Degenerate roots of three transcendental equations involving spherical Bessel functions
Authors: Pexton Robert L. , Steiger Arno D. 
Published: June 2010

Roots of the three transcendental equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row j Subscript l Baseline left-parenthesis alpha lamda right-parenthesis y Subscript l Baseline left-parenthesis lamda right-parenthesis equals j Subscript l Baseline left-parenthesis lamda right-parenthesis y Subscript l Baseline left-parenthesis alpha lamda right-parenthesis comma 2nd Row left-bracket x j Subscript l Baseline left-parenthesis x right-parenthesis right-bracket prime Subscript x equals alpha eta Baseline left-bracket x y Subscript l Baseline left-parenthesis x right-parenthesis right-bracket Subscript x equals eta Superscript prime Baseline equals left-bracket x j Subscript l Baseline left-parenthesis x right-parenthesis right-bracket prime Subscript x equals eta Baseline left-bracket x y Subscript l Baseline left-parenthesis x right-parenthesis right-bracket Subscript x equals alpha eta Superscript prime Baseline comma EndLayout">

j
l

(
α
λ
)

y
l

(
λ
)
=

j
l

(
λ
)

y
l

(
α
λ
)
,

[
x

j
l

(
x
)

]

x
=
α
η

′

[
x

y
l

(
x
)

]

x
=
η

′

=
[
x

j
l

(
x
)

]

x
=
η

′

[
x

y
l

(
x
)

]

x
=
α
η

′

,

\begin {array}{*{20}{c}} {{j_l}(\alpha \lambda ){y_l}(\lambda ) = {j_l}(\lambda ){y_l}(\alpha \lambda ),} \\ {[x{j_l}(x)]_{x = \alpha \eta }^\prime [x{y_l}(x)]_{x = \eta }^\prime = [x{j_l}(x)]_{x = \eta }^\prime [x{y_l}(x)]_{x = \alpha \eta }^\prime ,} \\ \end {array}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket j Subscript l Baseline left-parenthesis x right-parenthesis right-bracket prime Subscript x equals alpha mu Baseline left-bracket y Subscript l Baseline left-parenthesis x right-parenthesis right-bracket Subscript x equals mu Superscript prime Baseline equals left-bracket j Subscript l Baseline left-parenthesis x right-parenthesis right-bracket prime Subscript x equals mu Baseline left-bracket y Subscript l Baseline left-parenthesis x right-parenthesis right-bracket prime Subscript x equals alpha mu">

[

j
l

(
x
)

]

x
=
α
μ

′

[

y
l

(
x
)

]

x
=
μ

′

=
[

j
l

(
x
)

]

x
=
μ

′

[

y
l

(
x
)

]

x
=
α
μ

′

[{j_l}(x)]_{x = \alpha \mu }^\prime [{y_l}(x)]_{x = \mu }^\prime = [{j_l}(x)]_{x = \mu }^\prime [{y_l}(x)]_{x = \alpha \mu }^\prime

\]
that are degenerate for certain values of the parameter

α
∈
(
0
,
1
)

\alpha \in (0,1)

are presented. The symbols

j
l

{j_l}

and

y
l

{y_l}

denote the spherical Bessel functions of the first and second kind. Root degeneracies are discussed for each equation individually as well as for pairs of equations. Only positive roots are considered, since the equations are invariant under the transformations

λ
→
−
λ

\lambda \to - \lambda

,

η
→
−
η

\eta \to - \eta

, and

μ
→
−
μ

\mu \to - \mu

. When

l
=
0

l = 0

, only the third equation has nontrivial roots. These roots are identical with the roots of the first equation for

l
=
1

l = 1

, i.e.

μ

0
n

=

λ

1
n

(
n
=
1
,
2
,
…
)

{\mu _{0n}} = {\lambda _{1n}}(n = 1,2, \ldots )

. Various graphs of

λ

l
n

{\lambda _{ln}}

,

η

l
n

{\eta _{ln}}

, and

μ

l
n

{\mu _{ln}}

display root-degeneracies as intersections of curves. Accurate values of degenerate roots with the corresponding values of

α
\alpha

are exhibited in tables. Roots of the third equation for

l
=
1
(
1
)
15

l = 1(1)15

,

n
=
1
(
1
)
30

n = 1(1)30

,

α
=
0.1
(
0.1
)
0.7

\alpha = 0.1(0.1)0.7

, together with their minima and associated values of the parameter

α
\alpha

, are given in the microfiche supplement accompanying this issue. Roots of the first and the second equation and minima of roots of the second equation are published in v. 31 and v. 32 of this journal. The roots of the first two equations determine the eigenfrequencies of the transverse electric and the transverse magnetic normal modes of an ideal cavity resonator bounded by two concentric spheres (

r
=
α
R

r = \alpha R

and

r
=
R

r = R

). The roots of the third equation determine the frequencies of the irrotational magnetic eigenfields.