How do I find a complex prisoner number for f(z)=z^2+3?
I have tried all the numbers but they all seem to be going up
Huh ??
It has to do with Julias set
To find the complex prisoner number for a given function, you need to understand the concept of fixed points and the behavior of the function under iteration. In the case of the function f(z) = z^2 + 3, a complex prisoner number is a complex number z such that when the function is iterated repeatedly (i.e., applying f(z) to the result multiple times), the values of the iterates remain bounded.
To find the complex prisoner number, you can follow these steps:
1. Start with an initial complex number z0.
2. Apply the function f(z) = z^2 + 3 to z0: z1 = f(z0).
3. Repeat step 2, applying f(z) to the previous iterate: z2 = f(z1), z3 = f(z2), and so on.
4. Observe the values of the iterates (z1, z2, z3, ...) and check if they approach infinity or become too large. If the values "blow up" or "diverge" towards infinity, then the complex number z0 is not a prisoner number.
5. However, if the values remain bounded, even after many iterations, then z0 can be considered a prisoner number for the given function.
In the case of the function f(z) = z^2 + 3, you may notice that for many initial values, the iterates seem to become larger and larger, suggesting that there might not be a complex prisoner number. However, it is worth noting that complex dynamics can be quite intricate, and it may be difficult to manually find a complex prisoner number without some advanced mathematical tools or methods.
One technique you can employ is to plot the iterates on the complex plane using a computer software or programming language capable of generating complex fractals, such as Python with libraries like Matplotlib or Julia. This visual representation can reveal patterns and help identify whether there are any complex prisoner numbers.
Another approach is to use numerical methods such as the Newton-Raphson or Halley's method to solve the equation f(z) - z = 0, assuming there is a fixed point where f(z) = z.
Keep in mind that finding complex prisoner numbers can be a challenging task, and it may require advanced mathematical techniques or computational tools.