PlanetPhysics/Bessel Functions and Their Applications to Diffraction by Helical Structures

The linear differential equation $$\begin{matrix} x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-p^2)y = 0, \end{matrix}$$ in which $$p$$ is a constant (non-negative if it is real), is called the Bessel's equation .\, We derive its general solution by trying the series form $$\begin{matrix} y = x^r\sum_{k=0}^\infty a_kx^k = \sum_{k=0}^\infty a_kx^{r+k}, \end{matrix}$$ due to Frobenius.\, Since the parameter $$r$$ is indefinite, we may regard $$a_0$$ as distinct from 0.

We substitute (2) and the derivatives of the series in (1): $$ x^2\sum_{k=0}^\infty(r+k)(r+k-1)a_kx^{r+k-2}+ x\sum_{k=0}^\infty(r+k)a_kx^{r+k-1}+ (x^2-p^2)\sum_{k=0}^\infty a_kx^{r+k} = 0. $$ Thus the coefficients of the powers $$x^r$$, $$x^{r+1}$$, $$x^{r+2}$$ and so on must vanish, and we get the system of equations $$\begin{matrix} \begin{cases} {[}r^2-p^2{]}a_0 = 0,\\ {[}(r+1)^2-p^2{]}a_1 = 0,\\ {[}(r+2)^2-p^2{]}a_2+a_0 = 0,\\ \qquad \qquad \ldots\\ {[}(r+k)^2-p^2{]}a_k+a_{k-2} = 0. \end{cases} \end{matrix}$$ The last of those can be written $$(r+k-p)(r+k+p)a_k+a_{k-2} = 0.$$ Because\, $$a_0 \neq 0$$,\, the first of those (the indicial equation) gives\, $$r^2-p^2 = 0$$,\, i.e. we have the roots $$r_1 = p,\,\, r_2 = -p.$$ Let's first look the the solution of (1) with\, $$r = p$$;\, then\, $$k(2p+k)a_k+a_{k-2} = 0$$,\, and thus\, $$a_k = -\frac{a_{k-2}}{k(2p+k).}$$ From the system (3) we can solve one by one each of the coefficients $$a_1$$, $$a_2$$, $$\ldots$$\, and express them with $$a_0$$ which remains arbitrary.\, Setting for $$k$$ the integer values we get $$\begin{matrix} \begin{cases} a_1 = 0,\,\,a_3 = 0,\,\ldots,\, a_{2m-1} = 0;\\ a_2 = -\frac{a_0}{2(2p+2)},\,\,a_4 = \frac{a_0}{2\cdot4(2p+2)(2p+4)},\,\ldots,\,\, a_{2m} = \frac{(-1)^ma_0}{2\cdot4\cdot6\cdots(2m)(2p+2)(2p+4)\ldots(2p+2m)} \end{cases} \end{matrix}$$ (where\, $$m = 1,\,2,\,\ldots$$). Putting the obtained coefficients to (2) we get the particular solution $$\begin{matrix} y_1 := a_0x^p \left[1\!-\!\frac{x^2}{2(2p\!+\!2)}\! +\!\frac{x^4}{2\!\cdot\!4(2p\!+\!2)(2p\!+\!4)} \!-\!\frac{x^6}{2\!\cdot\!4\!\cdot\!6(2p\!+\!2)(2p\!+\!4)(2p\!+\!6)}\!+-\ldots\right] \end{matrix}$$

In order to get the coefficients $$a_k$$ for the second root\, $$r_2 = -p$$\, we have to look after that $$(r_2+k)^2-p^2 \neq 0,$$ or\, $$r_2+k \neq p = r_1$$.\, Therefore $$r_1-r_2 = 2p \neq k$$ where $$k$$ is a positive integer.\, Thus, when $$p$$ is not an integer and not an integer added by $$\frac{1}{2}$$, we get the second particular solution, gotten of (5) by replacing $$p$$ by $$-p$$: $$\begin{matrix} y_2 := a_0x^{-p}\!\left[1 \!-\!\frac{x^2}{2(-2p\!+\!2)}\!+\!\frac{x^4}{2\!\cdot\!4(-2p\!+\!2)(-2p\!+\!4)} \!-\!\frac{x^6}{2\!\cdot\!4\!\cdot\!6(-2p\!+\!2)(-2p\!+\!4)(-2p\!+\!6)}\!+-\ldots\right] \end{matrix}$$

