16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde o (x) d This is a different x. Note the relationship between this and the quantity x previous defined. We use x again here as this is how a is usually written Not only this function but also its first several derivatives which appear in analytic work are tabulated 3. Relation to tabulated functions Even more generally available are the closely related functions Error function: erf(x) Complementary error function cerf(x)=2 Φ(x)=1 p(1) 1 e/m(cos(to y)+jsin(toy)e Differentiation of this form will yield correctly the first 2 moments of the distribution 9/30/2004955AM Page 5 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 5 of 10 2 2 1 ( ) 2 x v x e dv π − −∞ Φ = ∫ This is a different x. Note the relationship between this and the quantity x previous defined. We use x again here as this is how Φ is usually written. Not only this function but also its first several derivatives which appear in analytic work are tabulated. 3. Relation to tabulated functions Even more generally available are the closely related functions: Error function: 2 0 2 ( ) x u erf x e du π − = ∫ Complementary error function: 2 2 ( ) u x cerf x e du π ∞ − = ∫ 1 () 1 2 2 x x erf ⎡ ⎤ ⎛ ⎞ Φ= + ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ () () ( ) 2 2 2 2 2 2 2 2 2 ( ) 2 ( ) 2 2 2 2 2 1 ( ) 2 1 , where 2 1 (cos sin ) 2 2 cos 2 2 2 2 x m jtx y jt m y y jtm y jtm t jtm t jtm t e e dx x m e e dy y e t y j t y e dy e t y e dy e e e σ σ σ σ φ πσ π σ σ σ π σ π π π ∞ − − −∞ ∞ − + −∞ ∞ − −∞ ∞ − −∞ − ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠ = − = = = + = = = ∫ ∫ ∫ ∫ Differentiation of this form will yield correctly the first 2 moments of the distribution