No but when I do, I get a result:

out=Integrate[BesselJ[2n+1/2,L]BesselJ[2*m+1/2,L]/Sqrt[L^2+K],{L,0,\[Infinity]},Assumptions->n\[Element]Integers&&n>=0&&m\[Element]Integers&&m>=0&&K\[Element]Reals&&K>0]
Series[out,{K,\[Infinity],4}]

outputs a very large expression. Note that you can not substitute values directly into the output, first you have to apply Normal to it. See the detailed documentation of Series.

I have used N[] in Manipulate[] to decrease the time for graph manipulation!

Of course you can do series expansions around infinity:

Series[function,{x,Infinity,order}]

But getting any kind of precision might be tricky, you might need a lot of terms!

Hi,

Well, precision is one issue, but what about timings when K=10^4 or more? From my tests, I see a dramatic increase of computing times for K=10^4, compared to K= 10^3. Why actually it is so? How the hypergeometric series are evaluated internally by MATHEMATICA? If it is plain summation, then certainly there is no hope to get reasonable times.

If we now have a formula which is actually a series in powers of K, then is there any method to obtain symbolically an equivalent series in 1/K ? Then the summation could be faster for large K. Of course, I realise that this is not a trivial issue, but MATHEMATICA is supposed to encapsulate the whole current mathematical knowledge, so maybe it has some method to achieve the goal?

Leslaw

That last plot just uses machine precision, to correctly plot it use:

Plot[f[7, 0, K], {K, 0, 1000}, WorkingPrecision -> 32]

No need for N[ ], as K is numerical value anyhow...

Hi again,

It seems that I have solved the problem with the expression of the integral in terms of the hypergeometric series. By a detailed expansion of the second series in the symbolic formula returned by Integrate, some rearranging of the terms, and taking a limit of Sin[pi*z]/z (with z being the factor in the denominator that is zero) when z tends to zero, one obtains the tripartite formula:

F[m,n,K_]:=(2^(-2 (1+m+n)) * K^(1/2+m+n) * Gamma[-(1/2)-m-n] * Gamma[1+m+n] * HypergeometricPFQ[{1+m+n,1+m+n},{3/2+2 m,3/2+2 n,2+2 m+2 n},K])/(Sqrt[?] * Gamma[3/2+2 m] * Gamma[3/2+2 n])+(HypergeometricPFQ[{1/2,1/2,1},{ 1/2-m-n,1+m-n,1-m+n,3/2+m+n},K])/(1+2 *(m+ n)) /; m[Equal]n

F[m,n,K_]:=(2^(-2 (1+m+n)) * K^(1/2+m+n)Gamma[-(1/2)-m-n] * Gamma[1+m+n] * HypergeometricPFQ[{1+m+n,1+m+n},{3/2+2 m,3/2+2 n,2+2 m+2 n},K])/(Sqrt[?] Gamma[3/2+2 m] * Gamma[3/2+2 n])+(Pochhammer[1/2,m- n]^2/(Pochhammer[1/2-m-n,m-n] * Pochhammer[1+m-n,m- n]*Pochhammer[1-m+n,m-n-1] * Pochhammer[3/2+m+n,m- n])) * (K^(m-n)) * ((-1)^(m-n+1)) * (HypergeometricPFQ[{1/2+m-n,1/2+ m-n,1},{1/2-2 * n,1+2 * (m-n),1,3/2+2*m},K])/((m-n) * (1+2 * (m+ n))) /; m>n

F[m,n,K_]:=F[n,m,K] /; m<n

Unfortunately, I am now surprised to see that such defined function does not get evaluated faster than the former expansion I had (at least for K < 1000). For K > 1000 both formulae are damned slow, and this is very bad news, because I need to do calculations for K > 10^4. So, perhaps the only advantage of the above formulation is that I don't have to worry about the number of terms to be summed in the expansions.

So, my new question is: Is there any method to speed up the evaluations (32 digits or more, and for K > 10^4)? Ideally, it would be nice to derive some asymptotic expansion valid for large K, and use it for calculations. However, attempts to apply a relevant Series[] command to the above expressions lead to error messages.

Leslaw

This is a sign for a pause :)

Sure, this formula is time consuming. I have tried it to Manipulate...

f[z_, m_, n_] := 1/2 \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(\[Mu] = 1\), \(\[Infinity]\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n + m + \[Mu]\)] 
\*SuperscriptBox[\(Gamma[
\*FractionBox[\(\[Mu]\), \(2\)] + 
\*FractionBox[\(1\), \(2\)]]\), \(2\)] 
\*SuperscriptBox[\((
\*SqrtBox[\(z\)])\), \(\[Mu]\)]/\((Gamma[
\*FractionBox[\(\[Mu]\), \(2\)] + n - m + 1]*Gamma[
\*FractionBox[\(\[Mu]\), \(2\)] - n + m + 1]*Gamma[
\*FractionBox[\(\[Mu]\), \(2\)] - n - m + 
\*FractionBox[\(1\), \(2\)]]*Gamma[
\*FractionBox[\(\[Mu]\), \(2\)] + n + m + 
\*FractionBox[\(3\), \(2\)]])\)\)\)

Manipulate[
 Plot[f[z, m, n], {z, 0, 150}], {m, 0, 20, 1}, {n, 0, 20, 1}]

Unfortunately, it requests a huge ammount of time...