The power series of (5) and (6) converge for all values of $$x$$ and are linearly independent (the ratio $$y_1/y_2$$ tends to 0 as\, $$x\to\infty$$).\, With the appointed value $$a_0 = \frac{1}{2^p\,\Gamma(p+1)},$$ the solution $$y_1$$ is called the Bessel function of the first kind and of order $$p$$ and denoted by $$J_p$$.\, The similar definition is set for the first kind Bessel function of an arbitrary order\, $$p\in \mathbb{R}$$ (and $$\mathbb{C}$$). For\, $$p\notin \mathbb{Z}$$\, the general solution of the Bessel's differential equation is thus $$y := C_1J_p(x)+C_2J_{-p}(x),$$ where\, $$J_{-p}(x) = y_2$$\, with\, $$a_0 = \frac{1}{2^{-p}\Gamma(-p+1)}$$.

The explicit expressions for $$J_{\pm p}$$ are $$\begin{matrix} J_{\pm p}(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\,\Gamma(m\pm p+1)}\left(\frac{x}{2}\right)^{2m\pm p}, \end{matrix}$$ which are obtained from (5) and (6) by using the last formula for gamma function.

E.g. when\, $$p = \frac{1}{2}$$\, the series in (5) gets the form $$y_1 = \frac{x^{\frac{1}{2}}}{\sqrt{2}\,\Gamma(\frac{3}{2})}\left[1\!-\!\frac{x^2}{2\!\cdot\!3}\!+\!\frac{x^4}{2\!\cdot\!4\!\cdot\!3\!\cdot\!5}\!-\!\frac{x^6}{2\!\cdot\!4\cdot\!6\!\cdot\!3\!\cdot\!5\!\cdot\!7}\!+-\ldots\right] = \sqrt{\frac{2}{\pi x}}\left(x\!-\!\frac{x^3}{3!}\!+\!\frac{x^5}{5!}\!-+\ldots\right).$$ Thus we get $$J_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}}\sin{x};$$ analogically (6) yields $$J_{-\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}}\cos{x},$$ and the general solution of the equation (1) for\, $$p = \frac{1}{2}$$\, is $$y := C_1J_{\frac{1}{2}}(x)+C_2J_{-\frac{1}{2}}(x).$$

In the case that $$p$$ is a non-negative integer $$n$$, the "+" case of (7) gives the solution $$J_{n}(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\,(m+n)!}\left(\frac{x}{2}\right)^{2m+n}, $$ but for\, $$p = -n$$\, the expression of $$J_{-n}(x)$$ is $$(-1)^nJ_n(x)$$, i.e. linearly dependent of $$J_n(x)$$.\, It can be shown that the other solution of (1) ought to be searched in the form\, $$y = K_n(x) = J_n(x)\ln{x}+x^{-n}\sum_{k=0}^\infty b_kx^k$$.\, Then the general solution is\, $$y := C_1J_n(x)+C_2K_n(x)$$.\\

