How to plot this modified Bessel function?

Question

I'm new to TeX and I wish to plot in my master's dissertation the graph of the function $E_2:\mathbb{R}^2 \setminus{0} \to \mathbb{R}$ defined by

$$E_2(x,y)= (2 \pi)^{-1}K_0(\sqrt(x²+y²)), $$

Where K_0 is the modified Bessel function of second kind of order 0.

Edit:

I have added graphical renderings to make it easier to grasp the purpose of the TeX code above. To get the \sqrt function displayed, I used braces in place of the parentheses in the code above. To get the "where" clause to work, i used $ signs in place of the backquotes (`). The use of \setminus does not seem to me to be mathematically correct, but I have not changed it since I do not know the author's intent.

Answer 1

The computer algebra system (CAS), Sage, knows about Bessel functions and, through the sagetex package it can calculate the points that LaTeX needs to plot with the tikz package.

I don't have experience with the Bessel function, so after reading the documentations I went to a Sage Cell Server to see what the result should look like. Copy/paste the code below into the Sage Cell server:

var('x,y')
f = (1/(2*pi))*Bessel(0,type='K')(sqrt(x^2+y^2))
plot3d(f,(x,-20,20),(y,-20,20))
#plot3d(f,(x,-20,20),(y,-20,20),aspect_ratio=[1,1,25])

and press Evaluate then you will get a plot that, if you reach in and rotate it, will look something like:

Now remove the hashtag before line 4 and add a hashtag before line 3. Press Evaluate, rotate the plot, and you can get something like this:

In summary, the accurate plot will have a small height and playing with the aspect ratio will show useful detail.

With that in mind the LaTeX code below will plot the modified Bessel function.

\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
\usepackage{sagetex}
\pgfplotsset{compat=1.15} 
\begin{document}
\begin{sagesilent}
var('x,y')
minX = -20
maxX = 20
minY = -20
maxY = 20
xcoords = [i for i in srange(minX,maxX,1)]
ycoords = [i for i in srange(minY,maxY,1)]
f = (1/(2*pi))*Bessel(0,type='K')(sqrt(x^2+y^2))
output = ""
output += r"\begin{tikzpicture}[scale=1.0]"
output += r"\begin{axis}[view={20}{45},xmin=%d, xmax=%d, ymin=%d, ymax=%d]"%(minX,maxX,minY,maxY)
output += r"\addplot3[colormap/viridis,surf,opacity=0.5,mesh/rows=%d] coordinates {"%(len(ycoords))
the length of ycoords is the number of y values
for y in ycoords:
    for x in xcoords:
        output += r"(%f, %f, %f) "%(x,y,f(x=x,y=y))
output += r"};"
output += r"\end{axis}"
output += r"\end{tikzpicture}"
\end{sagesilent}
\sagestr{output}
\end{document}

The output is:

We know this plot has exaggerated the z-axis. This can be corrected to what you want by forcing a maximum z value. For example, I added zmax=2 on line 18 to get:

Sage is not part of LaTeX so you will either have to download it from the Sagemath site or open up a free Cocalc account. With the Cocalc account you can be up and running in 5-10 minutes; all you need to do is create a LaTeX document, copy/paste the code in, and press the Build button.

EDIT: With respect to your comment below, there are several methods for getting .png output. With Sage, it's trivial. The picture from the Sage Cell server has an "i" in a circle to the bottom right of the picture. If you left click on it, a menu opens up with an option to create a .png.

With respect to LaTeX, you can get a similar look with the following code:

\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
\usepackage{sagetex}
\pgfplotsset{compat=1.15} 
\begin{document}
\begin{sagesilent}
var('x,y')
minX = -7
maxX = 7
minY = -7
maxY = 7
xcoords = [i for i in srange(minX,maxX,.5)]
ycoords = [i for i in srange(minY,maxY,.5)]
f = (1/(2*pi))*Bessel(0,type='K')(sqrt(x^2+y^2))
output = ""
output += r"\begin{tikzpicture}[scale=1.0]"
output += r"\begin{axis}[hide axis,view={20}{45},xmin=%d, xmax=%d, ymin=%d, ymax=%d]"%(minX,maxX,minY,maxY)
output += r"\addplot3[surf,fill=white,faceted color = black,opacity=0.7,mesh/rows=%d] coordinates {"%(len(ycoords))
the length of ycoords is the number of y values
for y in ycoords:
    for x in xcoords:
        output += r"(%f, %f, %f) "%(x,y,f(x=x,y=y))
output += r"};"
output += r"\end{axis}"
output += r"\end{tikzpicture}"
\end{sagesilent}
\sagestr{output}
\end{document}

The output is shown below:

You'll have to play around with the code to get the look you want. The pdf can be converted to png via an external program, such as GIMP.