Solving Bessel Functions Homework Questions

[skrat]

Homework Statement

Calculate:
a) $\frac{d}{dx}(xJ_1(x)-\int _0^xtJ_0(t)dt)$
b) $xJ_1(x)-\int _0^xtJ_0(t)dt$
c) let $\xi _{k0}$ be the $k$ zero of a function $J_0$. Determine $c_k$ so that $1=\sum _{k=1}^{\infty }c_kJ_0(\frac{x\xi _{k0}}{2})$.

Homework Equations

The Attempt at a Solution

a) $\frac{d}{dx}(xJ_1(x)-\int _0^xtJ_0(t)dt)=xJ_0(x)-xJ_0(x)=0$.

b) What do I do with the integral? Should I calculate $J_n(x)=\frac{1}{\pi }\int _0^{\pi }cos(tsin\varphi -n\varphi)d\varphi$ for n=0?

c) Hmmm, no idea here :/

 

[Curious3141]

Homework Statement

Calculate:
a) $\frac{d}{dx}(xJ_1(x)-\int _0^xtJ_0(t)dt)$
b) $xJ_1(x)-\int _0^xtJ_0(t)dt$
c) let $\xi _{k0}$ be the $k$ zero of a function $J_0$. Determine $c_k$ so that $1=\sum _{k=1}^{\infty }c_kJ_0(\frac{x\xi _{k0}}{2})$.

Homework Equations

The Attempt at a Solution

a) $\frac{d}{dx}(xJ_1(x)-\int _0^xtJ_0(t)dt)=xJ_0(x)-xJ_0(x)=0$.

b) What do I do with the integral? Should I calculate $J_n(x)=\frac{1}{\pi }\int _0^{\pi }cos(tsin\varphi -n\varphi)d\varphi$ for n=0?

c) Hmmm, no idea here :/

I'm no expert in the theory of Bessel functions, but isn't the expression in part b) just the integral of the entire expression in a) wrt x? Integrating 0 gives you a constant. The constant can easily be found by subbing in a suitable value of x, right?

c) exceeds my knowledge, someone else will have to help, sorry.