Exponential Function Forward Mode Theory
Keywords: exponential function forward mode theory exp expm1 derivatives taylor coefficients recursion

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Exponential Function Forward Mode Theory
Derivatives
If F ( x ) is exp ( x ) or expm 1 ( x ) the corresponding derivative satisfies the equation 1 * F ( 1 ) ( x ) - 1 * F ( x ) = d ( 0 ) = { 0 if F ( x ) = exp ( x ) 1 if F ( x ) = expm 1 ( x ) where the equation above defines d ( 0 ) . In the standard math function differential equation , A ( x ) = 1 , B ( x ) = 1 , and D ( x ) = d ( 0 ) . We use a , b , d , and z to denote the Taylor coefficients for A [ X ( t ) ] , B [ X ( t ) ] , D [ X ( t ) ] , and F [ X ( t ) ] respectively.
Taylor Coefficients Recursion
For orders j = 0 , 1 , … , z ( 0 ) = F ( x ( 0 ) ) e ( 0 ) = d ( 0 ) + z ( 0 ) e ( j + 1 ) = d ( j + 1 ) + ∑ k = 0 j + 1 a ( j + 1 - k ) * z ( k ) = z ( j + 1 ) z ( j + 1 ) = 1 j + 1 1 b ( 0 ) (∑ k = 1 j + 1 k x ( k ) e ( j + 1 - k ) - ∑ k = 1 j k z ( k ) b ( j + 1 - k ) ) = x ( j + 1 ) d ( 0 ) + 1 j + 1 ∑ k = 1 j + 1 k x ( k ) z ( j + 1 - k )

Input File: omh/theory/exp_forward.omh