Calculate an Iteration

Calculator for iterations with one start value. An iteration is a repeated calculation with previously computed values. As iteration variable in the formula, z is used. Only this variable may occur in the iteration term. The basic arithmetic operations + - * / are allowed, as well as the power function pow(), like pow(2#z) for 2^z. Further allowed functions are sin(), cos(), tan(), asin(), acos(), atan(), log() for the natural logarithm and fac() for the factorial. Also, the constants e and pi can be used. Complex numbers can't be computed.
When all steps are shown, the sum Σ of the single iteration steps is also calculated.

Example: z = z² + c is the construction sequence for the Mandelbrot set. If you set c=-1.9, the formula to enter is z*z-1.9. This generates chaotic values in the interval ]-2;2[.

Repeated execution one after the other with the previous result can lead to completely different results for slightly different starting values ​​after a certain number of runs. This is the case, for example, with the Mandelbrot set mentioned above, where the boundary between values ​​that tend towards infinity and others is fractal and chaotic. The branch of mathematics that deals with such processes is called chaos theory. Regularities can also be recognized in chaos, but this is more complicated than would be possible by looking at individual iteration sequences. Nevertheless, such iterations are often the basis for complex processes. An example of a familiar chaotic process that is based on the values ​​it has achieved so far is the weather. Within a certain frame, this is easy to predict, but if this frame (often time) is exceeded, it becomes chaotic and unpredictable. Nevertheless, there are long-term regularities; for example, at a certain distance from the equator, it will always be warmer on average in summer than in winter.