The expansions here developed are required for the author’s discussion of "Meteorological Perturbations of Tides and Currents in an Unlimited Channel rotating with the Earth” (
v. supra
, p. 170). Let
η
(
x
) be a real differentiable function of
x
defined in the range 0 ≼
x
≼ 1, and satisfying the condition
η
(
x
) >
c
> 0 for all such
x
. Let
ϕ
λ
(
x
) and ψ
λ
(
x
) be functions of the real variable
x
and the complex parameter
λ
, defined in the above range by the equations
d
/
dx
[
η
(
x
)
dϕ
λ
(
x
)/
dx
] + (
λ
+
iγ
)
ϕ
λ
(
x
) = -1,
d
/
dx
[
η
(
x
)
dψ
λ
(
x
)/
dx
] + (
λ
+
iγ
)
ψ
λ
(
x
) = -1 (1) together with the boundary conditions
ϕ'
λ
(0) = 0,
ψ'
λ
(0) = 0,
ϕ'
λ
(1) = 0,
ψ
λ
(1) = 0, (2)
γ
being a prescribed constant.
Roots of differentiable functions of one real variable
