While studying the Casimir effect, I had no problem getting the result for the energy of a massive scalar field:
$$E(a,m)= -\frac{mc^2}{4} - \frac{\hbar c}{4 \pi a} \int_{2\mu}^{\infty} \frac{\sqrt{y^2-4{\mu}^2}}{e^y -1}\; dy, \tag1$$ where $\mu$ is the mass parameter.
Also, I obtained the expression for a massless field:
$$E(a,0)= - \frac{\hbar c}{4\pi a} \int_0^\infty \frac{y}{e^y -1}\; dy = -\frac{\pi \hbar c}{24a}.\tag2$$ However, I can't obtain the result for large masses ($\mu \gg1)$, which is
$$E(a,m) \approx -\frac{mc^2}{4} - \frac{\hbar c \sqrt{\mu}}{4\sqrt{\pi}a} e^{-2\mu}.\tag3$$
I tried a transformation of variables, so that I could write the second term on(1) as:
$$ \frac{\hbar c}{4\pi a} {\mu}^2 \int_1^\infty \frac{\sqrt{y^2 - 1}}{e^{\mu y} -1}\; dy $$
and I tried making $\mu \rightarrow \infty$, but I couldn't get anywhere.
Could someone please help me on this?
Thank you in advance.