This exercise outlines how to extend some of the results of this chapter to the Riemann sphere as defined in Section 3.2. Suppose G⊆C is a region that contains 0, let f be a continuous function on G, and let γ⊂G∖{0} be a piecewise smooth path in G avoiding the origin, parametrized as γ(t), a≤t≤b.
(a) Show that
∫γf(z)dz=∫σf(z1)z21dz(5.2)
where σ(t):=γ(t)1, a≤t≤b.
(b) Now suppose γ is closed and limz→0f(z1)z21=L is finite. Let H:={z1:z∈G∖{0}} and define the function g:H∪{0}→C by
g(z):={f(z1)z21Lif z∈H,if z=0.
Thus g is continuous on H∪{0} and (5.2) gives the identity ∫γf=∫σg. In particular, we can transfer certain properties between these two integrals. For example, if ∫σg is path independent, so is ∫γf. Here is but one application:
i) Show that ∫γzndz is path independent for any integer n=−1.
ii) Conclude (once more) that ∫γzndz=0 for any integer n=−1.