Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 11F / c

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- Solution c

Solution

Choose continuous argument lifts

θ_{1}, θ_{2}

for the two curves. Their sum is a lift for their pointwise product, because

γ_{1} (t) γ_{2} (t) = ∣ γ_{1} (t) ∣∣ γ_{2} (t) ∣ e^{i (θ_{1} (t) + θ_{2} (t))} .

Subtract its initial value from its final value to obtain

w (γ_{1} γ_{2}) = w (γ_{1}) + w (γ_{2}) .

In particular, winding numbers subtract under pointwise division of curves avoiding zero.

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