====== Convergences ======

In contrast to the set of real numbers, the complex plane often exhibits unexpected behaviour, for example when applying integral and differential calculus to complex functions. Another example is iterations.

Here, it becomes apparent that – in contrast to real space – iterations for points that are close together in the complex plane can lead to completely different results. If we plot the divergence rate, i.e. the number of iterations until a maximum difference from the previous value is reached, on the starting points in different colours, we obtain an irregularly shaped structure, a fractal.

CGRAPH offers the creation of fractals using Mandelbrot and Julia sets, in which the visible image area is examined for convergence or divergence and the convergence speed is displayed in colour. Complex functions can also be examined for convergence or divergence through iteration. A graphic shows the iteration process.

^Symbol^Graphic^Description^
|{{symbol:icon_Mandelbrot.png?64x64}}|[[Mandelbrot sets]]|Creates an iterative graphic using a function z<sub>n+1</sub> = f(z<sub>n</sub>, param) for a starting value z<sub>0</sub>, in which the parameter is varied.|
|{{symbol:icon_Julia.png?64x64}}|[[Julia sets]]|Creates an iterative graphic using a function z<sub>n+1</sub> = f(z<sub>n</sub>, param) for fixed parameters, where the starting value is varied.|
|{{symbol:icon_Konvergenz.png?64x64}}|[[Function convergence]]|Determines the iterative progression of complex functions z<sub>n+1</sub> = f(z<sub>n</sub>, param) for specified initial values z<sub>0</sub> and parameters and detects whether they converge or diverge.|

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