Are these analytic? (easy stuff)

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Homework Help Overview

The discussion revolves around determining the analyticity of the functions p(x) = 1/x and q(x) = 1/4x in the context of a differential equation. The original poster seeks to understand whether these functions can be expanded in a Taylor series around a point, specifically regarding their implications for using different solution methods.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions the analyticity of the functions based on their ability to be expressed as a Taylor series. Some participants agree that the functions are not analytic at x = 0 and discuss the implications of this for identifying singular points.

Discussion Status

Participants are exploring the nature of the points where the functions are defined, with some noting that they are regular singular points. There is a recognition of the relationship between these points and the applicability of the Frobenius method for solving the differential equation.

Contextual Notes

The discussion includes considerations of the definitions of analyticity and singular points, as well as the methods available for solving the differential equation in question. There is an emphasis on the implications of these definitions for the choice of solution techniques.

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Homework Statement



I have p(x)=1/x and q(x)=1/4x

I want to know if these are analytic.
My definition of analytic is this:
Analytic means can be expanded in a taylor series about x=x0, with non zero radius of convergence)

reason for asking is I want to determine whether I should use basic power series or method of frobenius for the DE:
y'' + (1/x)y'+(1/4x)y = 0

The Attempt at a Solution


I'm guessing that since p and q cannot be expanded in a taylor series, that makes them non-analytic?
 
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not at x = 0, i don't think so
 
which is what I thought, which makes them singular points.

Interestingly, if I use the Frobenius method, my solution ends up being the general power series :S
 
they are regular singular points though
 
yes sorry.
regular singular points which corresponds to the use of method of frobenius
 

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