# 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 [1]

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

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 [2] can be used for the evaluation.

The BesselJ1 is related to the BesselJ0 as follows:

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