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Laplace Transform - Coggle Diagram

- - - - \[(f \ast g)(t) = \int^{2\pi}_{0} f(y) g(x-y) \cdot dy = \int^{2\pi}_{0} g(y) f(x-y) dy\]
    - - \[(f \ast g)(t) = \int^{\infty}_{-\infty} f(y) g(x - y)\cdot dy= \int^{\infty}_{-\infty} g(y) g(x-y) \cdot dy \]
    - - \[ (f \ast g)(t) = \int^{t}_{0} f(\tau) g(t-\tau) \cdot d\tau \]
    - - \[L^{-1}[\phi_1(s) \cdot \phi_2(s)] = \int^{t}_{0}f_1(u) f_2(t-u) \cdot du \]
        
        \[ L^{-1}[\phi_2(s)] = f_2(t)\]
        
        \[ L^{-1}[\phi_1(s)] = f_1(t)\]
- - - - Digital signal processing
        
        the Laplace transform changes a
        signal in the time domain into a signal in the s-domain, also called the s-plane
      - Nuclear Physics
        
        In order to get the true form of radioactive decay a Laplace Transform is used.
      - System modeling
        
        Used to simplify calculations in system modeling, where large number of differential equations are used.
      - Process Control
        
        It helps to analyze the variables which when altered, produce desired manipulations in the result.
      - Analysis of electronic circuits
        
        Used to quickly solve differential equations
        occurring in the analysis of electronic circuits
- - - - Linear equation with constant coefficients
      - f(t) has a convenient transform