How should I interpret the difference between f^2(t) and |f(t)|^2? I don't see how they are different, but an application that I'm studying indicates that they are uniquely different. Don't both notatations indicate a squaring of the function?
How should I interpret the difference between f^2(t) and |f(t)|^2? I don't see how they are different, but an application that I'm studying indicates that they are uniquely different. Don't both notatations indicate a squaring of the function?
If f is a real valued function then there is no difference between $\displaystyle |f(x)|^2$ and $\displaystyle f^2(x)$. For real numbers, absolute value can be defined as $\displaystyle |x|= \sqrt{x^2}$ so that $\displaystyle |x|^2= x^2$.
If f is complex valued, there is a difference: If f(x)= ix, then $\displaystyle |f(x)|= |x| so |f(x)|^2= |x|^2$, while $\displaystyle (f(x))^2= (ix)^2= -x^2$.