       Re: complex output for real integral

• To: mathgroup at smc.vnet.net
• Subject: [mg115860] Re: complex output for real integral
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Sun, 23 Jan 2011 05:35:14 -0500 (EST)

```
----- Original Message -----
> From: "Ted Sariyski" <tsariysk at craft-tech.com>
> To: mathgroup at smc.vnet.net
> Sent: Saturday, January 22, 2011 2:23:34 AM
> Subject: [mg115843] complex output for real integral
> Hi,
>
> I get complex answer for an integral from Exp[1/x^3]/x^3 over the real
> axes:
>
> In[]:=f[n_] = Assuming[
> {Element[x, Reals], Element[n, Integers], n >= 1},
> Integrate[E^(1/x^3)/x^3, {x, n, Infinity}]
> ]
>
> Out[]:= 1/6 (-1)^(1/3) (-3 Gamma[5/3] + 2 Gamma[2/3, -(1/n^3)])
>
> The imaginary part of f[n] is everywhere ~10^-16 and I could ignore
> it but I guess there is a better approach.
>
> I'll appreciate any help.
> --Ted

Actually you are getting a real result. The imaginary parts cancel. When evaluated to finite precision you see this as either numerical fuzz (machine epsilonish values in machine arithmetic at fixed precision) or as zeros of some precision (evaluated with Mathematica's bignum arithmetic).

If you can find an equivalent representation that does not require subparts that are imaginary, that would be best. But I do not know if that can be done. Would require some special function identities, so could check Gamma entries at functions.wolfram.com.

For practical needs e.g. plotting with finite precision evaluations, best is just to wrap Re[] around the expression.

Daniel Lichtblau
Wolfram Research

```

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