I try to calculate the following integral

syms r sigma  
assume(sigma>0)
assume(k>0 & k<1)
L=int(r*besseli(0,a*r)*besselk(0,a*r)),r,sigma,1)

Invalid expression. When calling a function or indexing a variable, use parentheses. Otherwise, check for mismatched delimiters.

Answer 1

[Torsten]

Wolfram Alpha gives an answer in terms of the Meijer G-function:

https://www.wolframalpha.com/input?i=integration&assumption=%7B%22C%22%2C+%22integration%22%7D+-%3E+%7B%22Calculator%22%2C+%22dflt%22%7D&assumption=%7B%22F%22%2C+%22Integral%22%2C+%22integrand%22%7D+-%3E%22x*besseli%280%2Ck*x%29*besselk%280%2Ck*x%29%22

https://uk.mathworks.com/help/symbolic/meijerg.html

Answer 2

[Walter Roberson]

I had to guess about what k was in the code.

syms r sigma  
assume(sigma>0)
syms a; assume(a>0 & a<1)
L=int(r*besseli(0,a*r)*besselk(0,a*r),r,sigma,1)

L = 

[Shreen El-Sapa]

syms r sigma  
assume(sigma>0)
syms k; assume(k>0 & k<1)
L=int(r*besseli(0,k*r)*besselk(0,k*r),r,sigma,1)

L = 

[Walter Roberson]

And that is your solution.

There is probably no closed-form integral for that.

[Shreen El-Sapa]

I want to calculate it numerically

if I put k=1/2, sigma=.5

L=int(r*besseli(0,0.5*r)*besselk(0,0.5*r),r,.5,1)

[Walter Roberson]

format long g

syms r
sigma = 0.5;
k = 1/2;
L1 = int(r*besseli(0,k*r)*besselk(0,k*r),r,sigma,1)

L1 = 

L2 = double(L1)

L2 = 
         0.450428766292829

L3 = vpaintegral(r*besseli(0,k*r)*besselk(0,k*r),r,sigma,1)

L3 = 

0.450429

L4 = integral(@(r) r.*besseli(0,k*r).*besselk(0,k*r), sigma, 1)

L4 = 
         0.450428766292829

[Shreen El-Sapa]

Thanks so much