Bessel function of the first kind ($J_{\nu }(z)$)

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 integer $m\ge 0$, $J_{-m}(z)=(-1{)}^m{J}_m(z)$. Thus negative integer orders remain regular at zero. The familiar $J_m(z)\sim z^m/(2^{m}m!)$ formula is used for nonnegative order; a negative factorial is not a valid continuation.

For $x>0$, the Bessel differential equation of order $1/2$ has independent solutions $x^{-1/2}\sin x$ and $x^{-1/2}\cos x$. Substituting $y=x^{-1/2}u$ reduces it to $u^{\prime \prime }+u=0$. The normalization of the Bessel function of the first kind gives the displayed formulas for orders $1/2$ and $-1/2$. The Frobenius method obtains the same pair from exponents $1/2$ and $-1/2$: their even-coefficient series reduce to the sine and cosine power series. In the negative-exponent recurrence, the free coefficient at the next exponent can add the positive-exponent solution, an example of an undetermined coefficient at Frobenius resonance.

With the Fourier transform convention $\hat{f}(\omega )={\int }_{\mathbb{R}}f(t)e^{-i\omega t}dt$, the regular order-zero Bessel function of the first kind has the tempered distribution transformThe density is integrable despite its endpoint singularities. Fourier inversion gives the equivalent integral representationThe second formula follows by $\omega =\cos \theta$ and cancellation of the odd imaginary part. It is the standard real integral representation of $J_0$.^[ref]

Indeed, transforming the order-zero Bessel differential equation $t{y}^{\prime \prime }+y^{\prime }+ty=0$ gives $(1-{\omega }^2)Y^{\prime }-\omega Y=0$. Inside $(-1,1)$, its solutions are $C(1-{\omega }^2{)}^{-1/2}$. The compactly supported choice also solves the equation as a distribution: rewrite it as $[(1-{\omega }^2)Y{]}^{\prime }+\omega Y=0$, and observe that $(1-{\omega }^2)Y=C\sqrt{1-{\omega }^2}{1}_{(-1,1)}$ is continuous and zero at the two endpoints. Its distributional derivative has no endpoint Dirac delta terms. The inverse is regular at $t=0$, and $y(0)=1$ fixes $C=2$.

The transformed equation alone also allows separate constants on $\omega <-1$ and $\omega >1$. Selecting the compactly supported density amounts to selecting the solution regular at the origin, rather than every possible singular solution of the Bessel differential equation.

A Fourier series expansion of a sinusoidally modulated complex wave phase in Bessel functions of the first kind.

For real $z$, the probabilities $J_m(z{)}^2$ over integer angular-momentum modes satisfy ${\sum }_m{m}^2{J}_m(z{)}^2=z^2/2$. Differentiate the unit-modulus function $e^{iz\cos \theta }$ and use Parseval to obtain the identity; its zero first moment follows from the symmetry of opposite orders.

The regular power series has $J_0(\rho )=1-{\rho }^2/4+{\rho }^4/64-\cdots$. Its quartic truncation has a double zero at $\rho =\sqrt{8}$, giving a rough disc-well energy estimate $4{\hslash }^2/(\mu a^2)$. A truncation zero is not an exact Bessel zero or a rigorous eigenvalue bound. The double-root behavior particularly warns against treating a low-order local expansion as accurate far from its center.

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,