Trigonometric and Hyperbolic Sine and Cosine Forward Theory
Keywords: trigonometric hyperbolic sine cosine forward theory sin sinh cos cosh differential equation

| Prev | Next |  |  |  |  |  | sin_cos_forward |

Trigonometric and Hyperbolic Sine and Cosine Forward Theory
Differential Equation
The standard math function differential equation is B ( u ) * F ( 1 ) ( u ) - A ( u ) * F ( u ) = D ( u ) In this sections we consider forward mode for the following choices:

`` |  | F ( u ) |  | sin ( u ) |  | cos ( u ) |  | sinh ( u ) |  | cosh ( u )
|  | A ( u ) |  | 0 |  | 0 |  | 0 |  | 0
|  | B ( u ) |  | 1 |  | 1 |  | 1 |  | 1
|  | D ( u ) |  | cos ( u ) |  | - sin ( u ) |  | cosh ( u ) |  | sinh ( u )

We use a , b , d and f for the Taylor coefficients of A [ X ( t ) ] , B [ X ( t ) ] , D [ X ( t ) ] , and F [ X ( t ) ] respectively. It now follows from the general Taylor coefficients recursion formula that for j = 0 , 1 , … , f ( 0 ) = D ( x ( 0 ) ) e ( j ) = d ( j ) + ∑ k = 0 j a ( j - k ) * f ( k ) = d ( j ) f ( j + 1 ) = 1 j + 1 1 b ( 0 ) (∑ k = 1 j + 1 k x ( k ) e ( j + 1 - k ) - ∑ k = 1 j k f ( k ) b ( j + 1 - k ) ) = 1 j + 1 ∑ k = 1 j + 1 k x ( k ) d ( j + 1 - k ) The formula above generates the order j + 1 coefficient of F [ X ( t ) ] from the lower order coefficients for X ( t ) and D [ X ( t ) ] .

Input File: omh/theory/sin_cos_forward.omh