Re: Problem with ExpIntegralEi vs. LogIntegral

• To: mathgroup at smc.vnet.net
• Subject: [mg73092] Re: Problem with ExpIntegralEi vs. LogIntegral
• 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:
> >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
> >   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
>
> >   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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