Re: Problem with ExpIntegralEi vs. LogIntegral

*To*: mathgroup at smc.vnet.net*Subject*: [mg73049] Re: [mg73036] Problem with ExpIntegralEi vs. LogIntegral*From*: Carl Woll <carlw at wolfram.com>*Date*: Wed, 31 Jan 2007 00:08:42 -0500 (EST)*References*: <200701301204.HAA16379@smc.vnet.net>

xadrezus wrote: >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 > > The problem here is that Log[20^(1/2+14.135 I)] is not equal to (1/2+14.135 I) Log[20] Remember that the inverse of Exp is a multivalued function, and Log takes the principal value. For Log[20^(1/2+14.135 I)] 1.49787-1.63762 I the principal value is a real number plus an imaginary part that is constrained to lie between (-Pi, Pi). On the other hand, for Log[20] the principal value is just a real number, with no imaginary part: (1/2+14.135 I) Log[20] 1.49787+42.3447 I The difference in the two value is a multiple of 2 Pi I: (Log[20^(1/2+14.135 I)]-(1/2+14.135 I)Log[20])/(2Pi I) -7.+0. I Carl Woll Wolfram Research > 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. > >

**References**:**Problem with ExpIntegralEi vs. LogIntegral***From:*"xadrezus" <xadrezus@yahoo.com>