1
$\begingroup$

I have the integral:

$$\displaystyle\int_{0}^{r_{0}}\displaystyle\int_{\pi-\arctan(\frac{r_{ob}}{r_{2}})}^{\pi+\arctan(\frac{r_{ob}}{r_{2}})}r_{1}\exp(-\beta(r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta)))d\theta dr_{1}$$

and was hoping to use the modified Bessel function of the first kind to simplify it (the mod. Bessel function 1st kind is:

$$I_{0}(x)=\frac{1}{\pi}\displaystyle\int_{0}^{\pi}\exp(x\cos\theta)d\theta$$

If I use the argument $x=2r_{1}r_{2}\beta$ can I use this integral in order to simplify the first? i.e. some kind to linear scaling of the Bessel function? I'm not sure, might I get something like

$$a\int_{0}^{r_{0}}r_{1}\left[I_{0}(x)\right]\exp(-\beta(r_{1}^{2}+r_{2}^{2}))dr_{1}$$

Alex

$\endgroup$
1
$\begingroup$

$\int_0^{r_0}\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}r_1e^{-\beta(r_1^2+r_2^2-2r_1r_2\cos\theta)}~d\theta~dr_1$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\int_0^{r_0}r_1e^{-\beta(r_1^2-2r_1r_2\cos\theta+r_2^2)}~dr_1~d\theta$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\int_0^{r_0}r_1e^{-\beta(r_1^2-2r_1r_2\cos\theta+r_2^2\cos^2\theta+r_2^2-r_2^2\cos^2\theta)}~dr_1~d\theta$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\int_0^{r_0}r_1e^{-\beta((r_1-r_2\cos\theta)^2+r_2^2\sin^2\theta)}~dr_1~d\theta$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\int_{-r_2\cos\theta}^{r_0-r_2\cos\theta}(r_1+r_2\cos\theta)e^{-\beta(r_1^2+r_2^2\sin^2\theta)}~dr_1~d\theta$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\int_{-r_2\cos\theta}^{r_0-r_2\cos\theta}r_1e^{-\beta(r_1^2+r_2^2\sin^2\theta)}~dr_1~d\theta+\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\int_{-r_2\cos\theta}^{r_0-r_2\cos\theta}r_2\cos\theta~e^{-\beta(r_1^2+r_2^2\sin^2\theta)}~dr_1~d\theta$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\left[-\dfrac{e^{-\beta(r_1^2+r_2^2\sin^2\theta)}}{2\beta}\right]_{-r_2\cos\theta}^{r_0-r_2\cos\theta}~d\theta+\int_{-r_2\cos\theta}^{r_0-r_2\cos\theta}\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}r_2\cos\theta~e^{-\beta(r_1^2+r_2^2\sin^2\theta)}~d\theta~dr_1$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\dfrac{e^{-\beta((-r_2\cos\theta)^2+r_2^2\sin^2\theta)}}{2\beta}d\theta-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\dfrac{e^{-\beta((r_0-r_2\cos\theta)^2+r_2^2\sin^2\theta)}}{2\beta}d\theta+\int_{-r_2\cos\theta}^{r_0-r_2\cos\theta}\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}r_2~e^{-\beta(r_1^2+r_2^2\sin^2\theta)}~d(\sin\theta)~dr_1$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\dfrac{e^{-\beta(r_2^2\cos^2\theta+r_2^2\sin^2\theta)}}{2\beta}d\theta-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\dfrac{e^{-\beta(r_0^2-2r_0r_2\cos\theta+r_2^2\cos^2\theta+r_2^2\sin^2\theta)}}{2\beta}d\theta+\int_{-r_2\cos\theta}^{r_0-r_2\cos\theta}\int_{\sin\left(\pi-\tan^{-1}\frac{r_{ob}}{r_2}\right)}^{\sin\left(\pi+\tan^{-1}\frac{r_{ob}}{r_2}\right)}r_2~e^{-\beta(r_1^2+r_2^2\theta^2)}~d\theta~dr_1$

