Special techniques for height surfaces, real and complex functions

Special techniques for height surfaces, real and complex functions#

Wpf figures as standard display of complex functions#

This is a description of the standard display of complex functions, using the square function as an example:

The figures below are showing the real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex function \(z = \mathrm{sqr}(x + iy)\) with \(-2 \le x \le 2\) (blue axis), \(-2 \le y \le 2\) (red axis), \(-10 \le z \le 10\) (green axis).

01a_TestSquare_re1 \(\quad\) 01b_TestSquare_im1 \(\quad\) 01c_TestSquare_abs1

Note

Although the range of the green axis is stated in the form \(z_{\text{min}} \le z \le z_{\text{max}}\), with \(z_{\text{min}} \ne 0\) in general, this applies only for the figures showing the real and imaginary part. For the figure showing the absolute value we have always \(z_{\text{min}} = 0\). This note is omitted from the standard text describing complex functions in this manual for better readability.

Truncation vs loglog transformation#

TestTruncated \(\quad\) TestLogLog

3D wpf plot: Surface plot without cutting of the branch cut (left figure), Surface plot with cutting of the branch cut (right figure)

Displaying branch cuts#

TestWithoutCut \(\quad\) TestWithCut

3D wpf plot: Surface plot without cutting of the branch cut (left figure). Surface plot with cutting of the branch cut (right figure).

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