- #1

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Does anybody know if there is a closed-form solution to this rather simple-looking definite integral?

##F(\lambda) = \int_0^{\infty} \dfrac{e^{-x}}{1 + \lambda x} dx##

If ##\lambda > 0##, it definitely converges. It has a limit of 1 as ##\lambda \rightarrow 0##. But it doesn't seem to be analytic in ##\lambda##, since if you try to do a power series in ##\lambda##, you get a nonconvergent sequence:

##F(\lambda) = \sum_{j=0}^\infty (-1)^j \lambda^j (j!)##