Series solutions near a regular singular point

  • Thread starter phrankle
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Main Question or Discussion Point

For solving a series solution near a regular singular point with the Frobenius method, why is it that the indices of summation derivatives aren't shifted?

For example, in my textbook and lecture notes

y = [tex]\sum[/tex]A[tex]_{}n[/tex]x[tex]^{}n+r[/tex] from n=0 to infinity

y' = [tex]\sum[/tex](n+r)A[tex]_{}n[/tex]x[tex]^{}n+r-1[/tex] from n=0 to infinity

y'' = [tex]\sum[/tex](n+r)(n+r-1)A[tex]_{}n[/tex]x[tex]^{}n+r-2[/tex] from n=0 to infinity


But shouldn't the index for y' be from n=1 to infinity because it shifts up when you take the derivative of a summation? Shouldn't the index for y'' be from n=2 to infinity?

Thanks.
 

Answers and Replies

tiny-tim
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Welcome to PF!

Hi phrankle! Welcome to PF! :smile:

No, because then your An wouldn't match your x^(n+r-1), would it?

Of course, you could change it to ∑A(n+1)x^(n+r), and do it from n = -1 to ∞ …

but that would be unnecessarily complicated, and you could easily make a mistake … :frown:
 

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