$=\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\dfrac{e^{-\beta r_2^2}}{2\beta}d\theta-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\dfrac{e^{-\beta(r_0^2+r_2^2-2r_0r_2\cos\theta)}}{2\beta}d\theta+r_2\int_{-r_2\cos\theta}^{r_0-r_2\cos\theta}e^{-\beta r_1^2}~dr_1\int_{\sin\left(\pi-\tan^{-1}\frac{r_{ob}}{r_2}\right)}^{\sin\left(\pi+\tan^{-1}\frac{r_{ob}}{r_2}\right)}e^{-\beta r_2^2\theta^2}~d\theta$

$=\left[\dfrac{\theta e^{-\beta r_2^2}}{2\beta}\right]_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\dfrac{e^{-\beta(r_0^2+r_2^2)}e^{2\beta r_0r_2\cos\theta}}{2\beta}d\theta+\int_{-r_2\cos\theta}^{r_0-r_2\cos\theta}\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^nr_1^{2n}}{n!}dr_1\int_{\sin\tan^{-1}\frac{r_{ob}}{r_2}}^{-\sin\tan^{-1}\frac{r_{ob}}{r_2}}\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^nr_2^{2n+1}\theta^{2n}}{n!}d\theta$

$=\dfrac{\left(\pi+\tan^{-1}\dfrac{r_{ob}}{r_2}-\left(\pi-\tan^{-1}\dfrac{r_{ob}}{r_2}\right)\right)e^{-\beta r_2^2}}{2\beta}-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\dfrac{e^{-\beta(r_0^2+r_2^2)}}{2\beta}\left(\sum\limits_{n=0}^\infty\dfrac{2^{2n}\beta^{2n}r_0^{2n}r_2^{2n}\cos^{2n}\theta}{(2n)!}+\sum\limits_{n=0}^\infty\dfrac{2^{2n+1}\beta^{2n+1}r_0^{2n+1}r_2^{2n+1}\cos^{2n+1}\theta}{(2n+1)!}\right)d\theta+\left[\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^nr_1^{2n+1}}{n!(2n+1)}\right]_{-r_2\cos\theta}^{r_0-r_2\cos\theta}\left[\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^nr_2^{2n+1}\theta^{2n+1}}{n!(2n+1)}\right]_{\frac{r_{ob}}{\sqrt{r_{ob}^2+r_2^2}}}^{-\frac{r_{ob}}{\sqrt{r_{ob}^2+r_2^2}}}$

$=\dfrac{e^{-\beta r_2^2}}{\beta}\tan^{-1}\dfrac{r_{ob}}{r_2}-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\sum\limits_{n=0}^\infty\dfrac{2^{2n-1}\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}\cos^{2n}\theta}{(2n)!}d\theta-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\sum\limits_{n=0}^\infty\dfrac{4^n\beta^{2n}r_0^{2n+1}r_2^{2n+1}e^{-\beta(r_0^2+r_2^2)}\cos^{2n+1}\theta}{(2n+1)!}d\theta-\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^n((r_0-r_2\cos\theta)^{2n+1}+r_2^{2n+1}\cos^{2n+1}\theta)}{n!(2n+1)}\sum\limits_{n=0}^\infty\dfrac{2(-1)^n\beta^nr_2^{2n+1}r_{ob}^{2n+1}}{n!(2n+1)(r_{ob}^2+r_2^2)^{n+\frac{1}{2}}}$

For $\int\cos^{2n}\theta~d\theta$ , where $n$ is any non-negative integer,

$\int\cos^{2n}\theta~d\theta=\dfrac{(2n)!\theta}{4^n(n!)^2}+\sum\limits_{k=1}^n\dfrac{(2n)!((k-1)!)^2\sin\theta~\cos^{2k-1}\theta}{4^{n-k+1}(n!)^2(2k-1)!}+C$

This result can be done by successive integration by parts, e.g. as shown as http://hk.knowledge.yahoo.com/question/question?qid=7012022000808

