Bessel's Equation, Legendre Polynomials, and Special Functions

Two equations carry the Frobenius method into physics. Bessel's equation governs waves and diffusion in circular and cylindrical geometry (the vibrating drumhead, heat in a cylinder, the hanging chain), and its indicial roots $\pm \nu$ realize every case of the three-case theorem as $\nu$ varies. Legendre's equation plays the same role for spheres, and its bounded solutions are polynomials. Both families carry an orthogonality relation, and that is what makes them usable: an arbitrary function expands in Bessel functions or Legendre polynomials exactly as it expands in sines and cosines. One prerequisite comes first, a factorial that accepts non-integer arguments.

The gamma function

The Frobenius series for Bessel's equation of order $\nu$ has coefficients like $1/(n!(\nu +n)!)$, which are meaningless when $\nu$ is not an integer unless the factorial is extended.

The one non-obvious value worth memorizing comes from the Gaussian integral: substituting $t=s^2$,

Because $1/\Gamma$ vanishes at the poles, the convention $1/(-m)!=0$ for positive integers $m$ is consistent, and it is the convention the Bessel series below requires.¹

Bessel's equation

The gap between the exponents is $2\nu$, so the parameter sweeps through every case of the Frobenius theorem: equal roots at $\nu =0$, an integer gap at $\nu =\frac{1}{2},1,\frac{3}{2},\dots$, and the generic case otherwise.²

The first solution: $J_{\nu }$

Running the recurrence for the larger exponent $r=\nu$ kills the odd coefficients and steps the even ones by

which telescopes into factorials. With the conventional normalization $a_0=1/(2^{\nu }\Gamma (\nu +1))$:

$J_0$ and $J_1$ resemble damped versions of cosine and sine, and the resemblance is quantitative. Substituting $u=\sqrt{x}y$ transforms Bessel's equation into

which for large $x$ is approximately $u^{\prime \prime }+u=0$; undoing the substitution suggests $y\approx x^{-1/2}(\text{sinusoid})$, and a more careful analysis confirms it:³

with error $O(x^{-3/2})$. So every $J_{\nu }$ oscillates forever with amplitude decaying like $x^{-1/2}$, and has infinitely many positive zeros, asymptotically $\pi$ apart.

The first few positive zeros, which recur constantly in applications:

The zeros interlace: between consecutive positive zeros of $J_0$ there is exactly one zero of $J_1$, and conversely. This is an immediate consequence of the derivative identities

plus Rolle's theorem.⁴

The second solution: three cases realized

The second solution depends on the exponent gap $2\nu$, and Bessel's equation exhibits each behavior of the general theorem.²

• $2\nu$ not an integer. Both exponents give Frobenius series, and replacing $\nu$ by $-\nu$ in the series (the gamma function keeps it meaningful) yields an independent solution: $$J_{-\nu }(x)=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{n!\Gamma (n-\nu +1)}{\left(\frac{x}{2}\right)}^{2n-\nu },y=c_1{J}_{\nu }+c_2{J}_{-\nu }.$$ The leading term $x^{-\nu }$ makes $J_{-\nu }$ unbounded at the origin, so independence is visible at a glance.
• $\nu =\frac{1}{2}$ (integer gap, no logarithm). The exponents $\pm \frac{1}{2}$ differ by $1$, but the recurrence at the smaller root never fails: $a_0$ and $a_1$ both remain free, the constant $a$ of the theorem is $0$, and the solutions are elementary, $$J_{1/2}(x)=\sqrt{\frac{2}{\pi x}}\sin x,J_{-1/2}(x)=\sqrt{\frac{2}{\pi x}}\cos x.$$ The asymptotic sinusoid-over-$\sqrt{x}$ form is exact at half-integer orders.
• $\nu =0$ (equal roots, logarithm forced). The differentiated-coefficient method gives the second solution $$y_2(x)=J_0(x)\ln x+\sum _{m=1}^{\infty }\frac{(-1{)}^{m+1}{H}_m}{2^{2m}(m!{)}^2}{x}^{2m},H_m=1+\frac{1}{2}+\cdots +\frac{1}{m}.$$
• $\nu =1$ (integer gap, logarithm present). Here $J_{-1}=-J_1$ collapses onto the first solution — for any integer $m$, in fact, $J_{-m}=(-1{)}^m{J}_m$, because the gamma factors $1/(-m+n)!$ vanish for $n<m$ — so a genuinely new solution must carry the log: $$y_2(x)=-J_1(x)\ln x+\frac{1}{x}\left(1-\sum _{m=1}^{\infty }\frac{(-1{)}^m(H_m+H_{m-1})}{2^{2m}m!(m-1)!}{x}^{2m}\right).$$

In practice these raw $y_2$'s are replaced by a standard choice. The Bessel function of the second kind $Y_{\nu }$ is the combination

chosen so that the limit at integer orders exists and so that the large-$x$ behavior pairs with $J_{\nu }$: $Y_{\nu }(x)\approx \sqrt{2/(\pi x)}\sin (x-\pi /4-\nu \pi /2)$. For every $\nu \ge 0$,

is the general solution.⁵ The two behave oppositely at the origin: $J_{\nu }$ is bounded there ($J_0(0)=1$, $J_{\nu }(0)=0$ for $\nu >0$) while $Y_{\nu }$ always blows up ($Y_0$ like $(2/\pi )\ln x$, $Y_1$ like $-2/(\pi x)$). Any physical problem that requires a solution bounded on the axis of symmetry therefore discards $Y_{\nu }$ outright, exactly as boundedness selected solutions in the Frobenius examples.

