# Complex map

**Complex map** is the graphical representation of a holomorphic function with the isolines of its real part and those of its imaginary part.

For any holomorphic function, in any point, the isolines of the real part are orthogonal to those of the imaginary part; therefore the the complex maps have specific mesh-like structure.

In TORI, the complex maps are used for illustration many functions, and in particular, those that yet are underrepresented in the literature: tetration, arctetration, SuperFactorial, AbelFactorial, etc.

For plotting of complex maps, the isolines of the real part and those of the imaginary part can be plotted separately as contour plot, and then combined at the same graphic. The C++ routine for the drawing of contour plots is suggested, namely, conto.cin. Such a routine seems to be for orders of magnitude faster than the similar routines in Mathematica or Maple; in addition, at the similar resolution, the resulting PDF files are very compact compared to those produced by the commercial software mentioned.

The complex maps allows the visual comparison of the function with just overlapping. An alternative of the complex map is the encoding where the modulus of a function is represented with the intensity of light from the screen, and the phase is shown by the color.

## References

http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf