13E (Complex Analysis or Complex Methods)

Write

p (z) = c \prod_{j = 1}^{2 m} (z - a_{j})

, including root multiplicities. For

∣ z ∣ > R

the convergent logarithm series for every factor

1 - a_{j} / z

L_{j} (z) = - \sum_{n \geq 1} \frac{a _{j}^{n}}{n z ^{n}}, e^{L_{j}} = 1 - a_{j} / z .

h (z) = c z^{m} exp (\frac{1}{2} j \sum L_{j} (z)) .

Its square is

p

. The even degree makes

z^{m}

single-valued, avoiding the winding obstruction that an odd degree would create on the exterior annulus.

For the branch behaving as

z^{2}

z^{4} - z = z^{2} (1 - z^{- 3})^{1/2} = z^{2} - \frac{1}{2} z^{- 1} - \frac{1}{8} z^{- 4} - \dots .

It converges uniformly on the circle of radius two. Only the coefficient of

z^{- 1}

contributes to its anticlockwise integral, so

\int_{C} z^{4} - z d z = - πi

for this branch, and

+ πi

for the other. These are the two allowed answers.

 Ancestors (9)

There are no discussions about this article yet.

There are currently no matching articles.