For $\int\cos^{2n+1}\theta~d\theta$ , where $n$ is any non-negative integer,

$\int\cos^{2n+1}\theta~d\theta$

$=\int\cos^{2n}\theta~d(\sin\theta)$

$=\int(1-\sin^2\theta)^n~d(\sin\theta)$

$=\int\sum\limits_{k=0}^nC_k^n(-1)^k\sin^{2k}\theta~d(\sin\theta)$

$=\sum\limits_{k=0}^n\dfrac{(-1)^kn!\sin^{2k+1}\theta}{k!(n-k)!(2k+1)}+C$

$\therefore\dfrac{e^{-\beta r_2^2}}{\beta}\tan^{-1}\dfrac{r_{ob}}{r_2}-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\sum\limits_{n=0}^\infty\dfrac{2^{2n-1}\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}\cos^{2n}\theta}{(2n)!}d\theta-\int_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}\sum\limits_{n=0}^\infty\dfrac{4^n\beta^{2n}r_0^{2n+1}r_2^{2n+1}e^{-\beta(r_0^2+r_2^2)}\cos^{2n+1}\theta}{(2n+1)!}d\theta-\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^n((r_0-r_2\cos\theta)^{2n+1}+r_2^{2n+1}\cos^{2n+1}\theta)}{n!(2n+1)}\sum\limits_{n=0}^\infty\dfrac{2(-1)^n\beta^nr_2^{2n+1}r_{ob}^{2n+1}}{n!(2n+1)(r_{ob}^2+r_2^2)^{n+\frac{1}{2}}}$

$=\dfrac{e^{-\beta r_2^2}}{\beta}\tan^{-1}\dfrac{r_{ob}}{r_2}-\left[\sum\limits_{n=0}^\infty\dfrac{\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}\theta}{2(n!)^2}+\sum\limits_{n=0}^\infty\sum\limits_{k=1}^n\dfrac{2^{2k-3}\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}((k-1)!)^2\sin\theta~\cos^{2k-1}\theta}{(n!)^2(2k-1)!}\right]_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}-\left[\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^k4^n\beta^{2n}r_0^{2n+1}r_2^{2n+1}e^{-\beta(r_0^2+r_2^2)}n!\sin^{2k+1}\theta}{(2n+1)!k!(n-k)!(2k+1)}\right]_{\pi-\tan^{-1}\frac{r_{ob}}{r_2}}^{\pi+\tan^{-1}\frac{r_{ob}}{r_2}}-\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^n((r_0-r_2\cos\theta)^{2n+1}+r_2^{2n+1}\cos^{2n+1}\theta)}{n!(2n+1)}\sum\limits_{n=0}^\infty\dfrac{2(-1)^n\beta^nr_2^{2n+1}r_{ob}^{2n+1}}{n!(2n+1)(r_{ob}^2+r_2^2)^{n+\frac{1}{2}}}$

$=\dfrac{e^{-\beta r_2^2}}{\beta}\tan^{-1}\dfrac{r_{ob}}{r_2}-\sum\limits_{n=0}^\infty\dfrac{\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}\left(\pi+\tan^{-1}\dfrac{r_{ob}}{r_2}-\left(\pi-\tan^{-1}\dfrac{r_{ob}}{r_2}\right)\right)}{2(n!)^2}-\sum\limits_{n=0}^\infty\sum\limits_{k=1}^n\dfrac{2^{2k-3}\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}((k-1)!)^2\left(\sin\left(\pi+\tan^{-1}\dfrac{r_{ob}}{r_2}\right)\cos^{2k-1}\left(\pi+\tan^{-1}\dfrac{r_{ob}}{r_2}\right)-\sin\left(\pi-\tan^{-1}\dfrac{r_{ob}}{r_2}\right)\cos^{2k-1}\left(\pi-\tan^{-1}\dfrac{r_{ob}}{r_2}\right)\right)}{(n!)^2(2k-1)!}-\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^k4^n\beta^{2n}r_0^{2n+1}r_2^{2n+1}e^{-\beta(r_0^2+r_2^2)}n!\left(\sin^{2k+1}\left(\pi+\tan^{-1}\dfrac{r_{ob}}{r_2}\right)-\sin^{2k+1}\left(\pi-\tan^{-1}\dfrac{r_{ob}}{r_2}\right)\right)}{(2n+1)!k!(n-k)!(2k+1)}-\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^n((r_0-r_2\cos\theta)^{2n+1}+r_2^{2n+1}\cos^{2n+1}\theta)}{n!(2n+1)}\sum\limits_{n=0}^\infty\dfrac{2(-1)^n\beta^nr_2^{2n+1}r_{ob}^{2n+1}}{n!(2n+1)(r_{ob}^2+r_2^2)^{n+\frac{1}{2}}}$

