# Bessel's equation of the second kind with integer order.

• yungman
In summary, the conversation discusses the equation for Y, which involves Bessel functions and their derivatives. When p is an integer, the equation becomes more complicated and cannot be simplified using the same methods. The use of L'Hopital's rule is discussed in order to find the limit of Y as p approaches an integer, but the correct method is still being determined.
yungman
This is the equation given for the Y.

$$Y_{p}=\frac{J_{p}(x)cos(p\pi)-P_{-p}(x)}{sin(p\pi)}$$

In many books, if p is an integer n, they just said $$Y_{n}=lim(p\rightarrow n) Y_{p}$$

$$J_{p}(x)=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!\Gamma(k+p+1)}(\frac{x}{2})^{2k+p}$$ which give

$$J_{n}(x)=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!\Gamma(k+n+1)}(\frac{x}{2})^{2k+n}=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!(k+n)!}(\frac{x}{2})^{2k+n}$$

For p=integer n, $$J_{-n}(x)=(-1)^{n}J_{n}(x) \Rightarrow J_{-n}(x)=J_{n}(x),n=0,2,4...$$

and $$J_{-n}(x)=-J_{n}(x), n=1,3,5...$$

No books that I have gave the answer for Y when p is an integer. This is what I am doing so far and I cannot get the right answer:

$$\stackrel{lim}{p \rightarrow n}$$ $$Y_{p}=\frac{J_{p}(x)cos(p\pi)-P_{-p}(x)}{sin(p\pi)}=J_{n}(x)\frac{cos(p\pi)-1}{sin(p\pi)}, n=0,2,4,6...$$

$$=J_{n}(x)\frac{cos(p\pi)+1}{sin(p\pi)}, n=1,3,5...$$

So I can evaluate the lim by L'Hopital that $$\frac{f(x)}{g(x)}=\frac{f'(x)}{g'(x)}$$

$$\frac{cos(p\pi)-1}{sin(p\pi)}=\frac{[cos(p\pi)-1]'}{[sin(p\pi)]'}=\frac{-\pi sin(p\pi)}{\pi cos(p\pi)}$$

If we take p to an integer, the whole thing become zero! I can't get $$Y_{n}$$

also what is $$\stackrel{lim}{x\rightarrow 0} x^{0}?$$

Alan

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To suppliment to my question, the reason I asked the question is because I don't see from my approach, I can get the graph of the Y function which go to negative infinity when x goes to zero.

What I did is basically separate the $$J_{n}(x)$$ out and look at the $$\frac{cos(p\pi)+1}{sin(p\pi)}$$ alone.

$$J_{n}(x)$$ equal to 1 for n=0 and 0 for n>0. So I cannot prove the Y functions go to negative infinity as x approach zero.

I've gone through at least 6 books and most of them have different formulas for Y. None explain how they come up with the formulas.

Two books suggested using the formula used in Cauchy Euler equation to get the second solution:

$$y_{2}(x)=J_{n}(x)ln(x)+x^{n}\sum^{\infty}_{1}b_{k}x^{k}$$

That was actually the reason I asked the question about the Euler equation yesterday to really understand how the equation come from and not just copy and use it.

Alan

Thanks
Alan

I think the problem is that you're factoring out $J_n(x)$ when you're not really allowed to do that. You can't generally selectively apply the limiting procedure to different parts of the function.

$$\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{\lim_{x\rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)}$$

provided a) the top and bottom limits exist and b) the limit of the denominator is not zero. Because the denominator is zero in this case, you can't split the limit over top and bottom like that - you'll have to take the Bessel functions into account when you do L'Hopital's rule. Be careful, because you'll be taking derivatives with respect to the order of the Bessel function, not its argument. You can see the derivatives on this page

http://www.math.sfu.ca/~cbm/aands/page_362.htm

Note that it involves the digamma function.

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yungman said:
This is the equation given for the Y.

$$Y_{p}=\frac{J_{p}(x)cos(p\pi)-J_{-p}(x)}{sin(p\pi)}$$

In many books, if p is an integer n, they just said $$Y_{n}=lim(p\rightarrow n) Y_{p}$$

$$J_{p}(x)=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!\Gamma(k+p+1)}(\frac{x}{2})^{2k+p}$$ which give

$$J_{n}(x)=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!\Gamma(k+n+1)}(\frac{x}{2})^{2k+n}=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!(k+n)!}(\frac{x}{2})^{2k+n}$$

