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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.
>
>
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