So, do I understand properly that there is no way to obtain the integral values with 32 digits precision (or more)?

Sure, we can by using N[2,32]:

In[1]:= Integrate[
 BesselJ[N[2, 32]*n + 1/2, L]*
  BesselJ[N[2, 32]*m + 1/2, L]/Sqrt[L^2 + K], {L, 0, \[Infinity]}, 
 Assumptions -> 
  n \[Element] Integers && n >= 0 && m \[Element] Integers && m >= 0 &&
    K \[Element] Reals && K > 0]


Out[1]= (Gamma[
     0.50000000000000000000000000000000 + 
      1.0000000000000000000000000000000 m + 
      1.0000000000000000000000000000000 n] HypergeometricPFQ[{1, 1/2, 
      1/2}, {0.50000000000000000000000000000000 - 
       1.0000000000000000000000000000000 m - 
       1.0000000000000000000000000000000 n, 
      1.50000000000000000000000000000000 + 
       1.0000000000000000000000000000000 m + 
       1.0000000000000000000000000000000 n, 
      1.00000000000000000000000000000000 - 
       1.0000000000000000000000000000000 m + 
       1.0000000000000000000000000000000 n, 
      1.00000000000000000000000000000000 + 
       1.0000000000000000000000000000000 m - 
       1.0000000000000000000000000000000 n}, K])/(2 Gamma[
     1.00000000000000000000000000000000 + 
      1.0000000000000000000000000000000 m - 
      1.0000000000000000000000000000000 n] Gamma[
     1.00000000000000000000000000000000 - 
      1.0000000000000000000000000000000 m + 
      1.0000000000000000000000000000000 n] Gamma[
     1.5000000000000000000000000000000+ 
      1.0000000000000000000000000000000 m + 
      1.0000000000000000000000000000000 n]) + (K^(
    0.50000000000000000000000000000000 + 
     1.0000000000000000000000000000000 m + 
     1.0000000000000000000000000000000 n)
     Gamma[-0.50000000000000000000000000000000 - 
      1.0000000000000000000000000000000 m - 
      1.0000000000000000000000000000000 n] Gamma[
     1.00000000000000000000000000000000 + 
      1.0000000000000000000000000000000 m + 
      1.0000000000000000000000000000000 n]^2 Gamma[
     1.50000000000000000000000000000000 + 
      1.0000000000000000000000000000000 m + 
      1.0000000000000000000000000000000 n] \
HypergeometricPFQ[{1.50000000000000000000000000000000 + 
       1.0000000000000000000000000000000 m + 
       1.0000000000000000000000000000000 n, 
      1.00000000000000000000000000000000 + 
       1.0000000000000000000000000000000 m + 
       1.0000000000000000000000000000000 n, 
      1.00000000000000000000000000000000 + 
       1.0000000000000000000000000000000 m + 
       1.0000000000000000000000000000000 n}, \
{1.50000000000000000000000000000000 + 
       1.0000000000000000000000000000000 m + 
       1.0000000000000000000000000000000 n, 
      2.0000000000000000000000000000000 + 
       2.0000000000000000000000000000000 m + 
       2.0000000000000000000000000000000 n, 
      1.5000000000000000000000000000000 + 0.*10^-32 m + 
       2.0000000000000000000000000000000 n, 
      1.5000000000000000000000000000000 + 
       2.0000000000000000000000000000000 m + 0.*10^-32 n}, 
     K])/(2 \[Pi] Gamma[
     1.5000000000000000000000000000000 + 
      2.0000000000000000000000000000000 m + 0.*10^-32 n] Gamma[
     1.5000000000000000000000000000000 + 0.*10^-32 m + 
      2.0000000000000000000000000000000 n] Gamma[
     2.0000000000000000000000000000000 + 
      2.0000000000000000000000000000000 m + 
      2.0000000000000000000000000000000 n])

You use the composition Evaluate@Integrate. Will the code work the same if I first generate the formula using Integrate (and real coefficients), and next call Evaluate?

The meaning of Evaluate[] is to avoid repeated Integrate[] as you firstly Integrate[] and after Integration[] you assign the result to the function.

I still want to understand what is the reason of the error that occurs when evaluating the fully symbolic formula. Is there no method of finding out where the error actually arises? Some sort of tracing analysis?

I don't know exactly the properties of the function you investigate, but I think that it has discontinuity points which must be investigated by applying the Limit[]. When we use the approximate computations, we have the values near the discontinuity points. It's only my supposition! You must be aware of this!