Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 3 / c / i

Table of contents
- Solution i

Solution

On the unit circle use

z = e^{i θ}

with

0 < θ < 2 π

. The specified branch has

Log z = i θ

there and

d z = i e^{i θ} d θ

. Therefore

I = - \int_{0}^{2 π} θ e^{i θ} d θ = - [(1 - i θ) e^{i θ}]_{0}^{2 π} = 2 πi .

This is the circle integral of a logarithm with a radial branch cut, using the one-sided values at the cut crossing. The logarithm is discontinuous at that crossing, so the Cauchy integral theorem for a function holomorphic around a closed contour does not apply.

Figure 1.
The cut-crossing point determines the circle integral of a logarithm
. Original unit-circle illustrations of the two radial branch cuts. The formula is $2 πi$ times the cut-crossing point; the second circle represents the general radius- $R$ case by rescaling. Details.

 Ancestors (13)

 View article source

 Discussion (0) New discussion

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