The 5 ways to visualize complex functions | Essence of complex analysis #3

What people said

Recommends Daniel · 2021-12 · Mathematics Stack Exchange · on “Elementary Linear Algebra Question”

I googled 'visual complex functions' and, in addition to the book you mentioned, a good video appeared: [this video]

Recommends ConMan · 2024-02 · Mathematics Stack Exchange · on “Understanding and Visualizing Complex Roots According to the Fundamental Theorem of Algebra”

While FTOA tells us that every polynomial has a certain number of roots, it doesn't give us a method for finding them. In general, while the roots of any linear or quadratic polynomial can be found through basic algebra, finding the roots of a cubic or quartic is quite difficult and for quintics and higher there is no guarantee that the roots can be exactly calculated. Beyond that, many of the methods used for working with real roots can be extended to complex numbers, for example: Vieta's formulas [LINK] still apply. A version of the rational root theorem [LINK] can be written in terms of Gaussian rationals [LINK] . If you can find some of the roots exactly, you can perform polynomial long

Cites Sammy Black · 2022-07 · Mathematics Stack Exchange · on “Images of curves passing through zeros of a complex polynomials”

The "bouncing" of the graph of a polynomial of even multiplicity over the reals is a direct consequence of the fact that $(-x)^{2n} = x^{2n}$ for any natural $n$ . For a given polynomial $p(x)$ and root $r$ of multiplicity $2n$ , you can see this behavior more clearly by constructing a particular approximating polynomial $\hat{p}(x) = \hat{p}_r(x)$ obtained by evaluating $x=r$ in all but the factors of $(x-r)$ . In other words, write $p(x) = q(x) \, (x-r)^{2n}$ for some polynomial $q(x)$ such that $q(r) \neq 0$ . Then, we define the approximation $\hat{p}(x) = q(r) \, (x-r)^{2n}$ . Since $$ \lim_{x \to r} \frac{p(x)}{\hat{p}(x)} = \lim_{x \to r} \frac{q(x)}{q(r)} = 1, $$ we can see that $\ha