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Re: Power Series for LogIntegral

• To: mathgroup at smc.vnet.net
• Subject: [mg45212] Re: [mg45204] Power Series for LogIntegral
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 21 Dec 2003 03:42:08 -0500 (EST)
• References: <200312201056.FAA09567@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On 20 Dec 2003, at 19:56, Bobby R. Treat wrote:

> Why doesn't Series return a power series?
>
> Series[LogIntegral[x],
>   {x, 0, 1}]
>
> SeriesData[x, 0,
>   {6/Log[x]^4 + 2/Log[x]^3 +
>     Log[x]^(-2) + Log[x]^
>      (-1)}, 1, 2, 1]
>
> Bobby
>
>

LogIntegral has a brach discontinuity running from -Infinity to 1, so 
to get a power series you need to expend at a point outside this 
segment:


Normal[Series[LogIntegral[x], {x, 3/2, 2}]]


((-1 + Log[3/2])/(3*Log[3/2]^2) - 2/(3*Log[9/4]))*
    (x - 3/2)^2 + (2*(x - 3/2))/Log[9/4] +
   ExpIntegralEi[Log[3/2]]


Normal[Series[LogIntegral[x], {x, I, 2}]]


(1/Pi - (2*I + Pi)/Pi^2)*(x - I)^2 - (2*I*(x - I))/Pi +
   ExpIntegralEi[(I*Pi)/2]


Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/

• References:
  • Power Series for LogIntegral
    • From: drbob@bigfoot.com (Bobby R. Treat)