The exponential integral function $x \mapsto E_1(x)$ is strictly decreasing on the positive real axis and, so, is globally real analytically invertible there. Where can I find information concerning a maximal domain in the complex plane over which $E_1$ possesses a holomorphic global inverse that continues the real one?
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2 Answers
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Just two remarks: a) it is simply related with the integral logarithm, $$\int_{-\infty}^x\frac{e^t}{t}dt=\int_0^{e^x}\frac{dt}{\log t},$$ which is a better known function. b) There is a whole book on it:
Niels Nielsen, Theorie des Integrallogarithmus und verwandter Transzendenten. 37.0454.01 Leipzig: B. G. Teubner. VI u. 104 S. (1906).
answered Feb 13, 2021 at 0:55
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Just a remark: for all your elementary needs regarding special functions, the one-stop shop is the Digital Library of Mathematical Functions.
There is an index, and you find what you are looking for here.
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$\begingroup$ Thank you for the reference. But while it confirms that $E_1$ itself continues onto a slit $\endgroup$ Commented Feb 12, 2021 at 21:52
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$\begingroup$ Thank you for the reference. But while it confirms that $E_1$ itself continues well-definedly onto a slit region, it seems to say nothing about continuing the inverse off of the image of the positive real axis. For instance, might $E_1$ actually be one-to-one on that slit domain? (Sorry about the botched comment just above.) $\endgroup$ Commented Feb 12, 2021 at 22:07