Bessel Approximations in Mathematica

[Dustinsfl]

How do I use the Bessel Function at different orders to approximate the sine function?

I am plotting $\sin\pi x$ against the BesselJ function. However, from the example I saw in class, as I increase the number of terms, the $(0,1)$ coordinate is pulled down to (0,0). This isn't happening for me so I am not entering in my BesselJ correct plotted. How would I do that?

 

[chisigma]

How do I use the Bessel Function at different orders to approximate the sine function?

I am plotting $\sin\pi x$ against the BesselJ function. However, from the example I saw in class, as I increase the number of terms, the $(0,1)$ coordinate is pulled down to (0,0). This isn't happening for me so I am not entering in my BesselJ correct plotted. How would I do that?

Let be $\lambda_{1},\ \lambda_{2},\ ... ,\ \lambda_{k}$ the positive zeroes of $J_{n}(x)$, being n a non negative integer. In that case, given an f(x), is...

$\displaystyle f(x) = \sum_{k=1}^{\infty} a_{k}\ J_{n} (\lambda_{k}\ x)$ (1)

... where...

$\displaystyle a_{k}= \frac{2}{J^{2}_{n+1} (\lambda_{k})} \ \int_{0}^{1} x\ f(x)\ J_{n} (\lambda_{k}\ x)\ dx$ (2)

Kind regards

$\chi$ $\sigma$