Inverse Cosine and Hyperbolic Cosine Forward Mode Theory
Keywords: inverse cosine hyperbolic forward mode theory acos acosh derivatives taylor coefficients recursion

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Inverse Cosine and Hyperbolic Cosine Forward Mode Theory
Derivatives acos ( 1 ) ( x ) = - 1 / 1 - x * x acosh ( 1 ) ( x ) = + 1 / x * x - 1 If F ( x ) is acos ( x ) or acosh ( x ) the corresponding derivative satisfies the equation ∓ ( x * x - 1 ) * F ( 1 ) ( x ) - 0 * F ( u ) = ∓ 1 and in the standard math function differential equation , A ( x ) = 0 , B ( x ) = ∓ ( x * x - 1 ) , and D ( x ) = ∓ 1 . 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
We define Q ( x ) = ∓ ( x * x - 1 ) and let q be the corresponding Taylor coefficients for Q [ X ( t ) ] . It follows that q ( j ) = { ∓ ( x ( 0 ) * x ( 0 ) - 1 ) if j = 0 ∓ ∑ k = 0 j x ( k ) x ( j - k ) otherwise It follows that B [ X ( t ) ] = Q [ X ( t ) ] and from the equations for the square root that for j = 0 , 1 , … , b ( 0 ) = q ( 0 ) b ( j + 1 ) = 1 j + 1 1 b ( 0 ) (j + 1 2 q ( j + 1 ) - ∑ k = 1 j k b ( k ) b ( j + 1 - k ) ) It now follows from the general Taylor coefficients recursion formula that for j = 0 , 1 , … , z ( 0 ) = F ( x ( 0 ) ) e ( j ) = d ( j ) + ∑ k = 0 j a ( j - k ) * z ( k ) = { ∓ 1 if j = 0 0 otherwise z ( j + 1 ) = 1 j + 1 1 b ( 0 ) (∑ k = 0 j e ( k ) ( j + 1 - k ) x ( j + 1 - k ) - ∑ k = 1 j b ( k ) ( j + 1 - k ) z ( j + 1 - k ) ) z ( j + 1 ) = 1 j + 1 1 b ( 0 ) (∓ ( j + 1 ) x ( j + 1 ) - ∑ k = 1 j k z ( k ) b ( j + 1 - k ) )

Input File: omh/theory/acos_forward.omh