Question

1. The Airy functions (named after British astronomer George Airy) are power series solutions to the Airy differential equation $y'' - xy = 0$. This equation arises in the modeling of light diffraction. When one solves this differential equation with power series methods, the result is two possible solutions given by $Ai(x) = 1 + frac{x^3}{2 cdot 3} + frac{x^6}{(2 cdot 3)(5 cdot 6)} + frac{x^9}{(2 cdot 3)(5 cdot 6)(8 cdot 9)} + cdots$ $= 1 + sum_{n=1}^{infty} frac{x^{3n}}{(2 cdot 3)(5 cdot 6) cdots ((3n-1)(3n))}$ $Bi(x) = x + frac{x^4}{3 cdot 4} + frac{x^7}{(3 cdot 4)(6 cdot 7)} + frac{x^{10}}{(3 cdot 4)(6 cdot 7)(9 cdot 10)} + cdots$ $= x + sum_{n=1}^{infty} frac{x^{3n+1}}{(3 cdot 4)(6 cdot 7) cdots (3n)(3n+1)}$ These are the Airy functions of the first and second kind, respectively. (a) (5 pts) Find the domain of $Ai(x)$ and $Bi(x)$. That is to say, what is the interval of convergence of each power series?

          1. The Airy functions (named after British astronomer George Airy) are power series solutions to the Airy differential equation $y'' - xy = 0$. This equation arises in the modeling of light diffraction. When one solves this differential equation with power series methods, the result is two possible solutions given by

$Ai(x) = 1 + frac{x^3}{2 cdot 3} + frac{x^6}{(2 cdot 3)(5 cdot 6)} + frac{x^9}{(2 cdot 3)(5 cdot 6)(8 cdot 9)} + cdots$

$= 1 + sum_{n=1}^{infty} frac{x^{3n}}{(2 cdot 3)(5 cdot 6) cdots ((3n-1)(3n))}$

$Bi(x) = x + frac{x^4}{3 cdot 4} + frac{x^7}{(3 cdot 4)(6 cdot 7)} + frac{x^{10}}{(3 cdot 4)(6 cdot 7)(9 cdot 10)} + cdots$

$= x + sum_{n=1}^{infty} frac{x^{3n+1}}{(3 cdot 4)(6 cdot 7) cdots (3n)(3n+1)}$

These are the Airy functions of the first and second kind, respectively.
(a) (5 pts) Find the domain of $Ai(x)$ and $Bi(x)$. That is to say, what is the interval of convergence of each power series?

$1. The Airy functions (named after British astronomer George Airy) are power series solutions to the Airy differential equation y” - xy = 0. This equation arises in the modeling of light diffraction. When one solves this differential equation with power series methods, the result is two possible solutions given by Ai(x) = 1 + fracx^32 cdot 3 + fracx^6(2 cdot 3)(5 cdot 6) + fracx^9(2 cdot 3)(5 cdot 6)(8 cdot 9) + cdots = 1 + sumn=1^infty fracx^3n(2 cdot 3)(5 cdot 6) cdots ((3n-1)(3n)) Bi(x) = x + fracx^43 cdot 4 + fracx^7(3 cdot 4)(6 cdot 7) + fracx^10(3 cdot 4)(6 cdot 7)(9 cdot 10) + cdots = x + sumn=1^infty fracx^3n+1(3 cdot 4)(6 cdot 7) cdots (3n)(3n+1) These are the Airy functions of the first and second kind, respectively. (a) (5 pts) Find the domain of Ai(x) and Bi(x). That is to say, what is the interval of convergence of each power series?$

Added by Antonia A.

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