       • To: mathgroup at smc.vnet.net
• Subject: [mg73036] Problem with ExpIntegralEi vs. LogIntegral
• 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
I expected to get the very same result. Instead, it returns:

N[ ExpIntegralEi[ (1/2+14.135 I ) * ( Log ) ] ]

-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

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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