Series solutions near a regular singular point

In summary, the indices of summation derivatives are not shifted when solving a series solution near a regular singular point with the Frobenius method. This is because the indices need to match the coefficients and exponents in the series, and changing the indices would make the process more complicated and prone to errors.
  • #1
phrankle
6
0
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.
 
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  • #2
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:
 

1. What is a series solution near a regular singular point?

A series solution near a regular singular point is a mathematical method used to approximate the solution of a differential equation when the equation has a singularity at a particular point. In this method, the solution is expressed as a power series and the coefficients are determined by substituting the series into the original equation.

2. What is a regular singular point?

A regular singular point is a point in a differential equation where the equation becomes singular (i.e. it cannot be solved) but the equation can be rewritten in a form that allows for a series solution. This point is considered "regular" because the equation can be transformed into a form that can be solved using standard methods.

3. How do you determine the radius of convergence for a series solution near a regular singular point?

The radius of convergence for a series solution near a regular singular point can be determined by using the Cauchy-Hadamard theorem. This theorem states that the radius of convergence is equal to the reciprocal of the limit superior of the absolute value of the coefficients in the series. In simpler terms, the radius of convergence is the distance from the center of the series to the nearest singularity.

4. What is the significance of finding a series solution near a regular singular point?

Finding a series solution near a regular singular point allows us to approximate the solution of a differential equation when traditional methods fail. This method is particularly useful in physics, engineering, and other scientific fields where differential equations are commonly used to model complex systems.

5. Can a series solution near a regular singular point be used for all types of differential equations?

No, a series solution near a regular singular point can only be used for differential equations that have a singularity at a particular point and can be rewritten in a form that allows for a series solution. It is not applicable to all types of differential equations, such as those with non-singular points or those that cannot be transformed into a series form.

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