## Positive integrand giving negative answer

2

I'm integrating a positive function f(t) times sin(t) from 0 to pi/5 and get -38.

Actually f is slightly negative for a short time (smallest value ~ -0.0005), but far from enough to explain this. Here is the code for the function:

NormFact[k_, S_] :=
Sqrt[(4 S - 2 k - 1) (4 S - k) (4 S - k + 1) (k +
1) (k + 2)/(4*Pi*S)];

xi[k_, j_, S_] := (-1)^j*
Binomial[k, j]*(4*S - k - 1)!/((j + 2)!*(4*S - k - j - 1)!);

Ofunc[t_, S_, coeffs_] :=
1 - Cos[t/2]^(4*S) +
coeffs.Table[
NormFact[k, S]*
Sum[xi[k, j, S]*Sin[t/2]^(2*j + 2)*Cos[t/2]^(4*S - 2*j - 2), {j,
0, k}], {k, 0, Length[coeffs] - 1}];

cfs = {-2.6155133, 0.9614036, 0.100279, -0.432464,
0.39887624, -0.2508507, 0.10555472, -0.0007824, -0.0526054,
0.0703274, -0.0688152, 0.05138949, -0.03833719,
0.02062684, -0.0018939, -0.0027808};


And here's the result when I integrate:

In:= Integrate[Ofunc[t, 58.5, cfs]*Sin[t], {t, 0, Pi/5}]
Out:= -38.132 + 0. I


Can anyone see my mistake?

2On v9.0.1, I get 37.8958. What version are you using? Run "ReleaseID" /. ("Kernel" /. SystemInformation["Small"]) – rcollyer – 2013-08-08T16:27:28.983

Hm! I get 8.0.1.0 (2063990, 2063802). Note: I tried restarting the program and get the same negative answer. – jorgen – 2013-08-08T16:30:18.923

1I confirmed I get that answer on 8.0.1, but I don't get it on v9. So, it is likely a bug in v8. Although, I haven't tried v8.0.4. My only advice: upgrade. – rcollyer – 2013-08-08T16:36:26.593

Thanks for advice! Note: it seems to me the answer you get must also be wrong, since the function and sine are both too small on the interval to give 38 (try plotting it) – jorgen – 2013-08-08T16:37:53.720

1Hmm, I hadn't paid attention to that. NIntegrate gives an answer I'm comfortable with (0.13956), and it works fine on v8.0.1, too. So, I'd use that, especially considering the coefficients in your function have vastly different scales, and may be confusing things. – rcollyer – 2013-08-08T16:48:43.230

Good idea, thanks! That works for me too. PS I can't upvote your comments, I guess I don't have enough rep on this account yet. I'll try to remember when I get there. – jorgen – 2013-08-08T17:41:50.553

I get around 0.13956 when I use exact exponents, and around 37.896 when using exponents with decimals. Offhand I do not know if this is from arithmetic or from Integrate going through slightly different somersaults. Given the coefficient sizes and oscillations, huge numerical error here is not entirely a surprise. – Daniel Lichtblau – 2013-08-08T18:06:16.823

When I moreover rationalize the coefficients I get a longish exact result. It numericizes to around 0.13956. – Daniel Lichtblau – 2013-08-08T18:27:24.230