File:Factorialz.jpg

Factorial in the complex plane. Copy from http://en.citizendium.org/wiki/Image:Factorialz.jpg

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In the complex \(z\)-plane, the lines of constant \(u=\Re(z!)\) and the lines of constant \(v=\Im(z!)\) are shown. The levels \(u= -24, -20, -16, -12, -8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,12,16,20,24\) are drown with thick black lines. some of intermediate levels \(u=\)const are shown with thin blue lines for positive values and with thin red lines for negative values.
The levels \(v= -24, -20, -16, -12, -8,-7,-6,-5,-4,-3,-2,-1\) are shown with thick red lines.
The level \(v=0\) is shown with thick pink lines.
The levels \(v=1,2,3,4,5,6,7,8,12,16,20,24\) are drown with thick blue lines. some of intermediate levels \(v=\)const are shown with thin green lines.
The dashed blue line shows the level \(u=\mu_0\approx 0.88560319441\) and corresponds to the value of the principal local minimum of the factorial of the real argument.

The dashed red line shows the level \(u=\mu_1\approx -3.544643611\) and corresponds to the value of the first local maximum of the factorial of the real argument.