Difference between revisions of "BesselY0"

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m (Text replacement - "\$([^\$]+)\$" to "\\(\1\\)")
 
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[[File:BesselY0J0J1plotT060.png|500px|thumb|
 
[[File:BesselY0J0J1plotT060.png|500px|thumb|
 
[[Explicit plot]] of [[BesselY0]] (red), [[BesselJ1]] (green) and [[BesselJ0]] (blue)]]
 
[[Explicit plot]] of [[BesselY0]] (red), [[BesselJ1]] (green) and [[BesselJ0]] (blue)]]
[[File:Bessely0mapT064.png|500px|thumb|[[Complex map]] $u+\mathrm i v=Y_0(x+\mathrm i y)$ ]]
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[[File:Bessely0mapT064.png|500px|thumb|[[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
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'''BesselY0''', called also [[Neumann function]], and also \(\mathrm{BesselY}_0\) or simply \(Y_0\) is
 
kind of [[Bessel function]]
 
kind of [[Bessel function]]
 
(or [[Cylindric function]]),
 
(or [[Cylindric function]]),
solution $f=f(z)$ of the Bessel equation
+
solution \(f=f(z)\) of the Bessel equation
:$ \!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \displaystyle
+
:\( \!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \displaystyle
f''+f'/z+f=0$
+
f''+f'/z+f=0\)
   
 
with integral representation
 
with integral representation
   
: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!
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: \(\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\displaystyle Y_0(x)=\frac{-2}{\pi} \int_0^\infty \cos(x \cosh(t)) \mathrm d t$
+
\displaystyle Y_0(x)=\frac{-2}{\pi} \int_0^\infty \cos(x \cosh(t)) \mathrm d t\)
   
valid at least for $x>0$.
+
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;
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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.
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the [[complex map]] of \(Y_0\) is shown below.
   
   
 
==Various notations==
 
==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.
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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.
 
<!--
 
<!--
 
more integral representations can be deduced.
 
more integral representations can be deduced.
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</ref>
 
</ref>
   
==Relation to $J_0$==
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==Relation to \(J_0\)==
   
In the upper half–plane, id est, for $\Re(z)>0$, the following relation with [[BesselJ0]] takes place:
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In the upper half–plane, id est, for \(\Re(z)>0\), the following relation with [[BesselJ0]] takes place:
   
: $Y_0(z)=Y_0(-z)+2~ \mathrm i~ J_0(-z)$
+
: \(Y_0(z)=Y_0(-z)+2~ \mathrm i~ J_0(-z)\)
   
At $\Re(z)<0$, the similar relation is valid:
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At \(\Re(z)<0\), the similar relation is valid:
   
: $Y_0(z)=Y_0(-z)- 2~ \mathrm i~ J_0(-z)$
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: \(Y_0(z)=Y_0(-z)- 2~ \mathrm i~ J_0(-z)\)
   
These formulas are used for evaluation of $Y_0(z)$ at $\Re(z)<0$ in the [[C++]] complex(double) implementation [[Bessely0.cin]] .
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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==
 
==Behavior at zero==
   
:$ \!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle
+
:\( \!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle
 
Y_0(z)=\frac{2 (\log (z)+\gamma -\log (2))}{\pi
 
Y_0(z)=\frac{2 (\log (z)+\gamma -\log (2))}{\pi
 
}+\frac{z^2 (-\log (z)-\gamma +1+\log (2))}{2
 
}+\frac{z^2 (-\log (z)-\gamma +1+\log (2))}{2
 
\pi }+\frac{z^4 (2 \log (z)+2 \gamma -3-2 \log
 
\pi }+\frac{z^4 (2 \log (z)+2 \gamma -3-2 \log
 
(2))}{64 \pi }+O\left(z^6\right)
 
(2))}{64 \pi }+O\left(z^6\right)
  +
\)
$
 
where $~\gamma=$[[EulerGamma]]$~\approx 0.5772156649015329~$ is universal constant.
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where \(~\gamma=\)[[EulerGamma]]\(~\approx 0.5772156649015329~\) is universal constant.
   
 
==Expansion at large values of the argument==
 
==Expansion at large values of the argument==
At large $|z|$, the expansion of $Y_0$ can be written in analogy with that for $J_0$:
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At large \(|z|\), the expansion of \(Y_0\) can be written in analogy with that for \(J_0\):
:$ \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ \displaystyle
+
:\( \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ \displaystyle
 
J_0(z)=\sqrt{\frac{2}{\pi z}}
 
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( \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)
 
+\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)
 
\right)
  +
\)
$
 
:$ \!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ \displaystyle
+
:\( \!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ \displaystyle
 
Y_0(z)=\sqrt{\frac{2}{\pi z}}
 
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( \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)
 
+\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)
 
\right)
  +
\)
$
 
   
 
==Keywords==
 
==Keywords==

Latest revision as of 18:26, 30 July 2019

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 \(J_0\):

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

Keywords

Bessel function BesselJ0 BesselJ1 BesselH0

References