A..5 Fourier-Bessel $BE83+(B

Proof. $r\in(0,1)$ $B$r8GDj$7$F!"(B $\theta\mapsto f(r,\theta)$ $B$r9M$($k$H!"(B Fourier $B5i?tE83+(B

$$f(r,\theta)=\frac{a_0(r)}{2} +\sum_{n=1}^\infty (a_n(r)\cos n\theta+b_n(r)\sin n\theta),$$

$$a_n(r):=\frac{1}{\pi}\int_0^{2\pi}f(r,\theta)\cos n\theta \D\theta, \quad b_n(r):=\frac{1}{\pi}\int_0^{2\pi}f(r,\theta)\sin n\theta \D\theta$$

$B$,F@$i$l$k!#(B $a_n(r)$, $b_n(r)$ $B$O(B $(0,1)$ $B>e$N4X?t$@$+$i!"(B Fourier-Bessel $BE83+$,$G$-$k!#(B

$$a_n(r)=\sum_{j=1}^\infty A_{nj}J_n(\mu_{nj}r),\quad A_{nj}:=\frac{2}{J_{n+1}(\mu_{nj})^2}\int_0^1 r a_n(r)J_n(\mu_{nj}r) \D r,$$

$$b_n(r)=\sum_{j=1}^\infty B_{nj}J_n(\mu_{nj}r),\quad B_{nj}:=\frac{2}{J_{n+1}(\mu_{nj})^2}\int_0^1 r b_n(r)J_n(\mu_{nj}r) \D r.$$

$B0J>e$^$H$a$k$HDjM}$rF@$k!#(B $\qedsymbol$ ARRAY(0xea8ae0) $\qedsymbol$