#### 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