Spherical Bessel function ($j_{\ell }(x)=\sqrt{\pi /(2x)}{J}_{\ell +1/2}(x)$)

The spherical Bessel functions solve the radial wave equation in three dimensions. In particular,The apparent singularity at the origin is removable, as follows from the power series of the Bessel function of the first kind.

A scalar field solving the Helmholtz equation inside a sphere of radius $R$ and Laplace's equation outside has a regular interior mode $b{j}_{\ell }(\sqrt{s}r)P_{\ell }(\cos \theta )$ and a decaying exterior mode $a{r}^{-\ell -1}{P}_{\ell }(\cos \theta )$. Continuity of the field and its radial derivative gives, for $x=\sqrt{s}R>0$,The Spherical Bessel function recurrence turns this into $j_{\ell -1}(x)=0$, not $j_{\ell }(x)=0$. This elimination does not require division by a possibly vanishing boundary value.

For $\ell =0$, $j_0(x)=\sin x/x$ and $j_{-1}(x)=\cos x/x$, so the first positive root is $\pi /2$. For $\ell =1$, the matching condition is $j_0(x)=0$, whose first positive root is $\pi$. The distinction matters when a scalar model permits the monopole sector but a physical multipole model excludes it.

To compare angular sectors, put $u=r{j}_{\ell }(\sqrt{s}r)$. Its equation and boundary conditions are $-u^{\prime \prime }+\ell (\ell +1)u/r^2=su$, $u(0)=0$ and $u^{\prime }(R)+\ell u(R)/R=0$. Its Rayleigh quotient isBoth added terms are nonnegative and increase with $\ell$. Thus the lowest scalar eigenvalue is ${\pi }^2/(4R^2)$ in the $\ell =0$ sector; if only $\ell \ge 1$ is admitted, the lowest is ${\pi }^2/R^2$ in the $\ell =1$ sector. Zeros at $x=0$ caused by the vanishing normalization of $j_{\ell }$ are not additional eigenvalues.

For $x^2{y}^{\prime \prime }+2x{y}^{\prime }+[x^2-\ell (\ell +1)]y=0$ with integer $\ell \ge 0$, the Frobenius method gives $\sigma =\ell$ or $-\ell -1$. The general coefficient equation is $[(j+\sigma )(j+\sigma +1)-\ell (\ell +1)]a_j=-a_{j-2}$, taking negative-index coefficients as zero. The regular branch has $a_1=0$ and $a_j=-a_{j-2}/[j(j+2\ell +1)]$. The singular branch has $a_j=-a_{j-2}/[j(j-2\ell -1)]$ except at $j=2\ell +1$. That resonant equation is compatible because its earlier odd coefficient vanishes; the newly free coefficient adds a multiple of the regular solution. Setting it to zero gives a pure even series times $x^{-\ell -1}$, independent of the regular branch, with no logarithmic term. For $\ell =0$ the resonance occurs already at $j=1$.

The normalization $a_0=2^{\ell }\ell !/(2\ell +1)!$ on the regular branch gives the Spherical Bessel functionIn particular $j_0(x)=\sin x/x$, with removable value one at zero.

For integer $\ell \ge 1$, put $I_{\ell }={\int }_0^{\infty }{j}_{\ell }(x{)}^{2}dx/x$ and $J_{\ell }={\int }_0^{\infty }{j}_{\ell }(x)j_{\ell -1}(x)dx$. Integrating $j_{\ell }^{\prime }=j_{\ell -1}-(\ell +1)j_{\ell }/x$ gives $J_{\ell }=(\ell +1)I_{\ell }$. For $\ell >1$, integrating $j_{\ell -1}^{\prime }=(\ell -1)j_{\ell -1}/x-j_{\ell }$ also gives $J_{\ell }=(\ell -1)I_{\ell -1}$. For $\ell =1$, the boundary values $j_0(0)=1$ and $j_0(\infty )=0$ give $J_1=1/2$, hence $I_1=1/4$. The recurrence proves the displayed identity. The integral without the square is different; for example ${\int }_0^{\infty }{j}_1(x)dx/x=\pi /4$.

These combinations of Spherical Bessel functions solve the radial Helmholtz equation. For time dependence $e^{-i\omega t}$, $h_n^{(1)}(kr)$ is outgoing and $h_n^{(2)}(kr)$ incoming. At fixed order and large argument, $h_n^{(1)}(t)\sim (-i{)}^{n+1}{e}^{it}/t$. Their recurrence is $h_{n+1}=(2n+1)h_n/t-h_{n-1}$.

For fixed $t>0$ and large integer order, $y_n(t)\sim -(2n-1)!!/t^{n+1}$ while $j_n(t)$ is negligible. Stirling's formula gives $(2n-1)!!=(2n)!/(2^{n}n!)\sim \sqrt{2}(2n/e{)}^n$, proving the displayed asymptotic. In particular its magnitude is bounded above and below by constant multiples of $(2n/(et){)}^n$. This two-sided comparison is stronger than a one-sided big-O estimate and is what converts a finite spherical trace norm into a weighted condition on far-field coefficients.

For integer $\ell \ge 0$ and $r,R>0$, $r_{<}=\min (r,R)$ and $r_{>}=\max (r,R)$. A proof compares the multipole expansion of $1/|\mathbf{r}-\mathbf{R}|$ with its Fourier transform of inverse distance in three dimensions. Applying the Rayleigh plane-wave expansion to the Fourier representation gives coefficient $8{\int }_0^{\infty }{j}_{\ell }(kr)j_{\ell }(kR)dk$ multiplying $Y_{\ell m}(\hat{r})Y_{\ell m}^{\ast }(\hat{R})$; the Coulomb multipole coefficient is $4\pi r_{<}^{\ell }/[(2\ell +1)r_{>}^{\ell +1}]$. Equating the coefficients proves the identity. The powers also match continuously at $r=R$.