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>