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I want to simplify the following integral:

$$ \int_0^{\infty} \frac{E^{a}}{1+E^b (\tau(E))^2B^2}\cdot (\tau(E))^c\cdot \frac{\partial f_0}{\partial E} \ \mathrm{d}E $$

where $f_0 = \frac{1}{1+e^{\beta(E-\mu)}}$. Here $a,b,c, \mu, \beta$ are fixed constants and $\tau(E)$ is a general function of $E$. I basically want to solve for as general form as possible before making any approximations. If not, I would like to solve it in the region $B\ll1$ (possibly expand it in terms of $B$) or $B\gg1$. Finally we can make the approximation $\tau(E)=E^r$ where $r$ is again a constant, when we cannot generalize even further to get a more closed for answer. I tried partial integration but I am not able to move forward much. I looked up Somerfeld expansion to help me with this integral but it didn't help much. Any help or direction is appreciated.

Felix Marin
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Indeterminate
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1 Answers1

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The Sommerfeld expansion treats f as a Heavyside function to first approximation ( for large $\beta$). Since you have a derivative, you should treat it as a Dirac delta to first approximation. So basically you are expanding the integrand near $E=\mu$, at least for large $\beta$.

As an example, consider the large $\beta$ limit, where we may replace $f_0(E)\rightarrow \Theta(\mu-E)$, thus ${df_0(E)\over dE}\sim -\delta(\mu-E)$ and the integral becomes

$\int_0^{\infty} \frac{E^{a}}{1+E^b (\tau(E))^2B^2}\cdot (\tau(E))^c\cdot \frac{\partial f_0}{\partial E} \ \mathrm{d}E \rightarrow -\frac{\mu^{a}}{1+\mu^b (\tau(\mu))^2B^2}\cdot (\tau(\mu))^c$

Higher order terms in $\beta$ are more complicated, but the expansion is still straightforward.

user619894
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  • I understand your solution but I first want to evaluate the integral in terms of $f_0$, let's say by partial integration. We know that Fermi-Dirac integrals are PolyLog functions, so maybe get that in that form and then take the limit of high $\beta$. – Indeterminate Aug 05 '20 at 02:36
  • I am not sure what you are getting at. Are you looking for a solution that is not an expansion in $\beta$? – user619894 Aug 05 '20 at 13:21
  • Yes, I am looking for an expansion in $B$ and not $\beta$. – Indeterminate Aug 06 '20 at 16:26
  • Can you solve the large B or small B case at all? – user619894 Aug 06 '20 at 16:31
  • How exactly should I do that? – Indeterminate Aug 06 '20 at 17:05
  • You probably should abandon hoping for a generic $\tau$. If $\tau$ is indeed a power of E, you end up with a power of E, times the Fermi Dirac, or its derivative. If that isn't tractable, don't expect much for finite B – user619894 Aug 06 '20 at 17:21