$=\dfrac{e^{-\beta r_2^2}}{\beta}\tan^{-1}\dfrac{r_{ob}}{r_2}-\sum\limits_{n=0}^\infty\dfrac{\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}}{(n!)^2}\tan^{-1}\dfrac{r_{ob}}{r_2}-\sum\limits_{n=0}^\infty\sum\limits_{k=1}^n\dfrac{2^{2k-3}\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}((k-1)!)^2\left(\sin\tan^{-1}\dfrac{r_{ob}}{r_2}\cos^{2k-1}\tan^{-1}\dfrac{r_{ob}}{r_2}+\sin\tan^{-1}\dfrac{r_{ob}}{r_2}\cos^{2k-1}\tan^{-1}\dfrac{r_{ob}}{r_2}\right)}{(n!)^2(2k-1)!}-\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^k4^n\beta^{2n}r_0^{2n+1}r_2^{2n+1}e^{-\beta(r_0^2+r_2^2)}n!\left(-\sin^{2k+1}\tan^{-1}\dfrac{r_{ob}}{r_2}-\sin^{2k+1}\tan^{-1}\dfrac{r_{ob}}{r_2}\right)}{(2n+1)!k!(n-k)!(2k+1)}-\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^n((r_0-r_2\cos\theta)^{2n+1}+r_2^{2n+1}\cos^{2n+1}\theta)}{n!(2n+1)}\sum\limits_{n=0}^\infty\dfrac{2(-1)^n\beta^nr_2^{2n+1}r_{ob}^{2n+1}}{n!(2n+1)(r_{ob}^2+r_2^2)^{n+\frac{1}{2}}}$

$=\dfrac{e^{-\beta r_2^2}}{\beta}\tan^{-1}\dfrac{r_{ob}}{r_2}-\sum\limits_{n=0}^\infty\dfrac{\beta^{2n-1}r_0^{2n}r_2^{2n}e^{-\beta(r_0^2+r_2^2)}}{(n!)^2}\tan^{-1}\dfrac{r_{ob}}{r_2}-\sum\limits_{n=0}^\infty\sum\limits_{k=1}^n\dfrac{4^{k-1}\beta^{2n-1}r_0^{2n}r_2^{2n+2k-1}r_{ob}e^{-\beta(r_0^2+r_2^2)}((k-1)!)^2}{(n!)^2(2k-1)!(r_{ob}^2+r_2^2)^k}+\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^k2^{2n+1}\beta^{2n}r_0^{2n+1}r_2^{2n+1}r_{ob}^{2k+1}e^{-\beta(r_0^2+r_2^2)}n!}{(2n+1)!k!(n-k)!(2k+1)(r_{ob}^2+r_2^2)^{k+\frac{1}{2}}}-\sum\limits_{n=0}^\infty\dfrac{(-1)^n\beta^n((r_0-r_2\cos\theta)^{2n+1}+r_2^{2n+1}\cos^{2n+1}\theta)}{n!(2n+1)}\sum\limits_{n=0}^\infty\dfrac{2(-1)^n\beta^nr_2^{2n+1}r_{ob}^{2n+1}}{n!(2n+1)(r_{ob}^2+r_2^2)^{n+\frac{1}{2}}}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.