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MATH306 SUPPLEMENTARY MATERIAL
A BRIEF INTRODUCTION TO BESSEL and
RELATED SPECIAL FUNCTIONS
Edmund Y. M. Chiang
c Draft date December 1, 2008
Contents
Contents
i
1 Trigonometric and Gamma Functions
3
1.1
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Bessel Functions
9
2.1
Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Basic Properties of Bessel Functions . . . . . . . . . . . . . . . . . . . 12
2.3.1
Zeros of Bessel Functions . . . . . . . . . . . . . . . . . . . . . 12
2.3.2
Recurrence Formulae for Jν . . . . . . . . . . . . . . . . . . . 12
2.3.3
Generating Function for Jn . . . . . . . . . . . . . . . . . . . . 13
2.3.4
Lommel’s Polynomials . . . . . . . . . . . . . . . . . . . . . . 14
2.3.5
Bessel Functions of half-integer Orders . . . . . . . . . . . . . 14
2.3.6
Formulae for Lommel’s polynomials . . . . . . . . . . . . . . . 14
2.3.7
“Pythagoras’ Theorem” for Bessel Function . . . . . . . . . . 15
2.3.8
Orthogonality of the Lommel Polynomials . . . . . . . . . . . 15
2.4
Integral formaulae
2.5
Asymptotic Behaviours of Bessel Functions . . . . . . . . . . . . . . . 17
2.5.1
2.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Addition formulae . . . . . . . . . . . . . . . . . . . . . . . . . 18
Fourier-Bessel Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
i
CONTENTS
3 An New Perspective
1
21
3.1
Hypergeometric Equations . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2
Confluent Hypergeometric Equations . . . . . . . . . . . . . . . . . . 23
3.3
A Definition of Bessel Functions . . . . . . . . . . . . . . . . . . . . . 23
4 Physical Applications of Bessel Functions
27
5 Separation of Variables of Helmholtz Equations
31
5.1
What is not known? . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2
CONTENTS
Chapter 1
Trigonometric and Gamma
Functions
1.1
Trigonometric Functions
Pythagoras Theorem:
sin2 x + cos2 x = 1
(1.1)
holds for all real x.
Addition Theorem:
(1.2)
sin(A + B) = sin A cos B + sin B cos A
(1.3)
cos(A + B) = cos A cos B − sin A sin B
hold for all angles A, B.
1.2
Gamma Function
We recall that
(1.1)
k! = k × (k − 1) × · · · 3 × 2 × 1.
Euler was able to give a correct definition to k! when k is not a positive integer. He
invented the Euler-Gamma function in the year 1729. That is,
Z ∞
(1.2)
Γ(x) =
tx−1 e−t dt.
0
3
4
CHAPTER 1. TRIGONOMETRIC AND GAMMA FUNCTIONS
So the integral will “converge” for all positive real x. Since
Z ∞
(1.3)
Γ(1) =
e−t dt = 1,
0
so one can show
(1.4)
Γ(x + 1) = xΓ(x),
and so for each positive integer n
(1.5)
Γ(n + 1) = n!.
In fact, the infinite integral will not only “converge” for all positive real x but can
be “analytically continued” into the whole complex plane C. We have
Z ∞
(1.6)
Γ(z) =
tz−1 e−t dt, <(z) > 0,
0
which is analytic in C except at the simple poles at the negative integers including
0. So it is a meromorphic function.
Euler worked out that
(1.7)
1
2
!=
1√
π.
2
So
1
11 9 7 5 3 1 5 !=
, , ,
!.
2
2 2 2 2 2 2
One can even compute negative factorial:
(1.8)
(1.9)
1.3
Γ(1/2) = (−1/2)(−3/2)(−5/2)(−19/2)(−11/2)Γ(−11/2).
Chebyshev Polynomials
Theorem 1.3.1 (De Moivre’s Theorem). Let z = r(cos θ + i sin θ) be a complex
number. Then for every non-negative integer n,
(1.1)
z n = rn (cos nθ + i sin nθ).
