Bessel function

The two Hankel functions $H_{\nu }^{(1)}=J_{\nu }+i{Y}_{\nu }$ and $H_{\nu }^{(2)}=J_{\nu }-i{Y}_{\nu }$ solve the Bessel differential equation. Their large-argument forms distinguish outgoing and incoming oscillations and have sector-dependent asymptotic expansions.

For fixed order, the standard Hankel function expansions in $e^{\pm iz}{z}^{-1/2}$ are valid in different sectors on a chosen argument sheet. Analytic continuation can mix exponentials; the sectors must be specified rather than treating a formal inverse-power series as global.

For noninteger order, $Y_{\nu }=(\cos (\pi \nu )J_{\nu }-J_{-\nu })/\sin (\pi \nu )$; the integer-order function is defined by a limit. It gives a second independent solution of the Bessel differential equation. At order zero, $Y_0(z)\sim (2/\pi )[\log (z/2)+\gamma ]$ near positive zero and $Y_0(z)\sim \sqrt{2/(\pi z)}\sin (z-\pi /4)$ at large positive argument, where $\gamma$ is the Euler--Mascheroni constant.

The spherical second-kind function is $n_{\ell }(z)=\sqrt{\pi /(2z)}{Y}_{\ell +1/2}(z)$. Together with the first-kind Spherical Bessel function, it spans the radial free-wave solutions; $n_{\ell }(z)\sim -\cos (z-\ell \pi /2)/z$ at large positive $z$.

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.

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 $r=R$.

The Bessel function of the first kind $J_{\nu }$ is the solution of the Bessel differential equation selected by its regular power-series behavior at the origin. For $\mathrm{Re}\nu >-1/2$, it has the Poisson integral representation

For fixed $0<\alpha <\pi /2$, the Bessel function of the first kind has an interior nondegenerate stationary point in its cosine integral. The stationary phase method gives the displayed oscillatory leading term and an additive $O(n^{-3/2})$ remainder. The approximation loses uniformity near $\alpha =0$, where the stationary point reaches the endpoint and the curvature vanishes. For negative integer-order arguments, parity supplies a factor $(-1{)}^n$.

For fixed $\nu$, putThenas $|z|\to \infty$, uniformly for $|\arg z|\le \pi -\delta$. In particular,

The Bessel differential equation of order $\nu$ isFor noninteger $\nu$, $J_{\nu }$ and $J_{-\nu }$ form a linearly independent pair of solutions.

The modified Bessel equation of order zero isIts solution regular at zero and normalized by $I_0(0)=1$ is the modified Bessel function $I_0$.

For fixed real $\nu$ as $z\to +\infty$, the Modified Bessel function of the first kind and Modified Bessel function of the second kind have asymptotic expansionsThe expansion and its error bounds establish actual growing and decaying solutions, beyond formal matching of a differential equation. These formulas are documented in NIST DLMF, equations 10.40.1–2.

The solution of the Modified Bessel differential equation with power series $I_{\nu }(z)={\sum }_{j\ge 0}(z/2{)}^{2j+\nu }/[j!\Gamma (j+\nu +1)]$, interpreted through the reciprocal gamma function at its zeros. For fixed real order it grows exponentially along the positive real axis.

The differential equation defining modified Bessel functions of order $\nu$. For real fixed $\nu$ and positive $z$, its independent standard solutions are $I_{\nu }(z)$ and $K_{\nu }(z)$.

The modified Bessel function of the second kind is the solution $K_{\nu }(x)$ of the modified Bessel differential equation that decays exponentially as $x\to +\infty$.

For real $x>0$ and $\nu$, one useful representation isIt turns large-order asymptotics into a Laplace integral with a moving maximum.

For fixed $x>0$,as $\nu \to +\infty$. Applying Laplace's method to the integral representation of the modified Bessel function of the second kind proves the formula.

Laplace's integral method seeks $y(z)={\int }_C{e}^{zt}f(t)dt$. Substitution into the differential equation and integration by parts transfer multiplication by $z$ into differentiation with respect to $t$, producing a first-order equation for $f$ and endpoint conditions for $C$.

For $x>0$, substituting $y(x)={\int }_{\gamma }{e}^{xt}f(t)dt$ into $x{y}^{(3)}+2y=0$ and integrating by parts givesThe amplitude equation $(t^{3}f{)}^{\prime }=2f$ has solution $f(t)=C{t}^{-3}{e}^{-1/t^2}$. On $\gamma =(-\infty ,0)$ both endpoint terms vanish, and hence

For the Laguerre equationthe ansatz $y={\int }_{\gamma }{e}^{zt}f(t)dt$ givesThe contour must make the endpoint contributionvanish.

A finite contour based at an integrable branch point, or a closed contour around an integer-order pole, gives a solution analytic at $z=0$. It is therefore a constant multiple of the unique Frobenius power-series solution there.

For $\lambda =N\ge 0$, the amplitude $(t-1{)}^N/t^{N+1}$ has a pole at zero whose contour residue is a degree-$N$ Laguerre polynomial in $z$. For $\lambda =-m<0$, the amplitude $t^{m-1}/(t-1{)}^m$ has a pole at one whose residue is $e^z$ times a polynomial of degree $m-1$.

The normalized order-zero function has the representation