The question this time is I have two expressions, while they should be the same as I showed in the file. However, the numerical result shows they are pretty different. Integral expression gives result in the complex filed although the imaginary is negligible. Further, the real part of two expression are far away and only fits for \beta=1. I also tried to sum the series expression with more terms, say, to 10000 while the result does not change at all. Wondering which one is accurate?
Later, it turns out integral expression is really time efficiency. Please see my command in the last part of file.