Question: Dirichelet's discontinuous integral

Hi everybody,

 

I am doing the following:


Int(exp(-p*x)*cos(q*x),x=0..infinity):% = value(%) assuming p > 0 and q > 0;
                   /infinity                                
                  |                                     p   
                  |          exp(-p x) cos(q x) dx = -------
                  |                                   2    2
                 /0                                  p  + q 
Int(p/(p^2+q^2), q = a .. b):% = value(%) assuming a > 0 and b > a;
                    /b                                    
                   |      p                /a\         /b\
                   |   ------- dq = -arctan|-| + arctan|-|
                   |    2    2             \p/         \p/
                  /a   p  + q                             
So we can say that
Int(Int(exp(-p*x)*cos(q*x),x=0..infinity),q=a..b)=rhs(%);
       /b   /infinity                                                  
      |    |                                            /a\         /b\
      |    |          exp(-p x) cos(q x) dx dq = -arctan|-| + arctan|-|
      |    |                                            \p/         \p/
     /a   /0                                                           
Now, without checking if there are conditions to respect so I can change the order of integration, I do
Int(exp(-p*x)*Int(cos(q*x),q=a..b),x=0..infinity);
                   /infinity           /  /b            \   
                  |                    | |              |   
                  |          exp(-p x) | |   cos(q x) dq| dx
                  |                    | |              |   
                 /0                    \/a              /   
Int(exp(-p*x)*int(cos(q*x),q=a..b),x=0..infinity);
                /infinity                                     
               |            exp(-p x) (sin(a x) - sin(b x))   
               |          - ------------------------------- dx
               |                           x                  
              /0                                              
i1:=eval(%,b=0)=-arctan(a/p);
                /infinity                                     
               |            exp(-p x) sin(a x)             /a\
               |          - ------------------ dx = -arctan|-|
               |                    x                      \p/
              /0                                              
i2:=eval(i1,a/p=z);
                /infinity                                     
               |            exp(-p x) sin(a x)                
               |          - ------------------ dx = -arctan(z)
               |                    x                         
              /0                                              
Limit of i2 when p -> 0 (wich mean z =(plus or minus) infinity depanding of the sign of a)
with(Student[Calculus1]):
e1:=op(Integrand(lhs(i2)));
                              exp(-p x) sin(a x)
                            - ------------------
                                      x         
for a > 0
i3:=Int(limit(e1,p=0,right),x=0..infinity)=limit(rhs(i2),z=infinity);
                       /infinity                       
                      |            sin(a x)        1   
                      |          - -------- dx = - - Pi
                      |               x            2   
                     /0                                
and for a < 0 
i4:=Int(limit(e1,p=0,left),x=0..infinity)=limit(rhs(i2),z=-infinity);
                        /infinity                     
                       |            sin(a x)      1   
                       |          - -------- dx = - Pi
                       |               x          2   
                      /0                              
then for a = 0
i5:=Int(limit(e1,p=0,left),x=0..infinity)=limit(rhs(i2),z=0);
                          /infinity                  
                         |            sin(a x)       
                         |          - -------- dx = 0
                         |               x           
                        /0                           
So I obtain the Dirichlet's discontinuous integral starting with the Gamma function.

My question:  can someone specify the conditions for having the right to change the order of integration?

 

Mario Lemelin

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