7.86 bug in integration exp(I*a*t) (31.8.98)

7.86.1 Al Rosethal

What’s going on here?

>q:=int(exp(I*a*t)/(t^2+b^2),t=-infinity..infinity); 
 
                                q := 0 
 
> qr:=int(cos(a*t)/(t^2+b^2),t=-infinity..infinity); 
 
                            signum(a) Pi sinh(b a) 
                    qr := - ---------------------- 
                                      b
 

Why doesn’t Maple return the second result for the first integral?

7.86.2 John Roumeliotis(3.9.98)

Even more curious (?)

> q:=int(cos(a*t)/(t^2+b^2),t=-infinity..infinity); 
 
                          signum(a~) Pi sinh(b a~) 
                   q := - ------------------------ 
                                     b 
 
> simplify( subs(a=5,b=3,%) ); 
 
                          - 1/3 Pi sinh(15) 
 
> q:=int(cos(5*t)/(t^2+3^2),t=-infinity..infinity); 
 
               q := - 1/3 Pi sinh(15) + 1/3 Pi cosh(15)
 

7.86.3 HARALD PLEYM (3.9.98)

You can try this:

>q:=Int(exp(I*a*t)/(t^2+b^2),t=-infinity..infinity); 
                            infinity 
                           / 
                          |          exp(I a t) 
                    q :=  |          ---------- dt 
                          |            2    2 
                         /            t  + b 
                           -infinity 
 
> q:=map(evalc,q); 
 
                      infinity 
                     / 
                    |          cos(t a)   I sin(t a) 
              q :=  |          -------- + ---------- dt 
                    |           2    2      2    2 
                   /           t  + b      t  + b 
                     -infinity 
 
> value(%); 
 
                   (-1 - I) Pi sinh(b a) signum(a) 
                   ------------------------------- 
                                  b