Integral form of Bessel polynomial and orthogonality of Bessel differential equation

Integral form of Bessel polynomial:

We know, the generating function of Bessel polynomial as,

$$e^{\frac12 x(t-1/t)}=\sum_{n=-\infty}^\infty t^n J_n(x)$$

Put \(t=e^{i\theta}\) then,

$$or,\;\; e^{\frac12 x(e^{i\theta}-e^{-i\theta})}= \sum_{n=-\infty}^\infty e^{in\theta}J_n(x)$$

$$or,\;\; e^{ixsin\theta}=…+ e^{-4i\theta} J_{-4}(x)+ e^{-3i\theta} J_{-3} (x)+ e^{-2i\theta}J_{n-2}(x)+ e^{-1i\theta}J_{n-1}(x)+ J_0(x)+ e^{i\theta} J_1(x)+ e^{2j\theta}J_2(x)+e^{3i\theta}J_3(x)+ e^{4i\theta}J_4(x)+…$$

$$or,\;\; e^{ixsin\theta}=J_0(x)+ e^{i\theta}J_1(x)+ e^{2i\theta}J_2(x)+e^{-2i\theta}J_2(x)+ e^{3i\theta}J_3(x)- e^{-3i\theta}J_3(x)+ e^{4i\theta}J_4(x)+ e^{-4i\theta}J_4(x)+…$$

$$or,\;\; e^{ixsin\theta}= J_0(x)+(e^{i\theta}-e^{-i\theta}) J_1(x)+ (e^{2i\theta}+e^{-2i\theta})J_2(x)+ (e^{3i\theta}- e^{-3i\theta})J_3(x)+(e^{4i\theta}-e^{-4i\theta})J_4(x)+…$$

$$or,\;\; e^{ixsin\theta}= J_0(x)+ 2isin\theta J_1(x)+ 2cos2\theta J_2(x)+2isin3\theta-J_3(x)+2cos4\theta\cdot J_4(x)+…$$

$$or,\;\; cos(xsin\theta)+isin(xsin\theta)= J_0(x)+2cos2\theta\cdot J_2(x)+2cos4\theta\cdot J_4(x)+…+ i(2sin\theta\cdot J_1(x))+ 2sin3\theta\cdot J_3(x)+…$$

Comparing real and imaginary part, we get

For even n

$$ cos(xsin\theta) = J_0(x)+ 2cos2\theta\cdot J_2(x)+ 2cos4\theta\cdot J_4(x)+…+2cosn\theta\cdot J_n(x)$$

For odd n

$$ sin(xsin\theta)= 2sin\theta\cdot J_1(x)+ 2sin3\theta\cdot J_3(x)+…+2sinn\theta\cdot J_n(x)$$

Now multiplying equation (1) by \(cosm\theta\) on both side and integrating with respect to \(\theta\) from 0 to \(\pi\) we get,

For m\(\ne\)0

$$\int_0^\pi cos(xsin\theta)\cdot cosm\theta d\theta= \int_\pi^0 \pi J_n(x)\dotsm(3) $$

For m=n

$$\int_0^\pi 2cosn\theta\cdot J_n(x)\cdot cosm\theta d\theta= J_n(x)\int_0^\pi 2cosn\theta\cdot cosm\theta d\theta$$

$$=J_n(x)\int_0^\pi 2cosn\theta\cdot cosm\theta d\theta$$

$$=J_n(x) \int_0^\pi [cos(m+n) \theta+ cos(n-m) \theta] d\theta$$

$$=J_n(x) \int_0^\pi[cos2n\theta+1]d\theta$$

$$= \pi J_n(x)$$

Now, multiplying equation (2), by \(sinm\theta\) and integrate with respect to \(\theta\) from 0 to \(\pi\), we get,

$$\int_0^\pi sin(xsin\theta)\cdot sinm\theta d\theta= \int_\pi^0 J_n(x)\dotsm(4)$$

Now adding equation (3) and (4) we get,

$$\int_0^pi [cos(xsin\theta)\cdot cosn\theta + sin(xsin\theta)\cdot sinn\theta] d\theta= \pi J_n(x)$$

$$J_n(x)=\frac{1}{\pi}\int_0^\pi cos(xsin\theta- n\theta) d\theta$$

Orthogonality of Bessel differential equation:

We know the Bessel differential equation has the form

$$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-n^2)y=0\dotsm(1)$$

Since y=J_n(x) so,

$$ x^2\frac{d^2J_n(x)}{dx^2}+ x\frac{dJ_n(x)}{dx}+(x^2-n^2) J_n(x)=0\dotsm(2)$$

Put x= ar

$$ a^2r^2\frac{d^2J_n(ar)}{d(ar)^2}+ar\cdot \frac{dJ_n(ar)}{d(ar)}+(a^2 r^2-n^2)J_n(ar)=0$$

$$or,\;\; r^2\cdot\frac{d^2J_n(ar)}{dr^2}+r\cdot \frac{dJ_n(ar)}{dr}+(a^2r^2-n^2)J_n(ar)=0\dotsm(3)$$

For x=br,

$$ r^2\cdot \frac{d^2J_n(br)}{dr^2}+r\cdot \frac{dJ_n(br)}{dr}+(b^2r^2-n^2)J_n(br)=0\dotsm(4)$$

Now, multiplying equation (5) by \(\frac{J_n(br)}{r}\) and (4) by \(\frac{J_n(ar)}{r} \)and subtracting (3)-(4), we get,

$$or,\;\; r\cdot J_n(br)\frac{d^2J_n(ar)}{dr^2}- r J_n(ar)\cdot \frac{d^2J_n(br)}{dr^2}+J_n(br)\frac{dJ_n(ar)}{ar}-J_n(ar)\cdot \frac{dJ_n(br)}{dr}+ (a^2-b^2)r\cdot J_(ar)\cdot J_n(br)=0$$

Integrating above equation with respect to 'r' from 0 to P, we get,

$$or,\;\; \int_0^p\frac{d}{dr}\biggl[rJ_n(br)\cdot\frac{dJ_n(ar)}{dr}-rJ_n(ar)\cdot \frac{dJ_n(br)}{dr}\biggr]dr+(a^2-b^2)\int_0^p r\cdot J_n(ar) \cdot J_n(br)dr=0$$

$$or,\;\; \biggl| r. J_n(br)\cdot \frac{dJ_n(ar)}{dr}-r J_n(ar)\cdot \frac{dJ_n(br)}{dr}\biggr|_0^p+ (a^2-b^2)\int_0^p r\cdot J_n(ar)\cdot J_n(br) dr=0$$

$$or,\;\; (a^2-b^2)\int_0^p r\cdot J_n(ar)\cdot J_n(br)dr=0$$

For a\(\ne\)b

$$\int_0^p rJ_n(ar) J_n(br) dr=0$$

This is the orthogonality property of Bessel differential equation.

Reference:

1. Dass, H. K.Mathematical Physics. New Delhi: S. Chand, 2005.
2. Vaughn, Michael T.Introduction to Mathematical Physics. Weinheim: Wiley-VCH, 2007
3. Butkov, Eugene.Mathematical Physics. Reading, MA: Addison-Wesley Pub., 1968.
4. Carroll, Robert W.Mathematical Physics. Amsterdam: North-Holland, 1988.
5. Adhikari, Pitri Bhakta, and Dya Nidhi Chhatkuli.A Textbook of Physics. Third Revised Edition ed. Vol. III. Kathmandu: SUKUNDA PUSTAK BHAWAN, 2072.