Proof It is clear that that the formula holds when n = 1. We assume it holds at the
integer n. Then the addition formulae for the trigonometric theorem imply that
z n+1 = rn+1 (cos θ + i sin θ)(cos nθ + i sin nθ)
= rn+1 [cos(n + 1)θ + i sin(n + 1)θ].
Thus, the result follows by applying the principle of induction.
1.3. CHEBYSHEV POLYNOMIALS
5
Exercise 1.3.2. Justify the formula for negative integers n.
Remark 1.3.1. De Moivre’s formula actually works for all real and complex n. See
MATH304.
Example 1.3.3 (Multiple angle formulae). Let n = 2 in De Movire’s formula yields
z 2 = cos 2θ + i sin 2θ.
(1.2)
But
(1.3)
z 2 = (cos θ + i sin θ)2 = cos2 θ − sin2 θ + i(2 cos θ sin θ).
Comparing the real and imaginary parts of (1.2) and (1.3) yields
cos 2θ = cos2 θ − sin2 θ
= 2 cos2 θ − 1
= 1 − 2 sin2 θ
and
(1.4)
sin 2θ = 2 cos θ sin θ.
Similarly, we have
(1.5)
z 3 = cos 3θ + i sin 3θ.
But
(1.6)
z 3 = (cos θ + i sin θ)3
= cos3 θ + 3 cos2 θ(i sin θ) + 3 cos θ(i sin θ)2 + (i sin θ)3
= cos3 θ − 3 cos θ sin2 θ + i(3 cos2 θ sin θ − sin3 θ).
Comparing the real and imaginary parts of (1.5) and (1.6) yields
cos 3θ = cos3 θ − 3 cos θ sin2 θ
= 4 cos3 θ − 3 cos θ
and
sin 3θ = 3 cos2 θ sin θ − sin3 θ
= 3 sin θ − 4 sin3 θ.
6
CHAPTER 1. TRIGONOMETRIC AND GAMMA FUNCTIONS
More generally, we have
(1.7)
cos nθ + i sin nθ = z n = (cos θ + i sin θ)n
n X
n
=
cosn−k θ(i sin θ)k
k
k=0
n X
n k
=
i cosn−k θ sink θ.
k
k=0
Equating the real parts of the last equation and noting that i2l = (−1)l give
n
n
n
n−2
2
cos nθ = cos θ −
cos
θ sin θ +
cosn−4 θ sin4 θ + · · ·
2
4
n
[n/2]
+ (−1)
(1.8)
cosn−2[n/2] θ sin2[n/2] θ.
2[n/2]
But each of the sine function above has even power. Thus the substitution of
sin2 θ = 1 − cos2 θ in (1.8) yields
[n/2] (1.9)
cos nθ =
X
l=0
l
X
n
n−2l
k l
cos
θ
(−1)
cos2k θ .
2l
k
k=0
Finally, it is possible to re-write the summation formula as
(1.10)
cos nθ =
[n/2] X
k=0
(−1)
k
[n/2] X
j=k
n
2j
j
cosn−2k θ.
k
Writing θ = arccos x for 0 ≤ θ ≤ π, and Tn (x) = cos(n arccos x). We deduce from
the above formula that Tn (x) is a polynomial in x of degree n. The polynomial Tn
is called the Chebyshev1 polynomial of the first kind of degree n. The first
few polynomials are given by
T0 (x) = 1;
T1 (x) = x;
T2 (x) = 2x2 − 1;
T3 (x) = 4x3 − 3x;
T4 (x) = 8x4 − 8x2 + 1;
T5 (x) = 16x5 − 20x3 + 5x;
······
1
P. L. Chebyshev (1821–1894) Russian mathematician. The collection of the polynomials
{Tn (x)}∞
0 is a complete orthogonal polynomial set on [−1, 1]
1.3. CHEBYSHEV POLYNOMIALS
7
The set of Chebyshev polynomials {Tn (x)} can generate a L2 [−1, 1] space. That is,
it is a Hilbert space.
Interested reader should consult the handbook [1] and the book [5] about the
Chebyshev polynomials by T. J. Rivlin. Our library has a copy of this book.
