Graphics

Contents

• Real Argument
• The Psi Function
• Complex Argument

Real Argument

 -- 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)

You can construct this graph with the Axiom commands:

  -- 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")

The Psi Function ψ ( x )

This function is a special case of the polygamma function. In particular, ψ ( x ) is equal to polygamma(0,x).

  -- 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")

Complex Argument

  -- 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")

You can reproduce this image from Axiom with:

  -- 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")

To get these exact images with the colored background you need to use GIMP to set the background. The steps I used are:

1. Save the image as a pixmap
2. Open the saved file in gimp
3. Dialogs->Colors->ColorPicker button
4. Eyedrop the color of the web page
5. Set the color as the foreground on the FG/BG page
6. Dialogs->Layers
7. Duplicate Layer
8. Layer->Stack->Select bottom layer
9. Edit->Fill with Foreground color
10. (on Layers panel)Select image
11. (on Layers panel) Mode->Darken Only