Bessel function

Bessel function referes to a solution of the Bessel equation

$ \!\!\!\!\!\!\!\!\! (1) ~ ~ ~ f(z)+f'(z)/z+(1-\nu/z^2)f(x) =0$

BesselJ

Due to singularity of the equation at $z=0$, the regular solution should have specific behavior. This solution is called $J_\nu$. For $\nu=0$ and $\nu=1$, there are specific implementations BesselJ0 and BesselJ1. Many formulas about the Bessel functions below are borrowed from the handbook by Abramowirtz,Stegun ; the numeration of formulas from there is used below.

Integral representations

$ \!\!\!\!\!\!\!\!\!\! (9.1.20) ~ ~ ~ \displaystyle

J_\nu(z) = \frac{(z/2)^{\nu}}{\pi^{1/2} ~(\nu-1/2)!} ~ \int_0^\pi ~ \cos(z \cos(t)) \sin(t)^{2 \nu} ~t~ \mathrm d t $

Sonin representation

The Mehler,Sonin formulas suggest that

$\displaystyle J_\nu(z)=\frac{2}{\pi} \int_0 ^\infty \sin(x \cos(t) - \pi \nu/2) \cos(\nu t) \mathrm d t$

$\displaystyle Y_\nu(z)=\frac{-2}{\pi} \int_0 ^\infty \cos(x \cos(t) - \pi \nu/2) \cos(\nu t) \mathrm d t$

and, in particular,

$\displaystyle J_0(x)=\frac{2}{\pi} \int_0^\infty \sin(x \cosh(t)) \mathrm d t$

$\displaystyle Y_0(x)=\frac{-2}{\pi} \int_0^\infty \cos(x \cosh(t)) \mathrm d t$

Also,

$\displaystyle J_\nu(x)=\frac{2 (x/2)^{-\nu}}{\pi^{1/2} \Gamma(1/2-\nu)}

\int_1^\infty \frac{\sin(xt)~ \mathrm d t} { (t^2-1)^{\nu+1/2}}$

$\displaystyle Y_\nu(x)=-\frac{2 (x/2)^{-\nu}}{\pi^{1/2} \Gamma(1/2-\nu)}

\int_1^\infty \frac{\cos(xt)~ \mathrm d t} { (t^2-1)^{\nu+1/2}}$

However the reason of the suggested restriction $x>0$ is not clear. Peerhaps, these expressions can be used to deduce the expansion suitable for the numerical implementation.

Expansion of $J_\nu$ at zero

$\!\!\!\!\!\!\!\!\!\!\!\!\! (9.1.10) ~ ~ ~ \displaystyle

J_\nu(z)=\left(\frac{z}{2}\right)^{\!\nu}~ \sum_{k=0}^{\infty} ~ \frac{(-z^2/4)^k} {k!~ (\nu\!+\!k)!}$

Expansion of $Y_n$ at zero

The similar expansion for $Y_n$ at natural $n\!+\!1$ looks ugly:

$\!\!\!\!\!\!\!\!\!\!\!\!\! (\mathrm{GR} 8.403) ~ ~ ~ \displaystyle

\pi Y_n(z)= 2 J_n(z) ( \ln(z/2) + C ) - \sum_{k=0}^{n-1} \frac{(n\!-\!k\!-\!1)!}{k!} (z/2)^{2k-n} -$

$ \displaystyle

- (z/2)^n \frac{1}{n!} \sum_{k=1}^n \frac{1}{k} - \sum_{k=0}^{\infty} \frac{(-1)^k (z/2)^{n+2k}}{k! ~(n\!+\!k)!} \left( \sum_{m=1}^{n+k} \frac{1}{m} + \sum_{m=1}^k \frac{1}{m} \right) $ where $C$ is Euler constant, called also EulerGamma

$\displaystyle

C=- \int_0^\infty \exp(-t)~ \ln(t) ~\mathrm d t \approx 0.57721566490$

Up to year 2012, no beautiful representation for the expansion coefficients is available.

Expansion at infinity by Gradshtein,Ryzhik

Gradshtein,Ryzhik suggest the following expansions (See 8.4.5.5)

$ \displaystyle J_{\pm \nu}(z)=$

$ \displaystyle = \sqrt{\frac{2}{\pi z}} \cos\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big) ~ \left( ~ \sum_{k=0}^{n-1} ~

\left(\frac{-1}{4z^2}\right)^{\!\! k} \frac{ \Gamma(\nu+2k+1/2)}{(2k)!~ \Gamma(\nu-2k+1/2)} + R_1 \right) -$

$ \displaystyle - \sqrt{\frac{2}{\pi z}} \sin\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big)~ \left( \frac{1}{2z} ~ \sum_{k=0}^{n-1} ~

\left(\frac{-1}{4z^2}\right)^{\!\! k} \frac{ \Gamma(\nu+2k+3/2)}{(2k\!+\!1)! ~\Gamma(\nu-2k-1/2)} + R_2 \right)$

$ \displaystyle Y_{\pm \nu}(z)=$

$ \displaystyle = \sqrt{\frac{2}{\pi z}} \sin\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big) ~ \left( ~ \sum_{k=0}^{n-1} ~

\left(\frac{-1}{4z^2}\right)^{\!\! k} \frac{ \Gamma(\nu+2k+1/2)}{(2k)!~ \Gamma(\nu-2k+1/2)} + R_1 \right) +$

$ \displaystyle + \sqrt{\frac{2}{\pi z}} \cos\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big)~ \left( \frac{1}{2z} ~ \sum_{k=0}^{n-1} ~

