       The strange behaviour of NIntegrate

• To: mathgroup at smc.vnet.net
• Subject: [mg53508] The strange behaviour of NIntegrate
• From: Zaeem Burq <Z.Burq at ms.unimelb.edu.au>
• Date: Sat, 15 Jan 2005 01:44:07 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Dear all,

I'll describe the problem briefly first as follows:

I am trying to NIntegrate a nice function (which I know is integrable), of
the form

K[s] = h[s]/[(1-s)^1.5] from 0 to 1.

The function is real, but Mathematica spits out a COMPLEX answer!!
Admittedly, the complex part is very small, and if I truncate the integral
just below 1, the complex bit disappears and changes the real part very
slightly.

It seems obvious that Math'ca is choosing the last s value just beyond 1,
producing the Sqrt of a negative number in the denominator.

If interested, here are the calculations and results. Read on if you think
you can help. Thanx.

\!\(\(\[CurlyPhi][
x_] := \[ExponentialE]\^\(\(-x\^2\)/2\)\/\@\(2  \[Pi]\);\)\n
\(K[s_] := \(\(\ \)\(\((1 - \@s)\)\ \[CurlyPhi][\(1 - \@s\)\/\@\(1 -
s\)]\)\
\)\/\((1 - s)\)\^1.5`;\)\)

NIntegrate[K[s], {s, 0, 1}]

gives \!\(\(\(0.4206155890394436`\)\(\[InvisibleSpace]\)\) -
3.304807577181973`*^-49\ \[ImaginaryI]\)

and

NIntegrate[K[s], {s, 0, .999999}]

results in

0.420217

Any thoughts?

Zaeem.

Zaeem Burq
PhD Stochastic Processes,
Dept. of Mathematics and Statistics,
Unimelb.

Room 201, Richard Berry Building
University of Melbourne,
Parkville, VIC 3052.

ph: 8344 4248.

http://www.ms.unimelb.edu.au/~zab

```

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