We know that Taylor expansion is : $ f(x_0 + h) = f(x_0) + h f'(x_0) + .. \ $
I wish to expand the Airy function about it's first root , i.e ,
$Ai (c_1 - \epsilon ) = Ai (c_1) - \epsilon A_i'(c_1) + ...$ where $c_1$ is the first root of $Ai(z) = 0 $
I want to know that is it the correct expansion for the Airy function. If not that what prevents one from expanding Airy function in this way ?