16 Convolution
Definition¶
\[ f(t) \star g(t) = \int\limits_0^\infty f(t-\tau) g(\tau) \cdot d\tau \]
\[ f(t) \star g(t) = g(t) \star f(t) \]
Convolution Theorem¶
It is used for Laplace Transform
\[ L \{ f(t) \star g(t) \} = F(s) \cdot G(s) \]
\[ \begin{aligned} L^{-1} \{ F(s) \cdot G(s) \} &= f(t) \star g(t) \\ &= L^{-1}\{ F(s) \} \star L^{-1}\{ G(s) \} \end{aligned} \]
Trignometric¶
\[ \begin{aligned} \cos(x) &= \frac{ e^x \textcolor{orange}{+} e^{-x} }{2} \\ \sinh(x) &= \frac{ e^x \textcolor{orange}{-} e^{-x} }{2} \\ \cos(x) &= \frac{ e^{\textcolor{hotpink}{i} x} \textcolor{orange}{+} e^{-\textcolor{hotpink}{i} x} }{2i} \\ \sin(x) &= \frac{ e^{\textcolor{hotpink}{i} x} \textcolor{orange}{-} e^{-\textcolor{hotpink}{i} x} }{2i} \end{aligned} \]