File:Factorialz.jpg

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

Description:

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.