J F Ogilvie

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19 years, 185 days

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These are questions asked by J F Ogilvie

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.

 

Maple formerly accepted

rule1 := forall(string(y), TD(y) = D(y)):

but now that statement prodices an error message:

Error, (in forall) expecting bound variable(s) to be a name or list of unique names


The entire value of this forall structure is that it should apply to objects of specific types, but it is not obvious how to express those types; the Help page for 'forall' is essentially useless in relation to this question. 

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