I am confused about this notation: $z_{n+1}=z_n^2+i$

Question

I am confused about this notation:

$$z_{n+1}=z_n^2+i$$ (original image)

I am confused about this notation on the right side of z. What is it? Does it have something to do with Pascal’s triangle?

Answer 1

$$z_{n+1} = z_n^2 +i$$ looks as if it may be a complex recurrence relation with $i=\sqrt{-1}$ and $z_n^2$ being the square of $z_n$

Answer 2

$z_n^2$ is simply the square of the value of the variable $z_n$, that is $z_n \cdot z_n$.

Answer 3

It doesn't necessarily have anything to do with Pascals triangle.

If you have an indeterminate number of variable $x,y,z,w,u,v,.....$ it doesn't make sense to name them different letters because you don't know how many your have, you won't necessarily be able to keep track of which is which and, well it's just confusing.

What we do instead is index the variables and call the $x_1, x_2, x_3, x_4,.....$. Keep in mind variable names don't mean anything; these are just labels to keep them straight.

Now sometimes the variable represent a sequence and one term in the sequence determines the next. For example if $a_1 = 5$ and $a_2 = 7*a_1 + 3= 7*5 + 3= 38$ and $a_3 = 7*a_2 + 3 = 7*38 + 3 = 269$ and so on, we might write:

$a_{n+1} = 7*a_n + 3$ meaning that the next term (the $n+1$th term) is $7$ times the current term (the $n$th term) plus $3$.

In this case, our sequence is a bunch of $z_1, z_2, z_3,....$ which the formula where the $n+1$th term is the $n$th term squared plus $i$. That is $z_{n+1} = (z_n)^2 + i$.

....

I think the assumption is that the $z_1, z_2, z_3,...$ are all complex numbers and $i$ represents the imaginary unit (often, naively but not entirely accurately, introduced as $\sqrt{-1}$).

So if $z_1$ is some complex number $z_1 = a+ bi$ then $z_2 = z_1^2 + i = (a+bi)^2 + i = (a^2 - b^2 + 2abi) + i = [a^2-b^2] + (2ab + 1)i$ and so on.....

Answer 4

As someone else's answer has mentioned, this looks like a complex recurrence relationship with every subsequent value $z_{n+1}$ being calculated by taking the current value $z_n$, squaring it and adding a fixed complex constant $c$, in this simply $i$.

To provide a bit more context since your question didn't have a lot, this seems related to the algorithm used to generate the Quadratic Julia Set, a very famous fractal.