Integrate Bug...

• To: mathgroup at yoda.physics.unc.edu
• Subject: Integrate Bug...
• From: anna
• Date: Tue, 30 Nov 93 14:36:19 CST

```     Following integral has not been fixed in V2.2, and we still have
that bug:

In[1]:= Integrate[ u^(5/6) (1 - u)^(5/6) Cos[k u], {u,0,1}]

2             2
2 1/3                  4  Sqrt[k ]      Sqrt[k ]        11
(k )    Sqrt[Pi] BesselJ[-, --------] Cos[--------] Gamma[--]
3     2             2            6
Out[1]= -------------------------------------------------------------
1/3  2
2 2    k

In new V2.3  this bug has been fixed.

In[272]:= Integrate[ u^(5/6) (1 - u)^(5/6) Cos[k u], {u,0,1}]

11          4  k Sign[k]      k
Sqrt[Pi] Gamma[--] BesselJ[-, ---------] Cos[-]
6           3      2          2
Out[272]= If[Im[k] == 0, -----------------------------------------------,
2 2/3
(k )

5/6  5/6
>    Integrate[(1 - u)    u    Cos[k u], {u, 0, 1}]]

Now this result is right for all real and integer K

The result from Prudnikov, Brychkov, and Marichev

res = k^(-4/3) Sqrt[Pi] BesselJ[4/3, k/2] Cos[k/2] Gamma[11/6];

In[277]:= N[res/. k -> -1/4]

-16
Out[277]= 0.218449 - 1.63992 10    I

In[278]:= N[%272/. k -> -1/4]

Out[278]= 0.218449

In[279]:= u^(5/6) (1 - u)^(5/6) Cos[k u]/. k -> -1/4;

In[280]:= NIntegrate[%, {u, 0, 1}]

Out[280]= 0.218449

Anna Marichev

```

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