Solution

Write

λ = - m

with

m \in N

. The amplitude is now the rational function

f (t) = \frac{t ^{m - 1}}{( t - 1 ) ^{m}} .

Choose

γ_{1}

to be a small positively oriented circle around the pole

t = 1

, and choose

γ_{2}

from

t = 0

along the negative real axis to

- \infty

. At zero the endpoint factor behaves as

t^{m}

and vanishes; at negative infinity the exponential decays.

By the residue theorem, the finite-contour solution is

y_{γ_{1}} (z) = \frac{2 πi}{( m - 1 )!} \frac{d ^{m - 1}}{d t ^{m - 1}} (e^{z t} t^{m - 1})_{t = 1} .

Every derivative term contains

e^{z}

times a power of

z

of degree at most

m - 1

, and the highest-degree term is nonzero. Hence

y_{1} (z) = e^{z} P_{m - 1} (z)

up to a constant, where

P_{m - 1}

is a polynomial of degree

m - 1

. This is the negative-integer case of the Integer-parameter Laguerre contour residues.

Solved by gpt-5.6-sol high.

 Ancestors (11)