Many equations are Bessel's equation in disguise. The change of variables $z=a{x}^b$, $w=y{x}^c$ maps Bessel's equation in $(z,w)$ onto a two-parameter family in $(x,y)$; a notable member is Airy's equation $y^{\prime \prime }+xy=0$, whose general solution is $y=x^{1/2}\left[c_1{J}_{1/3}\right(\frac{2}{3}{x}^{3/2})+c_2{J}_{-1/3}(\frac{2}{3}{x}^{3/2}\left)\right]$.⁴

The drumhead

Separating variables in the wave equation for a circular membrane of radius $1$ (the PDE module does this in full) reduces the radial factor to Bessel's equation of order zero in $\lambda r$. Boundedness at the center forces the solution $J_0(\lambda r)$, and clamping the rim forces $J_0(\lambda )=0$: the admissible frequencies ${\lambda }_1<{\lambda }_2<\cdots$ are precisely the zeros of $J_0$. The $k$-th radial mode vibrates in $k$ annular zones separated by motionless nodal circles at the radii where $J_0({\lambda }_{k}r)=0$. Because the zeros are not integer multiples of one another ($5.520/2.405\approx 2.295$), the drum's overtones are inharmonic, unlike a string's.⁶

Legendre's equation and its polynomials

The equation appears when Laplace's equation is separated in spherical coordinates: the substitution $x=\cos \phi$ turns the polar-angle factor into Legendre's equation, and the physical interval $0\le \phi \le \pi$ becomes $-1\le x\le 1$ with the singular points sitting at the poles of the sphere.⁷ As with the drumhead, the physical problem requires boundedness, this time at $x=1$.

The ordinary-point method at $x=0$ gives two series with radius $1$; useful, but the bounded solutions are found faster through the hypergeometric form. Substituting $t=\frac{1}{2}(1-x)$ maps $x=1$ to $t=0$ and produces the hypergeometric equation with $a=-n$, $b=n+1$, $c=1$. The exponents at $t=0$ are both $0$ (since $1-c=0$), so this is the equal-roots case: one solution is analytic and the second carries $\ln t$, hence is unbounded. Up to constant multiples there is exactly one solution bounded near $x=1$, and because $a=-n$ the hypergeometric series terminates: it is a polynomial.⁸

The first few, from either formula:

Each $P_n$ has degree exactly $n$, contains only even or only odd powers (so $P_n(-x)=(-1{)}^n{P}_n(x)$), satisfies $P_n(1)=1$ and $P_n(-1)=(-1{)}^n$, and has all $n$ of its zeros real, simple, and inside $(-1,1)$.

Orthogonality and Legendre series

The $n\ne m$ case follows from the equation itself: writing Legendre's equation in the self-adjoint form $((1-x^2)y^{\prime }{)}^{\prime }=-n(n+1)y$, multiplying the equations for $P_n$ and $P_m$ crosswise, integrating by parts, and subtracting kills everything but $[n(n+1)-m(m+1)]\int P_n{P}_m\mathbf{d}x$. The normalization $2/(2n+1)$ comes from Rodrigues's formula and repeated integration by parts.⁹

Orthogonality converts function expansion into coefficient extraction, exactly as with Fourier series. Since $P_0,\dots ,P_n$ span the polynomials of degree $\le n$, any reasonable $f$ on $[-1,1]$ has a Legendre series

the coefficient formula obtained by multiplying by $P_m$ and integrating term by term. For polynomials the series terminates: $x^2=\frac{1}{3}{P}_0+\frac{2}{3}{P}_2$, $x^3=\frac{3}{5}{P}_1+\frac{2}{5}{P}_3$, and so on.

The same mechanism runs on the Bessel side: with respect to the weight $x$, the rescaled functions $J_0({\lambda }_{j}x)$ for successive zeros ${\lambda }_j$ are orthogonal on $[0,1]$, giving Fourier–Bessel expansions for radially symmetric data. Both families are instances of one theorem — Legendre's and Bessel's equations are Sturm–Liouville problems, and real eigenvalues with orthogonal eigenfunctions are the general property, not a coincidence of these two examples.

The special functions, side by side

One pattern runs through both families: a second-order equation from physics, a regular singular point imposed by the geometry, boundedness there selecting a one-dimensional family of solutions, a boundary condition quantizing the eigenvalues, and orthogonality assembling arbitrary data from the eigenfunctions. The same pattern organizes separation of variables and Sturm–Liouville theory for the classical PDEs.