For p=integer n, $$J_{-n}(x)=(-1)^{n}J_{n}(x) \Rightarrow J_{-n}(x)=J_{n}(x),n=0,2,4...$$

and $$J_{-n}(x)=-J_{n}(x), n=1,3,5...$$

No books that I have gave the answer for Y when p is an integer. This is what I am doing so far and I cannot get the right answer:

$$\stackrel{lim}{p \rightarrow n}$$ $$Y_{p}=\frac{J_{p}(x)cos(p\pi)-J_{-p}(x)}{sin(p\pi)}=J_{n}(x)\frac{cos(p\pi)-1}{sin(p\pi)}, n=0,2,4,6...$$

$$=J_{n}(x)\frac{cos(p\pi)+1}{sin(p\pi)}, n=1,3,5...$$

So I can evaluate the lim by L'Hopital that $$\frac{f(x)}{g(x)}=\frac{f'(x)}{g'(x)}$$

$$\frac{cos(p\pi)-1}{sin(p\pi)}=\frac{[cos(p\pi)-1]'}{[sin(p\pi)]'}=\frac{-\pi sin(p\pi)}{\pi cos(p\pi)}$$

If we take p to an integer, the whole thing become zero! I can't get $$Y_{n}$$

also what is $$\stackrel{lim}{x\rightarrow 0} x^{0}?$$

Alan

I type it wrong, there is no $$P_{-p}(x)$$, it is $$J_{-p}(x)$$.

Mute said:
I think the problem is that you're factoring out $J_n(x)$ when you're not really allowed to do that. You can't generally selectively apply the limiting procedure to different parts of the function.

$$\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{\lim_{x\rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)}$$

provided a) the top and bottom limits exist and b) the limit of the denominator is not zero. Because the denominator is zero in this case, you can't split the limit over top and bottom like that - you'll have to take the Bessel functions into account when you do L'Hopital's rule. Be careful, because you'll be taking derivatives with respect to the order of the Bessel function, not its argument. You can see the derivatives on this page

http://www.math.sfu.ca/~cbm/aands/page_362.htm

Note that it involves the digamma function.

I think in this case, the limit p approaching an integer, the limit of Y is zero/zero. that is the reason I use L'Hopital.

Do you mean applying L'Hopital of the whole equation by differentiate the top and bottom respect to p, not respect to x? This is what I am trying to do, is this correct? I am looking at the equations you posted and take the dirivative of J_p respect to p.

$$Y_{p}=\frac{J_{p}(x)cos(p\pi)-J_{-p}(x)}{sin(p\pi)}$$

$$J_{p}(x)=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!\Gamma(k+p+1)}(\frac{x}{2})^{2k+p}$$ and $$J_{-p}(x)=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!\Gamma(k-p+1)}(\frac{x}{2})^{2k-p}$$

L'Hopital rule stated that:

$$\stackrel{lim}{p \rightarrow n}Y_{p}=\stackrel{lim}{p \rightarrow n}\frac{J_{p}(x)cos(p\pi)-J_{-p}(x)}{sin(p\pi)}=\stackrel{lim}{p \rightarrow n}\frac{\frac{d}{dp}[J_{p}(x)cos(p\pi)-J_{-p}(x)]}{\frac{d}{dp}sin(p\pi)}$$

then substitute n into p. I'll work on this tomorrow with your equation.
Thanks
Alan

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## 1. What is Bessel's equation of the second kind with integer order?

Bessel's equation of the second kind with integer order is a differential equation that is used in mathematical physics to describe the behavior of certain physical phenomena, such as heat transfer and wave propagation. It is named after the mathematician Friedrich Bessel and is written as x2y'' + xy' + (x2-n2)y = 0, where n is an integer.

## 2. What is the significance of the integer order in Bessel's equation?

The integer order in Bessel's equation represents the number of times the solution oscillates or changes direction. This is important because it allows us to determine the behavior of the physical phenomena being studied. For example, in the case of heat transfer, the integer order can indicate the number of temperature peaks or nodes in the system.

## 3. How is Bessel's equation of the second kind solved?

Bessel's equation of the second kind is solved using Bessel functions, which are special functions that satisfy the equation. These functions are typically expressed as infinite series or integrals and can be calculated using numerical methods or special mathematical software.

## 4. What are some real-world applications of Bessel's equation of the second kind with integer order?

Bessel's equation of the second kind has many applications in physics and engineering. It is commonly used to describe heat transfer in circular or cylindrical objects, such as pipes and heat exchangers. It also plays a role in the study of wave propagation and vibration phenomena, such as in acoustic or mechanical systems.

## 5. Are there any variations of Bessel's equation of the second kind?

Yes, there are several variations of Bessel's equation of the second kind, such as the modified Bessel equation, the modified Bessel function, and the Hankel function. These variations have different forms and properties, but they are all related to Bessel's equation and are used in different applications in mathematics and physics.

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