8
CHAPTER 1. TRIGONOMETRIC AND GAMMA FUNCTIONS
Chapter 2
Bessel Functions
2.1
Power Series
We define the Bessel function of first kind of order ν to be the complex function
represented by the power series
(2.1)
Jν (z) =
+∞
+∞
X
X
(−1)k ( 21 z)ν+2k
(−1)k ( 21 )ν+2k 2k
= zν
z .
Γ(ν
+
k
+
1)
k!
Γ(ν
+
k
+
1)
k!
k=0
k=0
Here ν is an arbitrary complex constant and the notation Γ(ν) is the Euler Gamma
function defined by
Z ∞
(2.2)
Γ(z) =
tz−1 e−t dt, <(z) > 0.
0
2.2
Bessel Equation
A common introduction to the series representation for Bessel functions of first kind
of order ν is to consider Frobenius method: Suppose
y(z) =
∞
X
ck z α+k ,
k=0
where α is a fixed parameter and ck are coefficients to be determined. Substituting
the above series into the Bessel equation
(2.1)
z2
dy
d2 y
+ x + (z 2 − ν 2 )y = 0.
2
dz
dz
9
10
CHAPTER 2. BESSEL FUNCTIONS
and separating the coefficients yields:
0=
=
2
2d y
z
dz 2
∞
X
+x
dy
+ (z 2 − ν 2 )y
dz
ck (α + k)(α + k − 1)z α+k +
k=0
∞
X
ck (α + k)z α+k
k=0
+ (z 2 − ν 2 ) ·
∞
X
ck z α+k
k=0
=
∞
X
2
ck [(α + k) − ν ]z
k=0
α+k
+
∞
X
ck z α+k+2
k=0
= c0 (α2 − ν 2 )z α +
∞
X
{ck [(α + k)2 − ν 2 ] + ck−2 }z α+k
k=0
The first term on the right side of the above expression is
c0 (α2 − ν 2 )z α
and the remaining are
(2.2)
(2.3)
c1 [(α + 1)2 − ν 2 ] + 0
c2 [(α + 2)2 − ν 2 ] + c0
c3 [(α + 3)2 − ν 2 ] + c1
······
2
2
ck [(α + k) − ν ] + ck−2
······
= 0
= 0
= 0
= 0
Hence a series solution could exist only if α = ±ν. When k > 1, then we
require
ck [(α + k)2 − ν 2 ] + ck−2 = 0,
and this determines ck in terms of ck−2 unless α − ν = −2ν or α + ν = 2ν is an
integer. Suppose we discard these exceptional cases, then it follows from (2.2) that
c1 = c3 = c5 = · · · = c2k+1 = · · · = 0.
Thus we could express the coefficients c2k in terms of
c2k =
(−1)k c0
.
(α − ν + 2)(α − ν + 4) · · · (α − ν + 2k) · (α + ν + 2)(α + ν + 4) · · · (α + ν + 2k)
2.2. BESSEL EQUATION
11
If we now choose α = ν, then we obtain
∞
h
X
c0 z ν 1 +
(2.4)
k=0
i
(−1)k ( 12 z)2k
.
k!(ν + 1)(ν + 2) · · · (ν + k)
Alternatively, if we choose α = −ν, then we obtain
c00 z −ν
(2.5)
h
1+
∞
X
k=0
i
(−1)k ( 12 z)2k
.
k!(−ν + 1)(−ν + 2) · · · (−ν + k)
Since c0 and c00 are arbitrary, so we choose them to be
(2.6)
c0 =
1
2ν Γ(ν
+ 1)
,
c00 =
1
2−ν Γ(−ν
+ 1)
so that the two series (2.4) and (2.5) can be written in the forms:
(2.7)
Jν (z) =
+∞
X
(−1)k ( 21 z)ν+2k
,
Γ(ν
+
k
+
1)
k!
k=0
J−ν (z) =
+∞
X
(−1)k ( 21 z)−ν+2k
Γ(−ν + k + 1) k!
k=0
and both are called the Bessel function of order ν if the first kind.
It can be shown that the Wronskian of Jν and J−ν is given by (G. N. Watson
“A Treatise On The Theory Of Bessel Functions”, pp. 42–43):
(2.8)
W (Jν , J−ν ) = −
2 sin νπ
.
