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<meta http-equiv="Content-Type" content="text/html" charset="us-ascii"/>
<title>Axiom Documentation</title>
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<div align="center"><img align="middle" src="doctitle.png"/></div>
<hr/>
<div align="center">
<a href="http://dlmf.nist.gov">
Digital Library of Mathematical Functions
</a><br/>
The Gamma Function -- Graphics
</div>
<hr/>
<h3>Graphics</h3>
<h6>Contents</h6>
<ul>
<li>Real Argument</li>
<li>The Psi Function</li>
<li>Complex Argument</li>
</ul>
<h4>Real Argument</h4>
<img width="403" height="482" src="bitmaps/gammareal3.png"/>
<br/>
This graph shows the
<m:math display="inline">
<m:mrow>
<m:mi mathvariant="normal">Γ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:math> and
<m:math display="inline">
<m:mfrac bevelled="true">
<m:mn>1</m:mn>
<m:mrow>
<m:mi mathvariant="normal">Γ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mfrac>
</m:math>.
To create these two graphs in Axiom:
<pre>
-- Draw the first graph in a viewport
viewport1:=draw(Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
-- Draw the second graph in a viewport
viewport2:=draw(1/Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
-- Get the Gamma graph from the first viewport and layer it on top
putGraph(viewport2,getGraph(viewport1,1),2)
-- Remove the points and leave the lines
points(viewport2,1,"off")
points(viewport2,2,"off")
-- Show the combined graph
makeViewport2D(viewport2)
</pre>
<img width="300" height="176" alt="" src="bitmaps/loggamma.png"/>
<br/>
<m:math display="inline">
<m:mrow>
<m:mi>ln</m:mi>
<m:mrow>
<m:mi mathvariant="normal">Γ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
</m:math>. This function is convex on
<m:math display="inline">
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mn>0</m:mn>
<m:mo>,</m:mo>
<m:mi mathvariant="normal">∞</m:mi>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:math>;
<br/>
compare <a href="dlmffunctionalrelations.xhtml#bohrmolleruptheorem">
Functional Relations</a>
<p>
You can construct this graph with the Axiom commands:
<pre>
-- draw the graph of log(Gamma) in a viewport
viewport1:=draw(log Gamma(i), i=0..8, adaptive==true, unit==[1.0,1.0])
-- turn off the points and leave the lines
points(viewport1,1,"off")
</pre>
</p>
<br/>
<h4>The Psi Function
<m:math display="inline">
<m:mrow>
<m:mi>ψ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:math>
</h4>
<p> This function is a special case of the polygamma function.
In particular,
<m:math display="inline">
<m:mrow>
<m:mi>ψ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:math> is equal to polygamma(0,x).
</p>
<br/>
<br/>
<img width="522" height="556" alt="" src="bitmaps/psi.png"/>
<br/>
You can reconstruct this graph in Axiom by:
<pre>
-- first construct the psi function
psi(x)==polygamma(0,x)
-- draw the graph in a viewport
viewport:=draw(psi(y),y=-3.5..4,adaptive==true)
-- make the gradient obvious
scale(viewport,1,0.9,22.5)
-- and recenter the graph
translate(viewport,1,0,-0.02)
-- turn off the points and keep the line
points(viewport,1,"off")
</pre>
<h4>Complex Argument</h4>
<img width="400" height="400" alt="" src="bitmaps/gammacomplex.png"/>
<br/>
<m:math display="inline">
<m:mrow>
<m:mrow>
<m:mi mathvariant="normal">Γ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>+</m:mo>
<m:mrow>
<m:mi mathvariant="normal">ⅈ</m:mi>
<m:mi>y</m:mi>
</m:mrow>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
</m:math>.
<br/>
You can reconstruct this image in Axiom with:
<pre>
-- Set up the default viewpoint
viewPhiDefault(-%pi/4)
-- define the point set function
gam(x,y)==
g:=Gamma complex(x,y)
point [x,y,max(min(real g,4),-4), argument g]
-- draw the image and remember the viewport
viewport:=draw(gam, -4..4,-3..3,var1Steps==100,var2Steps==100)
-- set the color mapping for the image
colorDef(viewport,blue(),blue())
-- and smoothly shade it
drawStyle(viewport,"smooth")
</pre>
<img width="400" height="400" src="bitmaps/gammacomplexinverse.png"/>
<br/>
<m:math display="inline">
<m:mfrac bevelled="true">
<m:mn>1</m:mn>
<m:mrow>
<m:mrow>
<m:mi mathvariant="normal">Γ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>+</m:mo>
<m:mrow>
<m:mi mathvariant="normal">ⅈ</m:mi>
<m:mi>y</m:mi>
</m:mrow>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:mrow>
</m:mfrac>
</m:math>
<br/>
<p>
You can reproduce this image from Axiom with:
<pre>
-- Set up the default viewpoint
viewPhiDefault(-%pi/4)
-- Define the complex Gamma inverse function
gaminv(x,y)==
g:=1/(Gamma complex(x,y))
point [x,y,max(min(real g,4),-4), argument g]
-- draw the 3D image and remember the viewport
viewport:=draw(gaminv, -4..4,-3..3,var1Steps==100,var2Steps==100)
-- make the image a uniform color
colorDef(viewport,blue(),blue())
-- and make it pretty
drawStyle(viewport,"smooth")
</pre>
</p>
<p>
To get these exact images with the colored background you need
to use GIMP to set the background. The steps I used are:
<ol>
<li>Save the image as a pixmap</li>
<li>Open the saved file in gimp</li>
<li>Dialogs->Colors->ColorPicker button</li>
<li>Eyedrop the color of the web page</li>
<li>Set the color as the foreground on the FG/BG page</li>
<li>Dialogs->Layers</li>
<li>Duplicate Layer</li>
<li>Layer->Stack->Select bottom layer</li>
<li>Edit->Fill with Foreground color</li>
<li>(on Layers panel)Select image</li>
<li>(on Layers panel) Mode->Darken Only</li>
</ol>
Note that you may have to use "lighten only" first before it will
allow you to choose "darken only".
</p>
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