함수 $\displaystyle f \left (t \right) = \begin{cases} \cos 2 t, 0 \leq t < \frac{\pi} {4} \\ 0, t \geq \frac{\pi} {4} \end{cases}$ 를 단위계단함수로 나타내고 $f \left (t \right)$ 의 라플라스 변환 $\mathscr{L} \left \{f \left (t \right) \right \}$ 를 구하면?

• ①$\displaystyle f \left (t \right) = \cos 2 t - \cos 2 t u \left (t - \left (\frac{\pi} {4} \right) \right)$, $\displaystyle \mathscr{L} \left \{f \left (t \right) \right \} = \frac{s} {s ^ {2} + 4} + \frac{2} {s ^ {2} + 4} e ^ {- \left (\frac{\pi} {4} \right) s}$
• ②$\displaystyle f \left (t \right) = \cos 2 t - \cos 2 t u \left (t - \left (\frac{\pi} {4} \right) \right)$, $\displaystyle \mathscr{L} \left \{f \left (t \right) \right \} = \frac{s} {s ^ {2} + 4} - \left (\frac{2} {s ^ {2} + 4} \right) e ^ {- \left (\frac{\pi} {4} \right) s}$
• ③$\displaystyle f \left (t \right) = \cos 2 t + \cos 2 t u \left (t - \left (\frac{\pi} {4} \right) \right)$, $\displaystyle \mathscr{L} \left \{f \left (t \right) \right \} = \frac{s} {s ^ {2} + 4} + \frac{s} {s ^ {2} + 4} e ^ {- \left (\frac{\pi} {4} \right) s}$
• ④$\displaystyle f \left (t \right) = \cos 2 t + \cos 2 t u \left (t - \left (\frac{\pi} {4} \right) \right)$, $\displaystyle \mathscr{L} \left \{f \left (t \right) \right \} = \frac{s} {s ^ {2} + 4} - \left (\frac{s} {s ^ {2} + 4} \right) e ^ {- \left (\frac{\pi} {4} \right) s}$