In the article "Series evaluation of Tweedie exponential dispersion model densities" by Peter Dunn and Gordon Smyth it is stated that
$$-\log\Gamma(1+j)-\log\Gamma(-\alpha j)\approx j\lbrace(1-\alpha)+\alpha\log(-\alpha)-(1-\alpha)\log j\rbrace\\-\log(2\pi)-\frac{1}{2}\log(-\alpha)-\log j$$
By using Stirlings approximation which is: $$\log\Gamma(z)\approx (z-\frac{1}{2})\log z-z+\frac{1}{2}\log z+C$$ Where $C$ are higher order terms.
However, I just cant seem to use the Stirling approximation to get the same result.