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Hello all I am having a bit of confusion in regard to the following;

I understand that when we are working with ODE of the form

$$p_o(x)y''(x)+p_1(x)y'(x)+p_2(x)y(x)=0$$ and we are considering some singular point say $x_o$.

I know then for if all p polynomials, we can use the limit method to determine if it is regular or irregular singular point.

But also I know it must hold in general for all analytic functions and the limit method can only be used with polynomials.

So my question is, how can I know if something is analytic. I know the definition of it, i.e. , if it has a convergent power series at $x_o$, but surely we are not supposed to construct a taylor series for each function we see to do this?

Any advice? Thank you

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The same methods should be usable here for analytic functions (or more generally, meromorphic functions) as for polynomials. For example, the Wikipedia page "Regular singular point" deals with the coefficients being meromorphic functions. In your case, you want $p_1(z)/p_0(z)$ to have at most a pole of order $1$ and $p_2(z)/p_0(z)$ at most a pole of order $2$ at $x_0$, which will be true if $(z-x_0) p_1(z)/p_0(z)$ and $(z - x_0)^2 p_2(z)/p_0(z)$ are analytic and bounded in a deleted neighbourhood of $x_0$.

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Consider $p_{1}(x)/p_{0}(x)$ in the complex $z$ plane. This can then be broken into real and imaginary components $p_{1}(z)/p_{0}(z) = u(x,y)+iv(x,y)$.

Then inside some region where $u(x,y)$ and $v(x,y)$ satisfy the Cauchy Riemann equations:

\begin{eqnarray} \frac{\partial u}{\partial x} &=& \frac{\partial v}{\partial y} \\ \frac{\partial v}{\partial x} &=& - \frac{\partial u}{\partial y} \end{eqnarray}

$p_{1}(z)/p_{0}(z)$ will be analytic. So if your point of interest $z_0=x_0$ lies inside such region then $p_{1}(x)/p_{0}(x)$ will be analytic at the point $x_0$.

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