J F Ogilvie

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These are questions asked by J F Ogilvie

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. 

Some years ago it was promised that expansion of capabilities of Heun functions was imminent, but nothing has appeared.  Other functions long overdue for inclusion as special functions in Maple are the Lame functions, which arise as special cases of Heun's differential equation and therefore of Heun functions.  Lame's differential equation appears in Abramowitz and Stegun, but has long been neglected in Maple.  These spectial functions are much more generally useful to users of Maple than, for instance, esoteric parts of the physics package. 

The integral of y = Dirac(phi-m), in which phi is a continuously variable quantity and m is a positive integer, from -infinity to infinity yields 1 as an answer.   The analogous integral of y2 yields no answer.  Is it possible that the latter integral has some mathematical meaning that might yield an answer?

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