# BesselY0

Explicit plot of BesselY0 (red), BesselJ1 (green) and BesselJ0 (blue)
Complex map $$u+\mathrm i v=Y_0(x+\mathrm i y)$$

BesselY0, called also Neumann function, and also $$\mathrm{BesselY}_0$$ or simply $$Y_0$$ is kind of Bessel function (or Cylindric function), solution $$f=f(z)$$ of the Bessel equation

$$\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \displaystyle f''+f'/z+f=0$$

with integral representation

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

valid at least for $$x>0$$.

The explicit plot of $$Y_0$$ of real argument is shown at the upper right corner, for comparison, $$J_0$$ and $$J_1$$ are also plotted; the complex map of $$Y_0$$ is shown below.

## Various notations

In the literture, and in particular, in the first editions of the tables by Granshtein,Ryzhik, BesselY0 is called Neumann function and denoted $$N_0$$. In the latest editions and in the tables by Abramowits,Stegun, notation $$Y_0$$ is used. In Mathematica, BesselY0(z) can be accessed with BesselY[0,z] and is interpreted as "Bessel function of Second kind" [1]

## Relation to $$J_0$$

In the upper half–plane, id est, for $$\Re(z)>0$$, the following relation with BesselJ0 takes place:

$$Y_0(z)=Y_0(-z)+2~ \mathrm i~ J_0(-z)$$

At $$\Re(z)<0$$, the similar relation is valid:

$$Y_0(z)=Y_0(-z)- 2~ \mathrm i~ J_0(-z)$$

These formulas are used for evaluation of $$Y_0(z)$$ at $$\Re(z)<0$$ in the C++ complex(double) implementation Bessely0.cin .

## Behavior at zero

$$\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle Y_0(z)=\frac{2 (\log (z)+\gamma -\log (2))}{\pi }+\frac{z^2 (-\log (z)-\gamma +1+\log (2))}{2 \pi }+\frac{z^4 (2 \log (z)+2 \gamma -3-2 \log (2))}{64 \pi }+O\left(z^6\right)$$

where $$~\gamma=$$EulerGamma$$~\approx 0.5772156649015329~$$ is universal constant.

## Expansion at large values of the argument

At large $$|z|$$, the expansion of $$Y_0$$ can be written in analogy with that for $$J_0$$:

$$\!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ \displaystyle J_0(z)=\sqrt{\frac{2}{\pi z}} \left( \left(\frac{1}{8 z} -\frac{75}{1024 z^3}+O\left(\frac{1}{z^5}\right)\right) \sin \left(z-\frac{\pi }{4}\right) +\left(1-\frac{9}{128 z^2}+\frac{3675}{32768 z^4}+O\left(\frac{1}{z^6}\right)\right) \cos \left(z-\frac{\pi }{4}\right) \right)$$
$$\!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ \displaystyle Y_0(z)=\sqrt{\frac{2}{\pi z}} \left( \left(\frac{-1}{8 z}+\frac{75}{1024 z^3}+O\left(\frac{1}{z^5}\right)\right) \cos \left(z-\frac{\pi }{4}\right) +\left(1-\frac{9}{128 z^2}+\frac{3675}{32768 z^4}+O\left(\frac{1}{z^6}\right)\right) \sin \left(z-\frac{\pi }{4}\right) \right)$$