TITLE: Proving $\ln \lambda = \int_0^\infty \frac{\mathrm dt}t e^{-\lambda t}$
QUESTION [7 upvotes]: I am currently reading this paper. On page 5, it writes:

For each positive eigenvalue $\lambda$ of the operator $D$ we may write an identity
  $$\ln \lambda = \int_0^\infty \frac{\mathrm dt}t e^{-\lambda t}.\tag{1.17}$$
  This identity is "correct"  to an infinite constant, which does not depend on $\lambda$ and, therefore, may be ignored in what follows. Now we use $\ln \det(D)=\mathrm{Tr}\ln(D)$ and extend $(1.17)$ to the

I can not prove this identity. I tried
$$\int_0^{\infty} \frac{\mathrm dt}{t}e^{-t\lambda}=\int_0^{\infty} \frac{\mathrm d(\lambda t)}{(\lambda t)}e^{-t\lambda}=\int_0^{\infty} x^{-1}e^{-x}\ \mathrm dx=\Gamma(0),$$
where fort the third equality, I used the definition of Gamma function.

REPLY [10 votes]: Another approach is to explicitly identify what the author called an infinite constant. This amounts to finding a proper (though not unique) regularization, and in this case we can do as follows:
$$ \int_{0}^{\infty} \left( \frac{\mathrm{e}^{-\lambda t}}{t} - \frac{\mathrm{e}^{-t}}{t} \right) \, \mathrm{d}t = \log \lambda. \tag{*}$$
Now this is a quintessential example of what is called Frullani integral. The differentiating-under-the-integral-sign trick works perfectly with this regularization that leads to the identity $\text{(*)}$.