Complex Function Plotter (samuelj.li)
The color here denotes the argument. We see clearly that when we circle around the point 0 or 0.5, the image will circle around the origin(clockwise or counterclockwise).
[!Argument principle]
For a meromorphic function, the difference between the number of zeros enclosed by the curve and the number of singularities is precisely the winding number of the image curve around the origin. ![[Blogs/Pasted image 20230617185853.png|750]]
Rouché's Theorem Let $\Omega$ be a bounded region enclosed by a piecewise smooth curve, and let $f(z)$ and $g(z)$ be holomorphic functions on $\Omega$. If $$|f(z) - g(z)| < |g(z)|, \quad z \in \partial\Omega,$$ then $f(z)$ and $g(z)$ have the same number of zeros (counted with multiplicity) in $\Omega$.
proof Divide both sides by $g(x)$. Apply the argument principle. The conditions here are somewhat restrictive. It is not necessary for $f(x)$ and $g(x)$ to be holomorphic; the conclusion remains valid if the difference between the number of zeros and the number of poles is equal.
There is also a form of Rouché's Theorem as follows:
![[Blogs/Pasted image 20230617194216.png|i.e. the image set of f(x)/g(x) doesn't intersect with R+|700]]
How to interpret the image
[!tip] How to tell the pole from the zero? Look at the small squares around the point. The dense ones are poles, and the ones that can be distinguished are zeros. Bright colors represent poles, while dark colors represent zeros.
[!tip] How to check if the map is conformal? If the angles formed by the small squares are all 90 degrees, then it is a conformal map.