J F Ogilvie

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17 years, 26 days

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These are questions asked by J F Ogilvie

     I asked a similar question a few weeks ago and received a most helpful response, but unfortunately I had inaccurately transcribed thespecified form of that double integral from a file presented in Microsoft Word.  I shall be most grateful for the numerical evaluation of this difficult double integral here presented in unambiguous and correct Maple definition, for cases n=0 and n=1.

 

i1 := Int(exp(-(k^2*sinh(s)^2/(gam-I*k*cosh(s)) + I*k*(1-cosh(s)))*v)*Hypergeom([I/k],[1],I*k*v)*v^n, v=0..R/2);

Jn := 1/R*Int(sinh(s)*coth(s/2)^(2*I/k)/(gam-I*k*cosh(s))^5*i1, s=0..smax);

smax := 20;  R := 1680;  k := 1/4;  gam := 1/2;

The preceding response seemed to have several Maple statements that were not actually involved in the solution, making it difficult to follow the applied scheme of that solution.

      Thanks in advance,     John Ogilvie

Why is this integral so difficult -- so slow to execute?

seq(evalf(Int(1/R*sinh(s*coth(1/2*s))^(2*I/k)/(gam-I*k*cosh(s))^5*
     Int(exp(-(k^2*sinh(s)^2/(gam-I*k*cosh(s))-I*k*(1+cosh(s)))*v)*
     hypergeom([1/k*I],[1],k*v*I)*v^n,v = 0 .. 1/2*R), s=0..s_max)), n=0..1);

Hints to improve the efficiency of execution would be appreciated.

This simple set of integrations works correctly and quickly in Maple 17 and other releases, but not in Maple 2020.

psi(x) := (-1)^v*(alpha*GAMMA(k-v)/v!/GAMMA(k-2*v-1)/GAMMA(k-2*v))^(1/2)*exp(-1/2*k*exp(-alpha*x))*                 (k*exp(-alpha*x))^(1/2*k-v-1/2)*LaguerreL(v,k-2*v-1,k/exp(alpha*x))/binomial(k-v-1,v);

Int(eval(%, [k=30.5, v=j, alpha=2.5])^2, x=-1..infinity) =
   seq(evalf[12](Int(eval(%, [k=30.5, v=j, alpha=2.5])^2, x=-1..infinity)), j=0..5);

Every time that I try this simple integration in Maple 2020 I receive a message "kernel connection lost".  Why???

Maple seems to have difficulty computing this sum, which occurs in molecular physics of the H atom.

fd := j -> 2^8*j^5*(j-1)^(2*j-4)/(3*(j+1)^(2*j+4));

sum(fd(n)*ln(1-1/n^2), n=2..infinity);

Can anybody discover a solution, please?

restart;
deq1 := diff(u(x, y), x) - diff(u(x, y), y$2) = exp(x+y);
                              /  2         \             
              / d         \   | d          |             
      deq1 := |--- u(x, y)| - |---- u(x, y)| = exp(x + y)
              \ dx        /   |   2        |             
                              \ dy         /             

deq2 := diff(u(x, y), x) - diff(u(x, y), y$2) = exp(1)^(x+y);
                            /  2         \                  
            / d         \   | d          |           (x + y)
    deq2 := |--- u(x, y)| - |---- u(x, y)| = (exp(1))       
            \ dx        /   |   2        |                  
                            \ dy         /                  

pdsolve(deq1, u(x,y));  ### no result
pdsolve(deq2, u(x,y));
                        /                                        
                        |                                        
PDESolStrucApplyFunction|uApplyFunction(x,y)=_F1ApplyFunction(x)
                        \                                        

                        1
  _F2ApplyFunction(y) - -
                        2

                       x+y                                                                                   
  (expApplyFunction(1))    (_C1 expApplyFunction(uminus0y)+_C2 expApplyFunction(y)+_C3 expApplyFunction(y) y)
  -----------------------------------------------------------------------------------------------------------
                                            _C3 expApplyFunction(y)                                          

  ,[{diffApplyFunction(_F1ApplyFunction(x),x)=_c[1] _F1

  ApplyFunction(x),diffApplyFunction(diffApplyFunction(_F2

                                                    \
                                                    |
  ApplyFunction(y),y),y)=_c[1] _F2ApplyFunction(y)}]|
                                                    /

No solution appears when the differential equation is expressed in standard form, but when exp(x + y) is converted to

exp(1)^(x + y) the correct solution appears.

 

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