## 数学代写|复分析作业代写Complex function代考|Transferring Complex Functions to the Sphere

2023年4月6日

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## 数学代写|复分析作业代写Complex function代考|Transferring Complex Functions to the Sphere

Stereographic projection enables us to transfer the action of any complex function to the Riemann sphere. Given a complex mapping $z \mapsto w=f(z)$ of $\mathbb{C}$ to itself, we obtain a corresponding mapping $\widehat{z} \mapsto \widehat{w}$ of $\Sigma$ to itself, where $\widehat{z}$ and $\widehat{w}$ are the stereographic images of $z$ and $w$. We shall say that $z \mapsto w$ induces the mapping $\widehat{z} \mapsto \widehat{w}$ of $\Sigma$

For example, consider what happens if we transfer $f(z)=\bar{z}$ to $\Sigma$. Clearly [exercise],
Complex conjugation in $\mathbb{C}$ induces a reflection of the Riemann sphere in the vertical plane passing through the real axis.
For our next example, consider $z \mapsto \widetilde{z}=(1 / \bar{z})$, which is inversion in the unit circle C. Figure [3.21b] shows a vertical cross section of $\Sigma$ taken through $\mathrm{N}$ and the point $z$ in $\mathbb{C}$. This figure also illustrates the very surprising result of transferring this inversion to $\Sigma$ :
Inversion of $\mathbb{C}$ in the unit circle induces a reflection of the Riemann sphere in its equatorial plane, $\mathbb{C}$.
Here is an elegant way of seeing this. First note that not only are the pair of points $z$ and $\tilde{z}$ symmetric (in the two-dimensional sense) with respect to $C$, but they are also symmetric (in the three-dimensional sense) with respect to the sphere $\Sigma$. Now apply the three-dimensional preservation of symmetry result (3.13). Since $z$ and $\widetilde{z}$ are symmetric with respect to $\Sigma$, their stereographic images $\widehat{z}=\mathcal{J}_K(z)$ and $\widehat{\widetilde{z}}=\mathcal{J}_K(\widetilde{z})$ will be symmetric with respect to $\mathcal{I}_K(\Sigma)$. But $\mathcal{J}_K(\Sigma)=\mathbb{C}$. Done! A more elementary (but less illuminating) derivation may be found in Ex. 6.

By combining the above results, we can now find the effect of complex inversion on the Riemann sphere. In $\mathbb{C}$, we know that $z \mapsto(1 / z)$ is equivalent to inversion in the unit circle, followed by complex conjugation. The induced mapping on $\Sigma$ is therefore the composition of two reflections in perpendicular planes through the real axis-one horizontal, the other vertical.

## 数学代写|复分析作业代写Complex function代考|Behaviour of Functions at Infinity

Suppose two curves in $\mathbb{C}$ extend to arbitrarily large distances from the origin. Abstractly, one would say that they meet at the point at infinity. On $\Sigma$ this becomes a literal intersection at $\mathrm{N}$, and if each of the curves arrives at $\mathrm{N}$ in a well defined direction, then one can even assign an “intersection angle at $\infty$ “. For example, [3.20] illustrates that if two lines in $\mathbb{C}$ intersect at a finite point and contain an angle $\alpha$ there, then they intersect for a second time at $\infty$ and they contain an angle $-\alpha$ at that point.

Transferring a complex function to the Riemann sphere enables one to examine its behaviour “at infinity” exactly as one would at any other point. In particular, one can look to see if the function preserves the angle between any two curves passing through $\infty$. For example, the result (3.19) shows that complex inversion does preserve such angles at $\mathrm{N}$, and it is therefore said to be “conformal at infinity”. By the same token, this rotation of $\Sigma$ will also preserve the angle between two curves that pass through the singularity $z=0$ of $z \mapsto(1 / z)$, so complex inversion is conformal there too. In brief, complex inversion is conformal throughout the extended complex plane.

In this chapter we have found it convenient to depict $z \mapsto w$ as a mapping of $\mathbb{C}$ to itself, and in the above example we have likewise interpreted the induced mapping $\widehat{z} \mapsto \widehat{w}$ as sending points on the sphere to other points on the same sphere. However, it is often better to revert to the convention of the previous chapter, whereby the mapping sends points in the z-plane to image points residing in a second copy of $\mathbb{C}$, the $w$-plane. In the same spirit, the induced mapping $\widehat{z} \mapsto \widehat{w}$ may be viewed as mapping points in one sphere (the $z$-sphere) to points in a second sphere (the $w$-sphere). We illustrate this with an example.

Consider $z \mapsto w=z^n$, where $n$ is a positive integer. The top half of [3.22] illustrates the effect of the mapping (in the case $n=2$ ) on a grid of small “squares” abutting the unit circle and two rays containing an angle $\theta$. Very mysteriously, the images of these “squares” in the $w$-plane are again almost square. In the next chapter we will show that this is just one consequence of a more basic mystery, namely, that $z \mapsto w=z^n$ is conformal. Indeed, we will show that if a mapping is conformal, then any infinitesimal shape is mapped to a similar infinitesimal shape.

# 复分析代写

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## MATLAB代写

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