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Problem with ExpIntegralEi vs. LogIntegral

• To: mathgroup at smc.vnet.net
• Subject: [mg73036] Problem with ExpIntegralEi vs. LogIntegral
• From: "xadrezus" <xadrezus at yahoo.com>
• 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 
following
    inconsistence when computing the complex value of ExpIntegralEi 
for
    a complex argument, namely:

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

          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 
instead,
   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 
Mathamatica's
   documentation as well as MathWorld and other Internet resources, 
and
   all of them give the same definitions for LogIntegral and 
ExpIntegralEi,
   as well as series expansions, etc., which, when computed manually
   for that complex argument, result in the value given by 
LogIntegral.

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

   My question is:  how is ExpIntegralEi evaluating the above 
expression
   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.

• Follow-Ups:
  • Re: Problem with ExpIntegralEi vs. LogIntegral
    • From: Carl Woll <carlw@wolfram.com>