Complex functions behave very similarly to real functions. They take inputs (complex numbers), perform some operation on them, and then produce an output. For example:
f(z)= 9z takes inputs and multiples them by 9.
f(1+i) = 9(1+i) = 9 + 9i
For the example f(z) = z^2 :
f(z)= 9z takes inputs and multiples them by 9.
f(1+i) = 9(1+i) = 9 + 9i
For the example f(z) = z^2 :
In order to graph complex functions, four dimensions are necessary. This is because both the inputs and outputs require a two dimensional plane to be graphed (since they are complex). Thus, we add the dimensions together: 2D for inputs + 2D for outputs = 4D.
However, mathematicians can represent complex functions as mappings of inputs to outputs. The diagram below demonstrates this:
However, mathematicians can represent complex functions as mappings of inputs to outputs. The diagram below demonstrates this:
By creating a plane for the inputs and a separate one for the outputs, we can show the relationship between them with no more than two dimensions. z1 maps to w1, and z2 maps to w2.