BesselJ1

Explicit plot of BesselY0 (red), BesselJ1 (green) and BesselJ0 (blue)

\( u+\mathrm i v =J_1(x+\mathrm i y)\)

BesselJ1 or \(J_1\) is Bessel function of first order;

\(\!\!\!\! \mathrm{BesselJ1}(z)=J_1(z)= \mathrm{BesselJ}[1,z]\)

Function \(f=\mathrm{BesselJ1}\) satisfies the Bessel equation

\( \!\!\!\! f''(z)+f'(z)/z + (z^2\!-\!1)f(z) = 0\)

with boundary conditions

\(f(0) = 0~\) and \(~f'(0)=1/2\)

The complex map of \(f=J_1(x+\mathrm i y)\) is shown at right in the \(x,y\) plane with levels \(u=\Re(f)=\mathrm {const} ~\) and levels \(v=\Im(f)=\mathrm {const} ~\).

Relation to other Bessel functions

\( J_0'(z)=-J_1(z)\)

\(\displaystyle 1-J_0(z)=\int_0^z J_1(t)~\mathrm d t \)

Expansions

\(J_1\) is entire function, the series

\(\displaystyle J_1(z) = \frac{z}{2} \sum_{n=0}^\infty \frac{(-z^2/4)^n}{(n\!+\!1)~ \mathrm{Factorial}(n)^2}\)

converges for any complex \(z\) and can be used to plot the complex map at least for \(z<10\) withe the complex(double) arithmetics is available. For large values of \(|z|\), the asymptotic expansion can be used for the precise evaluation:

\(\displaystyle J_1(z)= \)

\(\displaystyle \sqrt{\frac{2}{\pi z}} \cos\left(\frac{\pi}{4}+z\right) \left( -1 - \frac{15}{128 z^2} +\frac{4725}{32768 z^4} - \frac{2837835}{4194304 z^6} + \frac{14783093325}{ 2147483648 z^8} - \frac{33424574007825}{274877906944 z^{10}} +O\left(\frac{1}{z^{12}}\right) \right) +\)

\(\displaystyle \sqrt{\frac{2}{\pi z}} \sin\left(\frac{\pi}{4}+z\right) \left( \frac{3}{8 z} - \frac{105}{1024 z^3} + \frac{72765}{262144 z^5} - \frac{66891825}{33554432 z^7} + \frac{468131288625}{17179869184 z^9} - \frac{1327867167401775}{2199023255552 z^{11}} +.. \right) \)

This asymptoric expansion is used for the numerical implenentation. However, \(z\) should not approach the negative part of the real axis. For the case \(\Re(z)<0\), the symmetry

\(J_1(z)= - J_1(-z)\)

is used.

Mathematica allows to calculate a dozen of term of this expansion; they can be extracted also from the complex(double) implementation Besselj1.cin.

Evaluation at real argument

At real values of the argument, the GNU library can be used for the evaluation.

The BesselJ1 is related to the BesselJ0 as follows:

\( J_1(z)=-J_0'(z)\)

Keywords

BesselJ0, Bessel function, Special function, Entire function, Cylindric function

References

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

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

http://www.mathworks.co.jp/help/techdoc/ref/besselj.html