- #1
- 2,802
- 605
Before stating the main question,which section should the special functions' questions be asked?
Now consider the Bessel differential equation:
<br /> \rho \frac{d^2}{d\rho^2}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+(\frac{\alpha_{\nu m}^2 \rho}{a^2}-\frac{\nu^2}{\rho})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})=0<br />
and a similar equation but with \alpha_{\nu m} replaced by \alpha_{\nu n} where \alpha_{\nu s} is the sth root of J_{\nu}(x).
Now if one multiplies the first equation by J_{\nu}(\alpha_{\nu n} \frac{\rho}{a}) and the second by J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) and then subtracts the second from the first,the following will be found upon integration of the whole equation from 0 to a:
<br /> \int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\frac{d}{d\rho}[\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]d\rho-\int_0^a J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\frac{d}{d\rho}[\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})]d\rho=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho<br />
Integrating the LHS by part and cancelling gives:
<br /> J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})|_0^a-J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})|_0^a=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho<br />
Using \frac{d}{dx}J_n(x)=\frac{n}{x}J_n(x)-J_{n+1}(x):
<br /> J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\rho[\frac{\nu a}{\alpha_{\nu m}\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})-J_{\nu+1}(\alpha_{\nu m}\frac{\rho}{a})]|_0^a-J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho[\frac{\nu a}{\alpha_{\nu n}\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})-J_{\nu+1}(\alpha_{\nu n}\frac{\rho}{a})]|_0^a=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho<br />
After placing \alpha_{\nu n}=\alpha_{\nu m}+\varepsilon and taking the limit as \varepsilon\rightarrow 0 and using \frac{d}{dx}J_n(x)=\frac{n}{x}J_n(x)-J_{n+1}(x) to replace terms involving Js with \varepsilon in their arguments and calculating the terms in the boundaries:
<br /> -J_{\nu+1}(\alpha_{\nu m})\varepsilon[-aJ_{\nu+1}(\alpha_{\nu m})](\alpha_{\nu m})=\frac{2\alpha_{\nu m} \varepsilon}{a^2}\int_0^a J^2_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho d\rho<br />
Which gives:
<br /> \int_0^a [J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]^2\rho d\rho=\frac{a^3}{2\alpha_{\nu m}}[J_{\nu+1}(\alpha_{\nu m})]^2<br />
But the correct equation is:
<br /> \int_0^a [J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]^2\rho d\rho=\frac{a^2}{2}[J_{\nu+1}(\alpha_{\nu m})]^2<br />
(This is what you find about normalization of Bessel functions everywhere)
What's wrong in my calculations?
Thanks
Now consider the Bessel differential equation:
<br /> \rho \frac{d^2}{d\rho^2}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+(\frac{\alpha_{\nu m}^2 \rho}{a^2}-\frac{\nu^2}{\rho})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})=0<br />
and a similar equation but with \alpha_{\nu m} replaced by \alpha_{\nu n} where \alpha_{\nu s} is the sth root of J_{\nu}(x).
Now if one multiplies the first equation by J_{\nu}(\alpha_{\nu n} \frac{\rho}{a}) and the second by J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) and then subtracts the second from the first,the following will be found upon integration of the whole equation from 0 to a:
<br /> \int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\frac{d}{d\rho}[\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]d\rho-\int_0^a J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\frac{d}{d\rho}[\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})]d\rho=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho<br />
Integrating the LHS by part and cancelling gives:
<br /> J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})|_0^a-J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho\frac{d}{d\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})|_0^a=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho<br />
Using \frac{d}{dx}J_n(x)=\frac{n}{x}J_n(x)-J_{n+1}(x):
<br /> J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})\rho[\frac{\nu a}{\alpha_{\nu m}\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})-J_{\nu+1}(\alpha_{\nu m}\frac{\rho}{a})]|_0^a-J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho[\frac{\nu a}{\alpha_{\nu n}\rho}J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})-J_{\nu+1}(\alpha_{\nu n}\frac{\rho}{a})]|_0^a=\frac{\alpha_{\nu n}^2-\alpha_{\nu m}^2}{a^2}\int_0^a J_{\nu}(\alpha_{\nu n} \frac{\rho}{a})J_{\nu}(\alpha_{\nu m} \frac{\rho}{a}) \rho d\rho<br />
After placing \alpha_{\nu n}=\alpha_{\nu m}+\varepsilon and taking the limit as \varepsilon\rightarrow 0 and using \frac{d}{dx}J_n(x)=\frac{n}{x}J_n(x)-J_{n+1}(x) to replace terms involving Js with \varepsilon in their arguments and calculating the terms in the boundaries:
<br /> -J_{\nu+1}(\alpha_{\nu m})\varepsilon[-aJ_{\nu+1}(\alpha_{\nu m})](\alpha_{\nu m})=\frac{2\alpha_{\nu m} \varepsilon}{a^2}\int_0^a J^2_{\nu}(\alpha_{\nu m} \frac{\rho}{a})\rho d\rho<br />
Which gives:
<br /> \int_0^a [J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]^2\rho d\rho=\frac{a^3}{2\alpha_{\nu m}}[J_{\nu+1}(\alpha_{\nu m})]^2<br />
But the correct equation is:
<br /> \int_0^a [J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})]^2\rho d\rho=\frac{a^2}{2}[J_{\nu+1}(\alpha_{\nu m})]^2<br />
(This is what you find about normalization of Bessel functions everywhere)
What's wrong in my calculations?
Thanks