Content: This page contains a Java applet illustrating the 2D-visualisation of complex functions and instruction for its use.


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Analysis for Computer Scientists, Chapter 4


Visualisation of a complex function

To visualise a complex function, a pattern has to be chosen and a function in the single variable z must be entered in the field f(z)=. Alternatively, one can load a predefined example by using the combo box Load example. By pressing the Map button the image of the selected pattern is computed and drawn. By moving the cursor over an object of the pattern (e.g. a line), the object and its image are highlighted. This way the images of selected grid lines can be studied.




The tab Pattern offers the option for choosing a number of different patterns and coordinate systems in the preimage. To this end, select one of the four options Grid, Polar coordinates, House or Cut from the combo box Pattern. Each pattern has a number of self-explanatory options. The default values are shown as soon as a selection is made from the combo box.


In the Options tab a number of adjustments can be made regarding the behaviour of the applet. With the slider Discretisation one can choose how many points are used to discretise a given object (e.g. a grid line). If the option Check for singularities is active, the number of discretisation points should be sufficiently large. The check box Origin in the preimage/Origin in the image determines if the origin (0,0) is included in the preimage/image. With Extended highlighting a more involved highlighting mode is activated. In this mode colour interpolation along the edges in the preimage is employed. The same colours are used in the image. For example, load the exponential function ("exp(z)") from the combo box Load example in the tab Function and place the cursor onto one of the grid lines parallel to the imaginary axis to see this option in action.
With the option Check for singularities you can dedect whether singularities occur. This avoids wrong connections of points in the image. For example, create a grid in the unit square [-1,1]x[-1,1] and map the logarithmic function with and without activated singularity checking.


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Financially supported by

University of Innsbruck: New Media and Learning Technologies
Austrian Federal Ministry of Education, Science and Research

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