Limit of spherical bessel function of the second kind

In summary: Thanks for the correction! In summary, the limit for the spherical Bessel function of the first kind when x<<1 is given by the formula j_{n}(x<<1)=\frac{x^n}{(2n+1)!}, and the formula for the spherical Bessel function of the second kind is y_{n}(x)=\frac{(-1)^{n+1}}{2^{n}x^{n+1}}\sum_{k=0}^{\infty}\frac{(-1)^{k}(k-n)!}{k!(2k-2n)!}x^{2k}. The correct asymptotic behavior for y_n(x) as x approaches 0 is given by the formula
  • #1
Mr. Rho
15
1
I know that the limit for the spherical bessel function of the first kind when $x<<1$ is:

[tex]j_{n}(x<<1)=\frac{x^n}{(2n+1)!}[/tex]

I can see this from the formula for $j_{n}(x)$ (taken from wolfram's webpage):

[tex]j_{n}(x)=2^{n}x^{n}\sum_{k=0}^{\infty}\frac{(-1)^{n}(k+n)!}{k!(2k+2n+1)!}x^{2k}[/tex]

and applying the relation: $(2n+1)!=(2n+1)!/2^{n}n!$ . So, the formula for the spherical bessel function of the second kind is (also taken from wolfram):

[tex]y_{n}(x)=\frac{(-1)^{n+1}}{2^{n}x^{n+1}}\sum_{k=0}^{\infty}\frac{(-1)^{k}(k-n)!}{k!(2k-2n)!}x^{2k}[/tex]

The result I get for the limit $x<<1$ in this case is:

[tex]y_n(x<<1)=\frac{(-1)^{n+1}(-n)!}{2^{n}(-2n)!}\frac{1}{x^{n+1}}[/tex]

But I don't know how to deal with these negative factorials. I think maybe I would need the relation $(2n-1)!=(2n)!/2^{n}n!$ because in Jackson's Classical Electrodynamics 2nd edition book they give a result for this in chapter 16, it is:

[tex]y_{n}(x<<1)=-\frac{(2n-1)!}{x^{n+1}}[/tex]

The question is: how to obtain this Jackson's result?
 
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  • #2
If I'm not way off here I think you can rewrite those negative factorials as gamma function, maybe that will help? That would give you ##(-n)!=\Gamma(1-n) ## and ##(-2n)! = \Gamma(1-2n) ## and then you could use some suitable notation for these and plug that in. I'm not 100% sure tho, its been a while since I did something similar but it might be worth a shot.
 
  • #3
  • #4
Alright, this ended up really bugging me all day, but I think it's because Mathworld gives a bad power series for the spherical Bessel function (how is k supposed to start at 0 when negative factorials are undefined?). Here's an alternative expression that gives the correct asymptotic behavior:

http://dlmf.nist.gov/10.53

(BTW, @Mr. Rho in your first post, I believe the asymptotic expression you wanted is $$y_n(x\rightarrow 0) \approx -\frac{(2n-1)!}{x^{n+1}}$$
with a double factorial, instead of a single factorial.)
 
  • #5
TeethWhitener said:
Alright, this ended up really bugging me all day, but I think it's because Mathworld gives a bad power series for the spherical Bessel function (how is k supposed to start at 0 when negative factorials are undefined?). Here's an alternative expression that gives the correct asymptotic behavior:

http://dlmf.nist.gov/10.53

(BTW, @Mr. Rho in your first post, I believe the asymptotic expression you wanted is $$y_n(x\rightarrow 0) \approx -\frac{(2n-1)!}{x^{n+1}}$$
with a double factorial, instead of a single factorial.)

yes, sorry for that, it is double factorial, thank you for the correction and thank you for those relations you give, the asymptotic limit it's very clear with them!

why would wolfram give a wrong relation?
 
Last edited:
  • #6
Mr. Rho said:
why would wolfram give a wrong relation?
It happens occasionally. There used to be a way to submit errors and corrections, but I don't see the link anymore.
 
  • #7
TeethWhitener said:
Alright, this ended up really bugging me all day, but I think it's because Mathworld gives a bad power series for the spherical Bessel function (how is k supposed to start at 0 when negative factorials are undefined?). Here's an alternative expression that gives the correct asymptotic behavior:

http://dlmf.nist.gov/10.53

(BTW, @Mr. Rho in your first post, I believe the asymptotic expression you wanted is $$y_n(x\rightarrow 0) \approx -\frac{(2n-1)!}{x^{n+1}}$$
with a double factorial, instead of a single factorial.)

Haha yeah, that was throwing me off too.
 

Related to Limit of spherical bessel function of the second kind

What is the limit of the spherical bessel function of the second kind as x approaches zero?

The limit of the spherical bessel function of the second kind as x approaches zero is equal to zero. This can be mathematically expressed as limx→0 yn(x) = 0, where yn(x) is the spherical bessel function of the second kind of order n.

What is the relationship between the spherical bessel function of the second kind and the spherical bessel function of the first kind?

The spherical bessel function of the second kind, denoted as yn(x), is related to the spherical bessel function of the first kind, denoted as jn(x), through the following formula: yn(x) = jn(x) * cot(x) - 1/x.

What is the significance of the limit of the spherical bessel function of the second kind?

The limit of the spherical bessel function of the second kind is important in the mathematical analysis of functions and their behavior as x approaches zero. It also has applications in fields such as physics, engineering, and signal processing.

Can the limit of the spherical bessel function of the second kind be evaluated for all values of n?

Yes, the limit of the spherical bessel function of the second kind can be evaluated for all integer values of n. However, for non-integer values of n, the limit may not exist or may require more advanced mathematical techniques to evaluate.

Is the limit of the spherical bessel function of the second kind always equal to zero?

No, the limit of the spherical bessel function of the second kind is not always equal to zero. It depends on the order of the function, n. For some values of n, the limit may approach infinity or negative infinity, while for others it may approach a finite value.

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