1. ## Clarification

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?

2. Originally Posted by kaiser0792
Don't both notatations indicate a squaring of the function?
One squares the function, the other squares the absolute value of the function.

3. Considering f(x) = -3x, f^2(x) = (-3x)^2 = 9x^2 and |-3x|^2 also equals 9x^2. How are the functions different?

4. They are not different. But before they were squared they were.

5. 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 $\displaystyle (f(x))^2= (ix)^2= -x^2$.

6. Thank You