Bessel’s equation

The linear differential equation

$x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+x{\displaystyle \frac{dy}{dx}}+(x^2-p^2)y=\mathrm{\hspace{0.33em}0},$ (1)

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

$y=x^r{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{a}_k{x}^k={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{a}_k{x}^{r+k},$ (2)

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):

$$x^2\sum _{k=0}^{\mathrm{\infty }}(r+k)(r+k-1)a_k{x}^{r+k-2}+x\sum _{k=0}^{\mathrm{\infty }}(r+k)a_k{x}^{r+k-1}+(x^2-p^2)\sum _{k=0}^{\mathrm{\infty }}{a}_k{x}^{r+k}=\mathrm{\hspace{0.33em}0}.$$

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

$\{\begin{array}{cc}[r^2-p^2]a_0=\mathrm{\hspace{0.33em}0},\hfill & \\ [{(r+1)}^2-p^2]a_1=\mathrm{\hspace{0.33em}0},\hfill & \\ [{(r+2)}^2-p^2]a_2+a_0=\mathrm{\hspace{0.33em}0},\hfill & \\ \mathrm{\dots }\hfill & \\ [{(r+k)}^2-p^2]a_k+a_{k-2}=\mathrm{\hspace{0.33em}0}.\hfill & \end{array}$ (3)

The last of those can be written

$$(r+k-p)(r+k+p)a_k+a_{k-2}=\mathrm{\hspace{0.33em}0}.$$

Because  $a_0\ne 0$,  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 $a_1$, $a_2$, $\mathrm{\dots }$  and express them with $a_0$ which remains arbitrary.  Setting for $k$ the integer values we get

$\{\begin{array}{cc}{a}_1=\mathrm{\hspace{0.33em}0},a_3=\mathrm{\hspace{0.33em}0},\mathrm{\dots },a_{2m-1}=\mathrm{\hspace{0.33em}0};\hfill & \\ a_2=-\frac{a_0}{2(2p+2)},a_4=\frac{a_0}{2\cdot 4(2p+2)(2p+4)},\mathrm{\dots },a_{2m}=\frac{{(-1)}^m{a}_0}{2\cdot 4\cdot 6\mathrm{\cdots }(2m)(2p+2)(2p+4)\mathrm{\dots }(2p+2m)}\hfill & \end{array}$ (4)

(where  $m=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }$). Putting the obtained coefficients to (2) we get the particular solution

$y_1:=a_0{x}^p[{\displaystyle \frac{x^2}{2(2p+2)}}+{\displaystyle \frac{x^4}{2\cdot 4(2p+2)(2p+4)}}-{\displaystyle \frac{x^6}{2\cdot 4\cdot 6(2p+2)(2p+4)(2p+6)}}+-\mathrm{\dots }]$ (5)

In order to get the coefficients $a_k$ for the second root  $r_2=-p$  we have to look after that

$${(r_2+k)}^2-p^2\ne \mathrm{\hspace{0.33em}0},$$

or  $r_2+k\ne p=r_1$.  Therefore

$$r_1-r_2=\mathrm{\hspace{0.33em}2}p\ne k$$

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 $-p$:

$y_2:=a_0{x}^{-p}[1-{\displaystyle \frac{x^2}{2(-2p+2)}}+{\displaystyle \frac{x^4}{2\cdot 4(-2p+2)(-2p+4)}}-{\displaystyle \frac{x^6}{2\cdot 4\cdot 6(-2p+2)(-2p+4)(-2p+6)}}+-\mathrm{\dots }]$ (6)

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  $x\to \mathrm{\infty }$).  With the appointed value

$$a_0=\frac{1}{2^p\mathrm{\Gamma }(p+1)},$$

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  $p\in ℝ$ (and $ℂ$). For  $p\notin \mathbb{Z}$  the general solution of the Bessel’s differential equation is thus

$$y:=C_1{J}_p(x)+C_2{J}_{-p}(x),$$

where  $J_{-p}(x)=y_2$  with  $a_0=\frac{1}{2^{-p}\mathrm{\Gamma }(-p+1)}$.

The explicit expressions for $J_{\pm p}$ are

$J_{\pm p}(x)={\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^m}{m!\mathrm{\Gamma }(m\pm p+1)}}{\left({\displaystyle \frac{x}{2}}\right)}^{2m\pm p},$ (7)

which are obtained from (5) and (6) by using the last for gamma function.

