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Mathematica is able to calculate InverseFunction[LogIntegral] and InverseFunction[RiemannR] (and many other inverse functions). How are these calculated? (I can see how LogIntegral and RiemannR are calculated if I go to the details section of the docs, but can't find a ref. for inverse functions.)

Note: I have tried looking at Series[LogIntegral[n], {n, 2, 3}], and InverseSeries[Series[LogIntegral[n], {n, 2, 3}]], but can't see clearly how this helps.

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    $\begingroup$ I believe it uses numerical inversion in some cases. InverseFunction[LogIntegral][1] gives a Root object in v9 which suggests it uses the technology described here. $\endgroup$ – Szabolcs May 21 '14 at 14:17
  • $\begingroup$ Great - thank you :) $\endgroup$ – martin May 21 '14 at 14:24
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You could look at

Trace[InverseFunction[LogIntegral], TraceInternal -> True]
Trace[InverseFunction[RiemannR], TraceInternal -> True]

they both give huge outputs. Maybe try to filter out some patterns to limit the output. Especially part of evaluation having to do with messages would be nice to filter out.

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  • $\begingroup$ Hmmmm, having looked at these, I am none the wiser I am afraid!! $\endgroup$ – martin May 21 '14 at 14:21
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    $\begingroup$ @martin yes, I must agree it looks quite hopeless. $\endgroup$ – Jacob Akkerboom May 21 '14 at 14:45

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