πz
This shows that the Jν and J−ν forms a fundamental set of solutions when ν is
not equal to an integer. In fact, when ν = n is an integer, then we can easily check
that
(2.9)
J−n (z) = (−1)n Jn (z).
Thus it requires an extra effort to find another linearly independent solution.
It turns out a second linearly independent solution is given by
(2.10)
Yν (z) =
Jν (z) cos νπ − J−ν (z)
sin νπ
when ν is not an integer. The case when ν is an integer n is defined by
(2.11)
Jν (z) cos νπ − J−ν (z)
.
ν→n
sin νπ
Yn (z) = lim
12
CHAPTER 2. BESSEL FUNCTIONS
The Yν so defined is linearly independent with Jν for all values of ν.
In particular, we obtain
n−1
(2.12)
(2.13)
−1 X (n − k − 1)! z 2k−n
Yn (z) =
π k=0
k!
2
∞
i
z
1 X (−1)k (z/2)n+2k h
2 log − ψ(k + 1) − ψ(k + n + 1)
+
π k=0 k!(n + k)!
2
for | arg z| < π and n = 0, 1, 2, · · · with the understanding that we set the sum
to be 0 when n = 0. Here the ψ(z) = Γ0 (z)/Γ(z). We note that the function is
unbounded when z = 0.
2.3
Basic Properties of Bessel Functions
The general reference for Bessel functions is G. N. Watson’s classic: “A Treatise on
the Theory of Bessel Functions”, published by Cambridge University Press in 1922
[6].
2.3.1
Zeros of Bessel Functions
See A. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with
formulas, graphs and mathematical tables”, 10th Edt., National Bureau of Standards, 1964. [1]. [3] and [6]
2.3.2
Recurrence Formulae for Jν
We consider arbitrary complex ν.
d
(−1)k z 2ν+2k
d ν
z Jν (z) =
dz
dz 2ν+2k k!Γ(ν + k + 1)
d (−1)k z 2ν−1+2k
=
dz 2ν−1+2k k!Γ(ν + k)
= z ν Jν−1 (z).
But the left side can be expanded and this yields
(2.1)
zJν0 (z) + νJν (z) = zJν−1 (z).
2.3. BASIC PROPERTIES OF BESSEL FUNCTIONS
Similarly,
d −nu
z Jν (z) = −z −ν Jν+1 (z).
dz
(2.2)
and this yields
zJν0 (z) − νJν (z) = −zJν+1 (z).
(2.3)
Substracting and adding the above recurrence formulae yield
2ν
Jν (z)
z
Jν−1 (z) − Jν+1 (z) = 2Jν0 (z).
Jν−1 (z) + Jν+1 (z) =
(2.4)
(2.5)
2.3.3
Generating Function for Jn
Jacobi in 1836 gave
(2.6)
e
1
z(t− 1t )
2
=
+∞
X
tk Jk (z).
k=−∞
Many of the forumulae derived above can be obtained from this expression.
(2.7)
e
1
z(t− 1t )
2
=
+∞
X
ck (z)tk
k=−∞
for 0 < |t| < ∞. We multiply the power series
zt
(2.8)
e2 =1+
(z/2)2 2
(z/2)
t+
t + ···
1!
1!
and
zt
e− 2 = 1 −
(2.9)
(z/2)
(z/2)2 2
t+
t − ···
1!
1!
Multiplying the two series and comparing the coefficients of tk yield
(2.10)
(2.11)
cn (z) = Jn (z), n = 0, 1, · · ·
cn (z) = (−1)n J−n (z), n = −1, −2, · · · .
Thus
(2.12)
e
1
z(t− 1t )
2
= J0 (z) +
+∞
X
k=1
Jk [tk + (−1)k t−k ].
13
14
CHAPTER 2. BESSEL FUNCTIONS
2.3.4
Lommel’s Polynomials
Iterating the recurrence formula
2ν
Jν (z) − Jν−1
z
with respect to ν a number of times give
(2.13)
Jν+1 (z) =
Jν+k (z) = P (1/z)Jν (z) − Q(1/z)Jν−1 .
