J F Ogilvie

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16 years, 298 days

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These are questions asked by J F Ogilvie

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.

 

Maple seems to have difficulty calculating this integral algebraically, although it seems not excessively complicated.

int( ((-A*omega*sin(omega*x+phi)*exp(-x/tau) - A*cos(omega*x+phi)*exp(-x/tau)/tau)^2 + 1)^(1/2), x=0..t ) assuming t>0, omega>0, tau>0, A>0

any suggestions to solve this integral?

thanks in advance

There are discrepancies between Maple's solution of Fourier transforms and the results printed in USA NIST Handbook of Mathematical Functions, page 30

fourier(exp(-a*abs(x))/sqrt(abs(x)),x,s) assuming a>0;
            /   /   (1/2)   (1/2)                (1/2)  
        1   |   |2 2      Pi      signum(s - _U1)       
       ---- |int|-------------------------------------,
       2 Pi |   |       /   2    \                      
            |   |       |_U1     |          (1/2)       
            |   |     a |---- + 1| (s - _U1)            
            |   |       |  2     |                      
            \   \       \ a      /                      

                                    \\
                                    ||
         _U1 = -infinity .. infinity||
                                    ||
                                    ||
                                    ||
                                    ||
                                    //


For this transform of
                 "exp(-a*abs(x))/sqrt(abs(x))"

 the result in the NIST table is
          "sqrt(a + sqrt(a^2 + s^2))/sqrt(a^2 + s^2)"

 .
fourier(sinh(a*t)/sinh(Pi*t),x,s) assuming a>-Pi, a<Pi;
                    2 sinh(a t) Pi Dirac(s)
                    -----------------------
                          sinh(Pi t)       

For this transform of sinh(a*x)/sinh(Pi*x)   the result in the NIST table is
                         "1/sqrt(2*Pi)"  "sin(a)/(cosh(s) + cos(a))"

 
fourier(cosh(a*t)/cosh(Pi*t),x,s) assuming a>-Pi, a<Pi;
                    2 cosh(a t) Pi Dirac(s)
                    -----------------------
                          cosh(Pi t)       

For this transform of cosh(a*x)/cosh(Pi*x) the result in the NIST table is  
                          "sqrt(2/Pi) cos(a/2)*cosh(s/2)/(cosh(s) + cos(a))"

These disparities are significant, apart from the fact that Maple failed to solve the first example above.

 

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