\left(\frac{-1}{4z^2}\right)^{\!\! k} \frac{ \Gamma(\nu+2k+3/2)}{(2k\!+\!1)! ~\Gamma(\nu-2k-1/2)} + R_2 \right)$

$ \displaystyle H_{\nu}(z)=$

$ \displaystyle = \sqrt{\frac{2}{\pi z}} \exp\!\Big(z-(\pm 2 \nu\!+\!1)\pi/4 \Big) ~ \left( ~ \sum_{k=0}^{n-1} ~

\left(\frac{\mathrm i}{2z}\right)^{\!\! k} \frac{ \Gamma(\nu+n+1/2)}{(2n)!~ \Gamma(\nu-n+1/2)} + \theta_1 \left(\frac{\mathrm i}{2z}\right)^{\!\! k} \frac{ \Gamma(\nu+n+1/2)}{(2n)!~ \Gamma(\nu-n+1/2)} \right) $

$ \displaystyle

|R_1|< \left| \frac{\Gamma(\nu+2n+1/2)}{(2z)^{2n} ~ (2n)! ~ \Gamma(\nu-2n+1/2)} \right|$

$ \displaystyle

|R_2|< \left| \frac{\Gamma(\nu+2n+3/2)}{(2z)^{2n+1} ~ (2n\!+\!1)! ~ \Gamma(\nu-2n-1/2)} \right|$

while $\Im(z)\le 0$, the esitmate $|\theta_1| < 1 $

For half–natural $\nu$, the singularity of $\Gamma$ terminates the series and they become finite sums.

Expansion at infinity from Abramowitz,Stegun

Let $\mu=4 \nu^2$. Define two series $P_\nu(z)$ and $Q_\nu(z)$ with

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.09) ~ ~ ~ \displaystyle

P_\nu(z)=1 -\frac{(\mu\!-\!1)(\mu\!-\!9)}{2! ~ (8z)^2} +\frac{(\mu\!-\!1)(\mu\!-\!9)(\mu\!-\!25)(\mu\!-\!49)}{4! ~ (8z)^4} - ..$

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.10) ~ ~ ~ \displaystyle

Q_\nu(z)= \frac{(\mu\!-\!1)}{1! ~ (8z)} +\frac{(\mu\!-\!1)(\mu\!-\!9)(\mu\!-\!25)}{3! ~ (8z)^3} - \frac{(\mu\!-\!1)(\mu\!-\!9)(\mu\!-\!25)(\mu\!-\!49)(\mu\!-\!81)}{5! ~ (8z)^5} +..$ Let $x=z-\Big(\nu/2+\pi/4\Big)\pi~$; then

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.05) ~ ~ ~ \displaystyle

J_\nu(z_)=\sqrt{\frac{2}{\pi z}}\Big(P_\nu(z) \cos(z)-Q_\nu(z) \sin(z) \Big)$

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.06) ~ ~ ~ \displaystyle

Y_\nu(z_)=\sqrt{\frac{2}{\pi z}}\Big(P_\nu(z) \sin(z)+Q_\nu(z) \cos(z) \Big)$

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.07) ~ ~ ~ \displaystyle

H_\nu(z_)=\sqrt{\frac{2}{\pi z}}\Big(P_\nu(z) + \mathrm i ~ Q_\nu(z) \Big) \mathrm e^{\mathrm i z}$

while $|\mathrm{Arg}(z)|<\pi$.

Asymptotic expansion for modulus and phase

$ \!\!\!\!\!\!\!\!\! (9.2.19) ~ ~ ~ J_\nu(z)= M \cos(\theta) ~~, ~~$ $~ Y_\nu(z)= M \sin(\theta) ~~, ~~$

$ \!\!\!\!\!\!\!\!\! (9.2.17) ~ ~ ~ M = \sqrt{J_\nu(z)^2 + Y_\nu(z)^2}$

In certain range, while $ |\Re(\theta)|<\pi$, also

$ \!\!\!\!\!\!\!\!\! (9.2.0) ~ ~ ~ \theta = \mathrm{atan2}( Y_\nu(z)/ J_\nu(z))$

$M$ is called "modulus" and $\theta$ is called "argument" . Let $\mu=4\nu^2$.

At large values of the argument, it worth to expand $M$ and $\theta$ instead of $J$:

$ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.28) \displaystyle ~ ~ ~ M^2 = \frac{2}{\pi z} \left( 1 +

\frac{1}{2} \frac{\mu \!-\!1}{(2z)^2} + \frac{1\cdot 3}{2 \cdot 4} \frac{(\mu \!-\!1)(\mu\!-\!9)}{(2z)^4} + \frac{1\cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{(\mu\!-\!1)(\mu\!-\!9)(\mu\!-\!25)}{(2z)^6} +.. \right) $

$ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9.2.29) \displaystyle ~ ~ ~ \theta = z - \left(\frac{\nu}{2}+\frac{1}{4}\right) \pi

+ \frac{\mu\!-\!1}{2(2z)} + \frac{ (\mu\!-\!1) (\mu-25)}{6(4z)^3} + \frac{ (\mu\!-\!1) (\mu^2-114\mu+1073)}{5(4z)^5} + \frac{ (\mu\!-\!1) (5\mu^3-1535\mu^2+54703\mu-375733) }{14(4z)^7} + ..$

References

http://en.wikipedia.org/wiki/Bessel_function

http://en.citizendium.org/wiki/Bessel_functions

Keywords

BesselJ0, BesselJ1, BesselY0, BesselY1,