(2.14)
Lommel (1871) [6, pp. 294–295] found that
Jν+k (z) = Rk,ν (z)Jν (z) − Rk−1,ν+1 Jν−1 .
(2.15)
2.3.5
Bessel Functions of half-integer Orders
One can check that
(2.16)
J− 1 (z) =
2
2 12
cos z,
πz
J 1 (z) =
2
2 12
sin z.
πz
Moreover,
r
(2.17)
n
Jn+ 1 (z) = (−1)
2
2 n+ 1 d n sin z
z 2
,
π
z dz
z
n = 0, 1, 2, · · · .
Thus applying a recurrence formula and using the Lommel polynomials yield
Jn+ 1 (z) = Rn,ν (z)J 1 (z) − Rn−1,ν+1 J− 1 (z)
(2.18)
2
2
2
That is, we have
(2.19)
2.3.6
2 12
2 12
sin z − Rn−1,ν+1
sin z.
Jn+ 1 (z) = Rn,ν (z)
2
πz
πz
Formulae for Lommel’s polynomials
For each fixed ν, the Lommel polynomials are given by
[n/2]
X (−1)k (n − k)!(ν)n−k 2 n−2k
(2.20)
Rn ν (z) =
k!(n − 2k)!(ν)k
z
k=0
where the [x] means the largest integer not exceeding x.
Lommel is a German mathematician who made a major contribution to Bessel
functions around 1870s.
2.3. BASIC PROPERTIES OF BESSEL FUNCTIONS
2.3.7
15
“Pythagoras’ Theorem” for Bessel Function
These Lommel polynomials have remarkable properties. Since
2 12
2 12
cos z,
J 1 (z) =
sin z
(2.21)
J− 1 (z) =
2
2
πz
πz
and sin2 x + cos2 x = 1; we now have
2
2
Jn+
(z) = 2(−1)n
1 (z) + J
−n− 1
(2.22)
2
2
R2n, 1 −n (z)
2
πz
.
That is, we have
2
Jn+
1 (z)
2
(2.23)
+
2
J−n−
1 (z)
2
n
2 X (2z)2n−2k (2n − k)!(2n − 2k)!
.
=
πz k=0
[(n − k)!]2 k!
A few special cases are
1. J 21 (z) + J−2 1 (z) =
2
2
2. J 23 (z) + J−2 3 (z) =
2
2
2
;
πz
2 πz
1+
2 1+
2
2
πz
2 4. J 27 (z) + J−2 7 (z) =
1+
2
2
πz
3. J 25 (z) + J−2 5 (z) =
2.3.8
1
;
z2
3
9
+
;
z2 z4
6
45 225 +
+ 6
z2 z4
z
Orthogonality of the Lommel Polynomials
Let us set
[n/2]
(2.24)
hn, ν (z) = Rn, ν (1/z) =
X (−1)k (n − k)!(ν)n−k z n−2k
,
k!(n
−
2k)!(ν)
2
k
k=0
then the set {hn ν (z) is called the modified Lommel polynomials. Since the Bessel
functions Jν (z) satisfies a three-term recurrence relation, so the Lommel polynomials
inherit this property:
(2.25)
2z(n + ν)hn ν (z) = hn+1, ν (z) + hn−1, ν (z)
16
CHAPTER 2. BESSEL FUNCTIONS
with initial conditions
(2.26)
h0, ν (z) = 1,
h1, ν (z) = 2νz.
If one start with a different set of initial conditions
(2.27)
h∗0, ν (z) = 0,
h∗1, ν (z) = 2ν,
then the sequence {h∗n, ν (z)} generated by the (2.25) is called the associated Lommel polynomials. It is known that a three-term recurrence relation for polynomials
with the coefficients of as in (2.25) will generate a set of orthogonal polynomials
on
R +∞(−∞, +∞). That is, there is a probability measure α onb (−∞, +∞) with
dα = 1
−∞
Theorem 2.3.1 (A. A. Markov, 1895). Suppose the set of {pn (z)}of orthogonal
polynomials with its measure α supported on a bounded internal [a, b], then
Z b
dα(t)
p∗n ν (z)
=
(2.28)
lim
n→∞ pn ν (z)
a z −t
holds uniformly for z 6∈ [a, b].
