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Unformatted text preview: of z = 1 of
length at least π . It follows that the solution to (5.31) must have inﬁnitely many
zeros in the region z > 1, and J0 (x) must have inﬁnitely many positive zeros,
as claimed.
The fact that Jn (x) has inﬁnitely many positive zeros could be proven in a
similar fashion. 149 Exercises:
5.8.1. Solve the following initial value problem for the heat equation in a disk:
Find u(r, θ, t), 0 < r ≤ 1, such that
∂u
1∂
=
∂t
r ∂r r ∂u
∂r u(r, θ + 2π, t) = u(r, θ, t), + 1 ∂2u
,
r2 ∂θ2 u wellbehaved near r = 0, u(1, θ, t) = 0,
u(r, θ, 0) = J0 (α0,1 r) + 3J0 (α0,2 r) − 2J1 (α1,1 r) sin θ + 4J2 (α2,1 r) cos 2θ.
5.8.2. Solve the following initial value problem for the vibrating circular membrane: Find u(r, θ, t), 0 < r ≤ 1, such that
∂2u
1∂
=
∂t2
r ∂r
u(r, θ + 2π, t) = u(r, θ, t), r ∂u
∂r + 1 ∂2u
,
r2 ∂θ2 u wellbehaved near r = 0, ∂u
(r, θ, 0) = 0,
∂t
u(r, θ, 0) = J0 (α0,1 r) + 3J0 (α0,2 r) − 2J1 (α1,1 r) sin θ + 4J2 (α2,1...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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