Complex numbers: Complex functions
Complex polynomial functions
The below dynamic figure illustrates the effect of the complex function \(f(z)=z^2+1\) on points in the complex plane; the red draggable point is again the original and the green point is the image under the given mapping \(f\). Experiment to get an idea of the behaviour of the function.
\(\phantom{x}\)
In the last example it is not so clear what kind of mapping in the complex plane we have. But maybe you have observed during your experimenting in the dynamic figure that the numbers \(\mathrm{i}\) and \(\mathrm{-i}\) are mapped onto \(0\) (What does this mean for the points in the complex plane?). That's not surprising when you realize that the function rule equals \(f(z)=z^2+1\); they are the only two complex numbers of which the square is equal to \(-1\). The function \(f(z)=z^2+1\) is an example of a complex quadratic function.
An expression of the form \[a_nz^n+a_{n-1}z^{n-1} + \cdots + a_2z^2+a_1z+a_0,\] in which \(a_0,a_1,\ldots, a_n\) arte complex numbers with \(a_n\neq 0\) and \(z\) is a variable, is called a polynomial. The parameters \(a_0, a_1, \ldots, a_n\) are called the coefficients of the polynomial. If all coefficients of a polynomial are reea, we speak of a real polynomial. The highest exponent \(n\) is called the degree of the polynomial. The term \(a_nz^n\) is called the leading term of the polynomial (for which the highest coefficient \(a_n\), also called eading coefficient, is not zero because otherwise you would have be omitted this term).
Let \(p(z)\) be a complex polynomial. The function that maps a complex number \(z\) to the value \(p(z)\) is called a complex polynomial function. If the degree of the polynomial \(p(z)\) is equal to 1, then we deal with a complex linear function; when the degree of the polynomial \(p(z)\) is equal to 2, then we are dealing with a complex quadratic function.
In the given example we see that we can also search for roots and fixed points for complex functions to
The complex number \(\alpha\) is a root of the complex function \(f\) if \(f(\alpha)=0\).
Note that if the polynomial is real, that is, if all coefficients are real, then the roots of the polynomial do not need to be real. The simplest example is the polynomial \(z^2+1\) with \(\ii\) and \(-\ii\) asroots.
A fixed point of a complex function \(f\) is a complex number \(\beta\) for which \(f(\beta)=\beta\) .
We can introduce concepts such as differentiability in a similar way as we have done for real functions and often the results are similar; for example, \(\dfrac{d}{dz}(z^3-z^2+z)=3z^2-2z+1\).
We do not go much further here. However, henceforth we use that all calculation rules for functions such as addition, multiplication, division and composition remain valid for complex functions and that all operations with algebraic expressions keep going well with expressions with complex numbers or with variables in which complex numbers can be substituted.