Since we know
Theorem 2.3.2 (Hurwitz). The limit
(2.29)
(z/2)ν+n Rn, ν+1 (z)
= Jν (z),
n→∞
Γ(n + ν + 1)
lim
holds uniformly on compact subsets of C.
So we have
Theorem 2.3.3. For ν > 0, the polynomials
R +∞{hn ν (z) are orthogonal with respect to
a discrete measure αν normalized to have −∞ dα(t) = 1, and
Z
dα(t)
Jν (1/z)
(2.30)
= 2ν
.
Jν−1 (1/z)
R z −t
One can use this formula to recover the measure α(t). Thus the Lommel “polynomials” was found to have very distinct properties to be complete in L2 (−1, +1).
2.4. INTEGRAL FORMAULAE
2.4
17
Integral formaulae
(2.1)
(z/2)ν
Jν (z) = 1
Γ( 2 )Γ(ν + 12 )
Z
1
(1 − t2 )ν−1/2 cos zt dt,
<(ν) > −1/2,
| arg z| < π.
<(ν) > −1/2,
| arg z| < π,
−1
Or equivalently,
(2.2)
Z π
(z/2)ν
cos(z cos θ) sin2ν θ dθ,
Jν (z) = 1
1
Γ( 2 )Γ(ν + 2 ) 0
where t = cos θ.
(1)
Writing Hν (z) = Jν (z) + iYν (z). Then
Theorem 2.4.1 (Hankel 1869).
(2.3)
Hν(1) (z)
2 12 ei[z−νπ/2−π/4] Z ∞·exp(iβ)
iu ν− 12
−u ν− 12
=
e u
du,
1+
πz
Γ(ν + 1/2) 0
2z
where |β| < π/2.
2.5
Asymptotic Behaviours of Bessel Functions
Expanding the integrand of Henkel’s contour integral by binomial expansion and
after some careful analysis, we have
Theorem 2.5.1. For −π < arg z < 2π,
(2.1)
Hν(1) (x)
=
2
πx
21
e
i(x− 12 νπ− 41 π)
X
p−1
( 12 − ν)m ( 12 + ν)m
(1)
+ Rp (x) ,
m m!
(2ix)
m=0
and for −2π < arg x < π,
(2.2)
Hν(2) (x)
=
2
πx
12
e
−i(x− 12 νπ− 14 π)
X
p−1
( 12 − ν)m ( 12 + ν)m
(2)
+ Rp (x) ,
(2ix)m m!
m=0
where
(2.3)
Rp(1) (x) = O(x−p )
and
Rp(2) (x) = O(x−p ),
as x → +∞, uniformly in −π + δ < arg x < 2π − δ and −2π + δ < arg x < π − δ
respectively.
18
CHAPTER 2. BESSEL FUNCTIONS
The two expansions are valid simultaneously in −π < arg x < π.
(1)
(2)
We thus have Jν (z) = 12 (Hν (z) + Hν (z)). So
Theorem 2.5.2. For | arg z| < π, we have
Jν (z) ∼
2
πz
21 h
∞
1 X (−1)k (ν, 2k)
1
cos z − νπ − π
2
4 k=0
(2z)2k
∞
1 X (−1)k (ν, 2k + 1) i
1
− sin z − νπ − π
2
4 k=0
(2z)2k+1
2.5.1
Addition formulae
Schläfli (1871) derived
(2.4)
Jν (z + t) =
∞
X
Jν−k (t)Jk (z).
k=−∞
Theorem 2.5.3 (Neumann (1867)). Let z, Z, R forms a triangle and let φ be the
angle opposite to the side R, then
(2.5)
∞
p
X
2
2
J0
k Jk (Z)Jk (z) cos kφ,
(Z + z − 2Zz cos φ) =
k=0
where 0 = 1, k = 2 for k ≥ 1.
We note that Z, z can assume complex values. There are generalizations to
ν 6= 0.
