stated for $x > 0$, where $t + 1 > 0$ on the path of integration. The same formula, with the principal branch of $\ln \Gamma$, defines $K$ as a holomorphic function on $\C \setminus (-\infty, 0]$.