Generating full sequence with complex numbers.

[smithnya]

Hello everyone,

I need some help with the following: I understand that by using x_n = ax_n-1+b we can generate a full sequence of numbers. For example, if x₁=ax₀+b, then x₂ = ax₁+b = a²x₀+ab+b, and so on and so forth to x_n. I need help applying this same concept to complex numbers (a+bi). Is it even possible? I think it is, but I can't figure it out. Can some one lend a hand?

 

[tiny-tim]

hello smithnya!

this is a recurrence relation

its solutions should be of the same form, whether the constants are real or complex

were you having a problem with any particular relation?​

 

[smithnya]

Well, my professor began to explain the relation among real numbers, and he explained for x₀, x₁, x₂, etc. He mentioned that the same could be done with complex numbers, but never went into detail, maybe he will explain later. It piqued my curiosity, but I can't figure out how to generate a full sequence using the same method above only with something of the form a+bi.

 

[HallsofIvy]

As tiny tim said, it is exactly the same thing: $x_{n+1}= ax_n+ b$ will give complex numbers if anyone or more of a, b, and $x_0$ is complex.

 

[blue_raver22]

I can confirm that it is indeed possible to generate a full sequence of complex numbers using a similar formula. The only difference is that instead of using a and b as constants, we will use complex numbers (a+bi) as our coefficients.

For example, if we start with x1 = (a+bi)x0 + (c+di), then x2 = (a+bi)x1 + (c+di) = (a+bi)((a+bi)x0 + (c+di)) + (c+di) = a2x0 + 2abi + b2i2 + (ac+bd)i + (ad-bc)d.

We can continue this process to generate a full sequence of complex numbers. It is important to note that the sequence may become more complex and difficult to calculate as the values of a, b, c, and d change. However, the concept remains the same and it is possible to generate a full sequence with complex numbers using this formula.

I hope this helps and good luck with your calculations!