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The complete Fermi Dirac integral (I'm purposely leaving off the gamma prefactor)... $$F_j(x) =\int\limits_{0}^{\infty} \frac{t^j}{e^{t-x}+1} \: dt$$ is generally defined for j>-1. Is there a way to evaluate this integral for j<0, such as j=-1 or -2?

If the gamma prefactor is included, it seems like one can use the polylogarithm or derivative property (seen here...https://en.wikipedia.org/wiki/Complete_Fermi%E2%80%93Dirac_integral) to find analytical equations, but can anything be done if they aren't included?

I looked into going to complex space and using closure and residue theorem, but it seems to become awfully complicated due to the real axis pole at t=0 and infinite poles on the imaginary axis.

Johnny
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1 Answers1

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That integral is defined for $\Re(j) > -1$ actually.

If you already put $j = -1$ you can easily that the integral does not converge. The same for $j = -2$ and so on.

Also, the integral will diverge even if you set more swell numbers like $j = -\frac{5}{3}$ and so on. The condition

$$\Re(j) > -1$$

must hold.