odd and even functions

[skan]

f(t) = 1 if -pi/2 <=t <=0
-1 if 0<=t<= pi/2
0 elsewhere

how does integral of f(t)*cost dt become and odd function with the integral limit from - pi/2 to pi/2 ?

thanks a lot

[Tom Mattson]

how does integral of f(t)*cost dt become and odd function with the integral limit from - pi/2 to pi/2 ?


It's not the integral of f(t)cos(t) that is odd (in fact it's just a number). It's the function f(t)cos(t) itself that is odd.

Let's see why.

Let y(t)=f(t)cos(t)
so, y(-t)=f(-t)cos(-t)

Note that f(-t)=-f(t) and cos(-t)=cos(t). So,

y(-t)=-f(t)cos(t),

and thus y(t) is odd. In general, the product of any even function and any odd function is odd.

[skan]

thanks a lot