Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 4 / d

Table of contents
- i d
  - Solution i
- ii d
  - Solution ii

i

Solution

Let

A = H \cap B (1, ϵ)

. Its normalized mapping-out map is

ψ_{A} (z) = z + \frac{ϵ ^{2}}{z - 1} + ϵ^{2},

so

ψ_{A} (0) = 0

and

ψ_{A}^{'} (0) = 1 - ϵ^{2} .

Using the restriction exponent

5/8

gives

P (γ [0, \infty) \cap (H \cap B (1, ϵ)) = \emptyset) = (1 - ϵ^{2})^{5/8} .

ii

Solution

For the vertical slit

A = (1, 1 + i ϵ]

, choose the square-root branch asymptotic to

z - 1

at infinity. The normalized map is

ψ_{A} (z) = (z - 1)^{2} + ϵ^{2} + 1 + ϵ^{2},

for which

ψ_{A} (0) = 0

and

ψ_{A}^{'} (0) = \frac{1}{1 + ϵ ^{2}} .

Therefore

P (γ [0, \infty) \cap (1, 1 + i ϵ] = \emptyset) = (1 + ϵ^{2})^{- 5/16} .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version