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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 ?

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Airy functions are analytic on the complex plane. You can form their Taylor series about $x=a$ just as you would any other function. So for $x\approx a$, $$ \text{Ai}(x)=\text{Ai}(a)+ \text{Ai}'(a)(x-a)+\frac{1}{2}\text{Ai}''(a)(x-a)^2 +\frac{1}{6} \text{Ai}'''(a)(x-a)^3+\cdots $$

For example, the Taylor series for $\text{Ai}(x)$ about $x=0$ is $$ \text{Ai}(x)={1\over 3^{2/3}\pi}\sum_{k=0}^\infty {1\over k!}\Gamma\left({k+1\over 3}\right)\sin\left({2\pi (k+1)\over 3}\right)\left(\root 3 \of 3\,z\right)^k. $$

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