Fdtxjr

This is a my proof of the normality of bessel functions .
But I want to check it and I prefer another proof if exist.
Help me please.
The Bessel function $J_nu(x)$ satisfies the following differential equation
$$quad x^2 fracd^2 J_nud x^2+x fracd J_nud x+left(x^2-nu^2right) J_nu=0$$
which can be written as
$$quad x fracdd xleft(x fracd J_nud xright)+left(x^2-nu^2right) J_nu=0$$
The variable $nu$ need not be an integer. It turns out to be useful to define a new variable $t$ by $x=s t$
where $s$ is a constant which we will take to be a zero of $J_nu,$ i.e. $J_nu(s)=0 .$ Let us define
$u(t)=J_nu(s t)$
which implies
$u(1)=0$
substituting beginequationt fracdd tleft(t fracd ud tright)+left(s^2 t^2-nu^2right) u=0
endequation
since $fracx dd x$ is equivalent to $frac t d d t $. We can also write down the equation obtained by picking another zero, $r$ say. Defining $$v(t)=J_nu(a t) text so quad v(a)=J_nu(ar)$$ we have
beginequationt fracdd tleft(t fracd vd tright)+left(r^2 t^2-nu^2right) v=0
endequation
To derive the orthogonality relation, we multiply Eq. (1) by $v,$ and Eq. (2) by $u$ . Subtracting
and dividing by $t$ gives
beginequationquad v fracdd tleft(t fracd ud tright)-u fracdd tleft(t fracd vd tright)+left(s^2-r^2right) t u v=0
endequation
Hence we have
$$fracdd tleft(v t fracd ud t-u t fracd vd tright)+left(s^2-r^2right) t u v=0$$
We next integrate this over the range of $t$ from 0 to $a,$ which gives
$$left[v t fracd ud t-u t fracd vd tright]_0^a+left(s^2-r^2right) int_0^a t u(t) v(t) d t=0$$
If $s=r$ Define $u=J_nu(s x), quad v=J_nu(r x)$
where $s$ is a zero of $J_nu$ but $r$ is not necessarily a zero.
In the first term the contribution at 0 vanishes because of the factor of $t,$ and the piece $u t v^prime$ also
vanishes at $t=1$ because $u$ is zero there. Hence we have
$$int_0^1 t u(t) v(t) d t=fracJ_p(b) a J_p^prime(a)b^2-a^2$$
where we used the fact that $$u^prime(t) equiv d u / d t=(d / d t) J_p(a t)=a(d / d(a x)) J_p(a x),$$ which is $a J_p^prime(a)$ at
$t=1 .$
Now let $b=a+delta,$ where we will let $delta rightarrow 0 .$ In this limit $b^2-a^2=2 a delta$ and $J_p(b)=J_p(a)+delta J_p^prime(a)=$
$delta J_p^prime(a) .$ Dividing out the factors of $delta$ in the numerator and denominator we get
$$int_0^1 t J_p^2(a t) d t=frac12 J_p^prime 2(a)$$
where $$J_p^prime(a) equiv d J_p(x) / dleft.xright|_x=a$$This can be written in different ways. From Eq. $(10)$ of the handout
on "The differential equation satisfied by Bessel functions" we have $J_p^prime(a)=J_p-1(a)$ (remember
that $J_p(a)=0 ),$ and from Eq. $(11)$ we have $J_p^prime(a)=-J_p+1(a) .$ Hence
$$int_0^1 t J_p^2(a t) d t=frac12 J_p^prime 2(a)=frac12 J_p-1^2(a)=frac12 J_p+1^2(a)$$
where $a$ is a zero of $J_p(x),$ i.e. $J_p(a)=0$