# NIntegrate fails while Integrate works

I have a function $f(t)$ defined as

$f(t)=\int\limits_0^t(t-\xi)^{\alpha-1}\ \cos(\xi)\ d\xi$

where $0<\alpha<1$. I now want to evaluate this integral at various values of time. Therefore, my code for the function definition reads:

f[t_] := NIntegrate[(t-x)^(a-1) Cos[x], {x, 0, t}]

For the sake of this example, let us choose $\alpha=.3$, and let us attempt to evaluate $f(t)$ at $t=0.2$. Calling f[.2] returns the following error:

NIntegrate::zeroregion: Integration region {{0.2,0.200000000000000011102230246246115891736237825076626353161415959706}} cannot be further subdivided at the specified working precision. NIntegrate assumes zero integral there and on any further indivisible regions.

NIntegrate::inumri: The integrand Cos[x]/(0.2 -x)^0.7 has evaluated to Overflow, Indeterminate, or Infinity for all sampling points in the region with boundaries {{0.200000000000000011102230246246115891736237825076626353161415959706,0.200000000000000011092200115363618871738896620143270802392916628941}}.

What is most puzzling is that there exists an analytical solution this integral in terms of the hyper-geometric function. Defining $f(t)$ as

f[t_] := Integrate[(t-x)^(a-1) cos[x], {x, 0, t}]

and then calling f[.2] returns the correct value of the integral with no errors: $f(.2)=2.02934$. In fact, the analytical solution is

f[t] = ConditionalExpression[3.33333 t^0.3 HypergeometricPFQ[{1}, {1.15, 0.65}, -(t^2/4)],
Re[t] > 0 && Im[t] == 0]

Does this error has something to do with WorkingPrecision? I am relatively new to Mathematica, so forgive me if this is trivial.

### UPDATE

My ultimate goal is to make this function Listable so that I can pass a list of times as the function argument. Consider again the function defined in terms of Integrate (and not NIntegrate). Here $\alpha-0.1$. I now define an array to form my list of times.

time = Array[#/5 - .2 &, 11];

Passing this to $f(t)$ as f[time] gives me the error

Power::infy: "Infinite expression 1/0^1.9 encountered."

Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.

I note that it does give me the output, despite the error. However, if I pass the list of times manually I get the desired output:

f[{0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2}]

{0, 8.36644, 8.50035, 8.06263, 7.20356, 6.00083, 4.52232, 2.83781, 1.02105, -0.851216, -2.70136}

Is this also related to WorkingPrecision and how Array defines its entries?

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Yes you need to increase the WorkingPrecision to 20 for instance. Also dont forget to give the reals to at least that precision for example a=0.320 – Spawn1701D Apr 21 '13 at 5:01
Thanks, that was helpful. I have updated my question with another related issue I had. – G. H. Hardly Apr 21 '13 at 5:29
You can of course make a definition as a function of two variables: hardly[a_, t_] := t^a HypergeometricPFQ[{1}, {(a + 1)/2, a/2 +1}, -t^2/4]/a – J. M. Apr 21 '13 at 6:52

f[t_] := NIntegrate[(t - x)^(0.3 - 1) Cos[x], {x, 0, t}, WorkingPrecision -> 10]

f[0.220]

Gives:

2.029340978

For the listable situation, it is due to WorkingPrecision as before. You can do the following:

time = Array[#/5 - .2`20 &, 11];

Then Map f across as follows:

f /@ time

Which gives:

{0, 2.029340978, 2.398210571, 2.524050347, 2.479729745, 2.294755393,
1.989951583, 1.585123960, 1.101227368, 0.5608006799, -0.01230648010}
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