Use float limits with Integrate?

Questions
1

I am a little confused with how Mathematica is integrating with Integrate. I am trying to compute an internal of the product of special functions (modified Bessel functions) with symbolic limits. After working for a minute, the computation stops with no success. However, if I carry out the same integral, but with floating point limits, it spits out an answer. What’s going on here? When I give Integrate floating point limits is it reverting to NIntegrate?

Thanks,
Jeff

Edit: Here’s an example. See how it fails even with rational limits?

In[1]:= (*Doesn't evaluate*)
        Integrate[BesselK[1, k r] BesselI[1, k r], {r, 1/2, 3/2}]

Out[1]//InputForm= Integrate[BesselI[1, k*r]*BesselK[1, k*r], {r, 1/2, 3/2}]

In[2]:= (*Evaluates with float limits*)
         Integrate[BesselK[1, k r] BesselI[1, k r], {r, .5, 1.5}]

Out[2]= -0.0705237 MeijerG[{{0.5, 0.5}, {}}, {{0., 1.}, {-1., -0.5}}, 0.5 k, 0.5] + 
 0.211571 MeijerG[{{0.5, 0.5}, {}}, {{0., 1.}, {-1., -0.5}}, 1.5 k, 0.5]
2

Has seemingly to do with argument k r:

In[5]:= Integrate[BesselK[1, k r] BesselI[1, k r], {r, 1/2, 3/2}, Assumptions -> k \[Element] Complexes]
Out[5]= Integrate[BesselI[1, k r] BesselK[1, k r], {r, 1/2, 3/2}, Assumptions -> k \[Element] Complexes]

stating that k \in \mathbb{C} does not help, so substitute the argument

    In[7]:= Integrate[BesselK[1, x] BesselI[1, x]/k, {x, k/2, 3 k/2}, 
                      Assumptions -> k \[Element] Complexes]
    
    Out[7]= (1/(4 k Sqrt[\[Pi]]))
            (-MeijerG[{{1, 1}, {}}, {{1/2, 3/2}, {-(1/2), 0}}, k/2, 1/2] + 
              MeijerG[{{1, 1}, {}}, {{1/2, 3/2}, {-(1/2), 0}}, (3 k)/2, 1/2]
            )
3

When I give Integrate floating point limits is it reverting to NIntegrate?

no, by no means: NIntegrate[] returns a number (possibly with conditions), but you got a symbolic result parametrized by reals:

In[1]:= Integrate[(r k)^2, {r, 1/2, 3/2}]
Out[1]= (13 k^2)/12

In[2]:= NIntegrate[(r k)^2 , {r, 0.5, 1.5}]

During evaluation of In[2]:= NIntegrate::inumr: The integrand k^2 r^2 has evaluated to non-numerical values for all sampling points in the region with boundaries {{0.5,1.5}}.

Out[2]= NIntegrate[(r k)^2, {r, 0.5, 1.5}]
4

Hi Dent de Lion,

I really appreciate your investigation of my problem. You are correct that they argument kr seems to be to blame for some of these computation failures. However, even with a simple argument, Integrate acts differently for float limits than for rational limits. Can you explain this?

For example,

In[1]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, 1/2, 3/2}] // InputForm

Out[1]//InputForm=Integrate[(BesselI[0, x]*BesselK[1, x])/x^2, {x, 1/2, 3/2}]

In[2]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, .5, 1.5}]

Out[2]= 1.45921

Thanks,
Jeff

5

This is what I get on my old-fashioned Mma 7

In[1]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, 1/2, 3/2}] // InputForm


Out[1]//InputForm=
16/9 - (Pi^(3/2)*(MeijerG[{{-1/2, 0}, {-1/4, 1/4, 1}}, 
     {{0, 0, 0}, {-1, -1, -1/4, 1/4}}, 3/2, 1/2] - 
    4*MeijerG[{{1/2, 1}, {3/4, 5/4, 2}}, {{1, 1, 1}, 
       {0, 0, 3/4, 5/4}}, 1/2, 1/2]))/2
6

And here is the latest and greatest

In[4]:= $Version
Out[4]= "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)"

In[5]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, 1/2, 3/2}]
Out[5]= 2/9 (8 + \[Pi]^(
    3/2) (9 MeijerG[{{1/2, 1}, {3/4, 5/4, 2}}, {{1, 1, 1}, {0, 0, 3/4,
           5/4}}, 1/2, 1/2] - 
      MeijerG[{{1/2, 1}, {3/4, 5/4, 2}}, {{1, 1, 1}, {0, 0, 3/4, 5/
         4}}, 3/2, 1/2]))

In[6]:= % // N
Out[6]= 1.45921

In[7]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, .5, 1.5}]
Out[7]= 1.45921

Can you explain this?

It’s a neverending story to explain software (implementations). Nevertheless Mathematica has an overwhelming strong relation to science and there it’s quite usual to switch to numerical results (in contrast to symbolic ones) if the user chooses to give numerical input. Compare

In[8]:= 1/2
Out[8]= 1/2

In[9]:= 1/2.
Out[9]= 0.5

nothing obscure in that.

7

Wow. sigh I’m using Mathematica 9. So it works in both an earlier and a later version. Thanks guys.