Initial value theorem

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, the initial value theorem is a theorem used to relate <a href="/facts/Frequency_domain/pyTVrUSq">frequency domain</a> expressions to the <a href="/facts/Time_domain/1w6mP2MU">time domain</a> behavior as time approaches <a href="/facts/Zero/4NBRtKHc">zero</a>.
Let

F
        (
        s
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        f
        (
        t
        )
        
          e
          
            −
            s
            t
          
        
        
        d
        t
      
    
    {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}

be the (one-sided) <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> of ƒ(t). If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is bounded on 
 
 
 
 (
 0
 ,
 ∞
 )
 
 
 {\displaystyle (0,\infty )}
 
 (or if just 
 
 
 
 f
 (
 t
 )
 =
 O
 (
 
 e
 
 c
 t
 
 
 )
 
 
 {\displaystyle f(t)=O(e^{ct})}
 
) and 
 
 
 
 
 lim
 
 t
 →
 
 0
 
 +
 
 
 
 
 f
 (
 t
 )
 
 
 {\displaystyle \lim _{t\to 0^{+}}f(t)}
 
 exists then the initial value theorem says

lim
          
            t
            
            →
            
            0
          
        
        f
        (
        t
        )
        =
        
          lim
          
            s
            →
            ∞
          
        
        
          s
          F
          (
          s
          )
        
        .
      
    
    {\displaystyle \lim _{t\,\to \,0}f(t)=\lim _{s\to \infty }{sF(s)}.}

Initial value theorem open-in-new

Initial value theorem