Question:Dirichelet's discontinuous integral

Question:Dirichelet's discontinuous integral

Maple

Hi everybody,

I am doing the following:

```
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```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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