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Problem with ExpIntegralEi vs. LogIntegral

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  • Subject: [mg73036] Problem with ExpIntegralEi vs. LogIntegral
  • From: "xadrezus" <xadrezus at>
  • Date: Tue, 30 Jan 2007 07:04:20 -0500 (EST)

Hi, best regards:

    I'm using an old version of Mathematica (2.2) and have found the 
    inconsistence when computing the complex value of ExpIntegralEi 
    a complex argument, namely:

    If I compute LogIntegral[ 20^( 1/2+14.135 I )], Mathematica 

          N[ LogIntegral[ 20^( 1/2+14.135 I ) ] ]

                1.99917 - 3.9127 I

   But as LogIntegral[z] == ExpIntegralEi[Log[z]], when I computed the
   previous value using ExpIntegralEi on the Log of the argument 
   I expected to get the very same result. Instead, it returns:

            N[ ExpIntegralEi[ (1/2+14.135 I ) * ( Log[20] ) ] ]

                 -0.105387 + 3.1474 I

   which, disconcertingly, it's quite different ! I've searched 
   documentation as well as MathWorld and other Internet resources, 
   all of them give the same definitions for LogIntegral and 
   as well as series expansions, etc., which, when computed manually
   for that complex argument, result in the value given by 

   I've also tried to relate both values in some way, so as to be able 
   determine one from the other, but to no avail.

   My question is:  how is ExpIntegralEi evaluating the above 
   in order to get the  result -0.105387 + 3.1474 I instead of the
   expected  result 1.99917 - 3.9127 I ?

   I would need to get to know which series expansion or algorithm
   ExpIntegralEi's is using to reach that result (-0.105387 + 3.1474 
I )
   and, if possible, duplicate it manually. Or else, to know how both
   values are related so I can determine one from the other.

   Thanks in advance and best regards.

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