Getting two different values for f(0)

[Mr Davis 97]

Let ##\displaystyle f(t) = \int_0^{\infty} \frac{-\sin (tx) x}{x^2+1}~ dx##. It seems clear that ##f(0) = 0##.

Now, I will do some manipulations to get a different expression: ##\displaystyle f(t) = \int_0^{\infty} \frac{-\sin (tx) x}{x^2+1}~ dx = \int_0^{\infty} \frac{-\sin (tx) x^2}{(x^2+1)x}~ dx = \int_0^{\infty} \frac{-\sin (tx) (x^2+1) + \sin (tx)}{(x^2+1)x}~ dx = \int_0^{\infty} \frac{- \sin(tx)}{x} ~ dx + \int_0^{\infty} \frac{\sin (tx)}{(x^2+1)x}~ dx = -\frac{\pi}{2} + \int_0^{\infty} \frac{\sin (tx)}{(x^2+1)x}~ dx##. So here, we see that ##f(0) = - \frac{\pi}{2}##.

What am I doing wrong? Why am I getting two different values?

 

[tnich]

Let ##\displaystyle f(t) = \int_0^{\infty} \frac{-\sin (tx) x}{x^2+1}~ dx##. It seems clear that ##f(0) = 0##.

Now, I will do some manipulations to get a different expression: ##\displaystyle f(t) = \int_0^{\infty} \frac{-\sin (tx) x}{x^2+1}~ dx = \int_0^{\infty} \frac{-\sin (tx) x^2}{(x^2+1)x}~ dx = \int_0^{\infty} \frac{-\sin (tx) (x^2+1) + \sin (tx)}{(x^2+1)x}~ dx = \int_0^{\infty} \frac{- \sin(tx)}{x} ~ dx + \int_0^{\infty} \frac{\sin (tx)}{(x^2+1)x}~ dx = -\frac{\pi}{2} + \int_0^{\infty} \frac{\sin (tx)}{(x^2+1)x}~ dx##. So here, we see that ##f(0) = - \frac{\pi}{2}##.

What am I doing wrong? Why am I getting two different values?

You assume that ##\int_0^{\infty} \frac{- \sin(tx)}{x} ~ dx = -\frac{\pi}{2}## when ##t=0##. But ##\sin(0x) = 0##.

 

[FactChecker]

You assume that ##\int_0^{\infty} \frac{- \sin(tx)}{x} ~ dx = -\frac{\pi}{2}## when ##t=0##. But ##\sin(0x) = 0##.

I agree, especially considering the first line of the original post. How are the two different?

 

[mathman]

[tex]\int_0^{\infty}\frac{sin(tx)}{x}dx\ and\ \int_0^{\infty}\frac{sin(tx)}{(x^2+1)x}dx[/tex] are both discontinuous at t=0, and are both =0 at that point.