Explore BrainMass

Explore BrainMass

    Bessel and Legendre's Equations - Finding Lower Bound

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    The problem is to determine a lower bound for the radius of convergence for the two following equations. I am able to get p(x) and q(x) for both equations, but I'm confused on how to proceed. I would like to see the problem worked out and what the lower bound is.

    Bessel's Equation:

    Centered at 1

    what I believe are p(x) and q(x)

    p(x) = and q(x) =

    Legendre's Equation

    Centered at 0, and also try it centered at 1

    © BrainMass Inc. brainmass.com December 24, 2021, 4:50 pm ad1c9bdddf


    Solution Preview

    Please see the attached doc file and please do not hesitate to contact me through the BrainMass staff if there is any vagueness. Good Luck.

    We have:

    and we want the lower bound for the radius of convergence about x=1. First we divide the equation by x^2:

    now the radius is at least equal to the minimum between the radii of 1/x and (x^2-v)/x^2 about x=1:

    1/x=1/(1+(x-1))=1-(x-1)+(x-1)^2-(x-1)^3+... provided |x-1|<1 therefore the radius of convergence about x=1 for the first one is 1.

    we can ...

    Solution Summary

    The expert finds the lower bounds for Bessel and Legendre's equations.