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Re: Problem with ExpIntegralEi vs. LogIntegral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg73092] Re: Problem with ExpIntegralEi vs. LogIntegral
*From*: "xadrezus" <xadrezus at yahoo.com>
*Date*: Thu, 1 Feb 2007 04:40:10 -0500 (EST)
*References*: <200701301204.HAA16379@smc.vnet.net><epp6lr$db2$1@smc.vnet.net>
Hi, Mr. Woll:
You're right, that's exactly the problem, and now I can clearly
see why both results are different.
Thank you very much for your kind and accurate help, and
Best regards.
On Jan 31, 5:41 am, Carl Woll <c... at wolfram.com> wrote:
> 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.- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
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