\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
asin_forward¶
View page sourceInverse Sine and Hyperbolic Sine Forward Mode Theory¶
Derivatives¶
If \(F(x)\) is \(\R{asin} (x)\) or \(\R{asinh} (x)\) the corresponding derivative satisfies the equation
and in the standard math function differential equation , \(A(x) = 0\), \(B(x) = \sqrt{1 \mp x * x }\), 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 \(Q(x) = 1 \mp x * x\) and let \(q\) be the corresponding Taylor coefficients for \(Q[ X(t) ]\). It follows that
It follows that \(B[ X(t) ] = \sqrt{ Q[ X(t) ] }\) and from the equations for the square root that for \(j = 0 , 1, \ldots\),
It now follows from the general Taylor Coefficients Recursion Formula that for \(j = 0 , 1, \ldots\),