E.g. when  $p=\frac{1}{2}$  the series in (5) gets the form

$$y_1=\frac{x^{\frac{1}{2}}}{\sqrt{2}\mathrm{\Gamma }(\frac{3}{2})}[1-\frac{x^2}{2\cdot 3}+\frac{x^4}{2\cdot 4\cdot 3\cdot 5}-\frac{x^6}{2\cdot 4\cdot 6\cdot 3\cdot 5\cdot 7}+-\mathrm{\dots }]=\sqrt{\frac{2}{\pi x}}(x-\frac{x^3}{3!}+\frac{x^5}{5!}-+\mathrm{\dots }).$$

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

$$y:=C_1{J}_{\frac{1}{2}}(x)+C_2{J}_{-\frac{1}{2}}(x).$$

In the case that $p$ is a non-negative integer $n$, the “+” case of (7) gives the solution

$$J_n(x)=\sum _{m=0}^{\mathrm{\infty }}\frac{{(-1)}^m}{m!(m+n)!}{\left(\frac{x}{2}\right)}^{2m+n},$$

but for  $p=-n$  the expression of $J_{-n}(x)$ is ${(-1)}^n{J}_n(x)$, i.e. linearly dependent on $J_n(x)$.  It can be shown that the other solution of (1) ought to be searched in the form  $y=K_n(x)=J_n(x)\ln x+x^{-n}{\sum }_{k=0}^{\mathrm{\infty }}{b}_k{x}^k$.  Then the general solution is  $y:=C_1{J}_n(x)+C_2{K}_n(x)$.

Other formulae

The first kind Bessel functions of integer order have the generating function $F$:

$F(z,t)=e^{\frac{z}{2}(t-\frac{1}{t})}={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{J}_n(z)t^n$ (8)

This function has an essential singularity at  $t=0$  but is analytic elsewhere in $ℂ$; 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

$$c_n=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{e^{\frac{zt}{2}}{e}^{-\frac{z}{2t}}}{t^{n+1}}𝑑t=\frac{1}{2\pi i}{\left(\frac{z}{2}\right)}^n{\oint }_{\delta }\frac{e^{\zeta }{e}^{-\frac{z^2}{4\zeta }}}{{\zeta }^{n+1}}𝑑\zeta =\sum _{m=0}^{\mathrm{\infty }}\frac{{(-1)}^m}{m!}{\left(\frac{z}{2}\right)}^{2m+n}\frac{1}{2\pi i}{\oint }_{\delta }{\zeta }^{-m-n-1}{e}^{\zeta }𝑑\zeta .$$

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  $\frac{1}{(m+n)!}=\frac{1}{\mathrm{\Gamma }(m+n+1)}$,  the residue theorem (http://planetmath.org/CauchyResidueTheorem) gives

$$c_n=\sum _{m=0}^{\mathrm{\infty }}\frac{{(-1)}^m}{m!\mathrm{\Gamma }(m+n+1)}{\left(\frac{z}{2}\right)}^{2m+n}=J_n(z).$$

This that $F$ has the Laurent expansion (8).

By using the generating function, one can easily derive other formulae, e.g. the of the Bessel functions of integer order:

$$J_n(z)=\frac{1}{\pi }{\int }_0^{\pi }\cos (n\phi -z\sin \phi )𝑑\phi$$

Also one can obtain the addition formula

$$J_n(x+y)=\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{\nu }(x)J_{n-\nu }(y)$$

and the series of cosine and sine:

$$\cos z=J_0(z)-2J_2(z)+2J_4(z)-+\mathrm{\dots }$$

$$\sin z=\mathrm{\hspace{0.33em}2}{J}_1(z)-2J_3(z)+2J_5(z)-+\mathrm{\dots }$$

Title	Bessel’s equation
Canonical name	BesselsEquation
Date of creation	2013-03-22 16:34:57
Last modified on	2013-03-22 16:34:57
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	26
Author	pahio (2872)
Entry type	Definition
Classification	msc 34A05
Classification	msc 33C10
Synonym	Bessel’s differential equation
Synonym	Bessel equation
Related topic	LaplaceEquationInCylindricalCoordinates
Related topic	CauchyResidueTheorem
Related topic	PropertiesOfEntireFunctions
Related topic	FrobeniusMethod
Related topic	TableOfLaplaceTransforms
Related topic	BesselFunctionsAndHelicalStructureDiffractionPatterns
Defines	Bessel’s function
Defines	Bessel function