Other formulae

The first kind Bessel functions of integer order have the generating function $$F$$: $$\begin{matrix} F(z,\,t) = e^{\frac{z}{2}(t-\frac{1}{t})} = \sum_{n=-\infty}^\infty J_n(z)t^n \end{matrix}$$ This function has an essential singularity at\, $$t = 0$$\, but is analytic elsewhere in $$\mathbb{C}$$; thus $$F$$ has the Laurent expansion in that point.\, Let us prove (8) by using the general expression $$c_n = \frac{1}{2\pi i}\oint_{\gamma} \frac{f(t)}{(t-a)^{n+1}}\,dt$$ of the coefficients of Laurent series.\, Setting to this\, $$a := 0$$,\, $$f(t) := e^{\frac{z}{2}(t-\frac{1}{t})}$$,\, $$\zeta := \frac{zt}{2}$$\, gives $$c_n = \frac{1}{2\pi i} \oint_\gamma\frac{e^{\frac{zt}{2}}e^{-\frac{z}{2t}}}{t^{n+1}}\,dt = \frac{1}{2\pi i}\left(\frac{z}{2}\right)^n\! \oint_\delta\frac{e^\zeta e^{-\frac{z^2}{4\zeta}}}{\zeta^{n+1}}\,d\zeta = \sum_{m=0}^\infty\frac{(-1)^m}{m!}\left(\frac{z}{2}\right)^{2m+n}\! \frac{1}{2\pi i}\oint_\delta \zeta^{-m-n-1}e^\zeta\,d\zeta.$$ The paths $$\gamma$$ and $$\delta$$ go once round the origin anticlockwise in the $$t$$-plane and $$\zeta$$-plane, respectively.\, Since the residue of $$\zeta^{-m-n-1}e^\zeta$$ in the origin is\, $$\frac{1}{(m+n)!} = \frac{1}{\Gamma(m+n+1)}$$,\, the residue theorem gives $$c_n = \sum_{m=0}^\infty \frac{(-1)^m}{m!\Gamma(m+n+1)}\left(\frac{z}{2}\right)^{2m+n} = J_n(z).$$ This means that $$F$$ has the Laurent expansion (8).

By using the generating function, one can easily derive other formulae, e.g. the integral representation of the Bessel functions of integer order: $$J_n(z) = \frac{1}{\pi}\int_0^\pi\cos(n\varphi-z\sin{\varphi})\,d\varphi$$ Also one can obtain the addition formula $$J_n(x+y) = \sum_{\nu=-\infty}^{\infty}J_\nu(x)J_{n-\nu}(y)$$ and the series representations of cosine and sine: $$\cos{z} = J_0(z)-2J_2(z)+2J_4(z)-+\ldots$$ $$\sin{z} = 2J_1(z)-2J_3(z)+2J_5(z)-+\ldots$$

Applications of Bessel functions in Physics and Engineering
One notes also that Bessel's equation arises in the derivation of separable solutions to Laplace's equation, and also for the Helmholtz equation in either cylindrical or spherical coordinates. The Bessel functions are therefore very important in many physical problems involving wave propagation, wave diffraction phenomena--including X-ray diffraction by certain molecular crystals, and also static potentials. The solutions to most problems in cylindrical coordinate systems are found in terms of Bessel functions of integer order ($$\alpha = n$$), whereas in spherical coordinates, such solutions involve Bessel functions of half-integer orders ($$\alpha = n + 1/2$$). Several examples of Bessel function solutions are:

heat conduction in a cylindrical object
 * 1) the diffraction pattern of a helical molecule wrapped around a cylinder computed from the Fourier transform of the helix in cylindrical coordinates;
 * 2) electromagnetic waves in a cylindrical waveguide
 * 3) diffusion problems on a lattice.
 * 4) vibration modes of a thin circular, tubular or annular membrane (such as a drum, other membranophone, the vocal cords, etc.)

In engineering Bessel functions also have useful properties for signal processing and filtering noise as for example by using Bessel filters, or in FM synthesis and windowing signals.

Applications of Bessel functions in Physical Crystallography
The first example listed above was shown to be especially important in molecular biology for the structures of helical secondary structures in certain proteins (e.g. $$\alpha-helix$$) or in molecular genetics for finding the double-helix structure of Deoxyribonucleic Acid (DNA) molecular crystals with extremely important consequences for genetics, biology, mutagenesis, molecular evolution, contemporary life sciences and medicine. This finding is further detailed in a related entry.