this topic is kind of confusing me a lot. I am working with a text book that defined the Complete Gamma Function as follows:
$$\ln\Gamma(x) = \ln\sqrt 2\pi-x+(x-\frac{1}{2})\ln x +\frac{B_1}{1\cdot2}\frac{1}{x}+\frac{B_2}{3\cdot4}\frac{1}{x^3}+\frac{B_3}{5\cdot6}\frac{1}{x^5}+\frac{B_4}{7\cdot8}\frac{1}{x^7}+\frac{B_1}{9\cdot10}\frac{1}{x^9}+... $$
with Bj as the Bernoullian numbers
$$ B_1 = \frac{1}{6}; B_2 = \frac{1}{30};B_3 = \frac{1}{42};B_4 = \frac{1}{30};B_5 = \frac{5}{66}$$
This truncates the Gamma function according to the textbook as
$$ \ln\Gamma(x) = \ln\sqrt 2\pi-x+(x-\frac{1}{2})\ln x +\frac{1}{x}(\frac{1}{12}-\frac{1}{x^2}(\frac{1}{360}-\frac{1}{x^2}(\frac{1}{1260}-\frac{1}{x^2}(\frac{1}{1680}-\frac{1}{x^2}\frac{1}{1188})))) $$
I found this definition here: http://research.dnv.com/hci/ocean/bk/c/a26/sa.htm
Now, how is this Gamma function related to the Gamma function which is usually defined as
$$\Gamma(z) = \int_0^\infty x^{z-1} e^{-x}\,{\rm d}x $$
Please dont be too harsh, I'm an engineer