2.6
Fourier-Bessel Series
Theorem 2.6.1. Suppose the real function f (r) is piecewise continuous in (0, a)
and of bounded variation in every [r1 , r2 ] ⊂ (0, a). Then if
Z a
√
(2.1)
r|f (r)| dr < ∞,
0
2.6. FOURIER-BESSEL SERIES
19
then the Fouier-Bessel series
(2.2)
f (r) =
∞
X
ck Jν xνk
k=1
r
a
converges to f (r) at every continuity point of f (r) and to
(2.3)
1
[f (r + 0) + f (r − 0)]
2
at every discontinuity point of f (r). Here
Z 1
2
xf (x)Jν (λk x) dx
(2.4)
ck = 2
Jν (λk ) 0
is the Fourier-Bessel coefficients of f .
The xνk are the zero of Jν (x) k = 0, 1, · · · . The discussion of the orthogonality
of the Bessel functions is omitted. You may consult G. P. Tolstov’s “ Fourier Series”,
Dover 1962.
The situation here is very much like the classical Fourier series in terms of sine
and cosine functions.
20
CHAPTER 2. BESSEL FUNCTIONS
Chapter 3
An New Perspective
3.1
Hypergeometric Equations
We shall base our consideration of functions on the finite complex plane C in this
chapter. Let p(z) and q(z) be meromorphic functions defined in C. A point z0
belongs to C is called a regular singular point of the second order differential equation
d2 f (z)
df (z)
+ p(z)
+ q(z)f (z) = 0
2
dz
dz
(3.1)
if
(3.2)
lim (z − z0 )2 q(z) = B
lim (z − z0 )p(z) = A,
z→z0
z→z0
both exist and being finite. We start by recalling that any second order linear
equation with three regular singular points in the C can always be transformed ([2,
pp. 73–75]) to the hypergeometric differential equation (due to Euler, Gauss
and Riemann)
(3.3)
z(1 − z)
dy
d2 y
+ [c − (a + b + 1)z] − ab y = 0
2
dz
dz
which has regular singular points at 0, 1, ∞.
It is known that the hypergeometric equation has a power series solution
(3.4)
2 F1
+∞
X
(a)k (b)k k
a, b z
=
F
[a,
b;
c;
z]
=
z
2 1
c
(c)k k!
k=0
21
22
CHAPTER 3. AN NEW PERSPECTIVE
which converges in |z| < 1. It is called the Gauss hypermetric series which
depends on three parameters a, b, c. Here the notation is the standard Pochhammer
notation (1891)
(a)k = a · (a + 1) · (z + 2) · · · (a + k − 1),
(ν)0 = 1
for each integer k.
Theorem 3.1.1 (Euler 1769). Suppose <(c) > <(b) > 0, then we have
(3.5)
2 F1
Z 1
Γ(c)
a, b z =
tb−1 (1 − t)c−b−1 (1 − xt)−a dt,
c Γ(b)Γ(c − b) 0
defined in the cut-plane C\[1, +∞), arg t = arg(1 − t) = 0, and that (1 − xt)−a
takes values in the principal branch (−π, π).
Thus Euler’s integal representation of 2 F1 thus continues the function analytically into the cut-plane.
Special cases of (3.4) include: ...
In 1812 Gauss gave a complete set of 15 contiguous relations ...
We note that one can write familiar functions in terms of p Fq :
1.
2.
3.
4.
5.
− e = 0 F0
x ;
−
a −a
(1 − x) = 1 F0
x ;
−
1, 1 log(1 + z) = x 2 F1
−x ;
2 − x2
cos x = 0 F1
− 4 ;
1/2 1/2, 1/2 2
−1
sin x = x 2 F1
x .
3/2 x
3.2. CONFLUENT HYPERGEOMETRIC EQUATIONS
3.2
23
Confluent Hypergeometric Equations
Writing the (3.1) in the variable x = z/b results in the equation
(3.1)
x(1 − x/b)
d2 y
dy
+ [c − (a + b + 1)x/b] − a y = 0
2
dx
dx
which has regular singular points at 0, b, ∞. Letting b tends to ∞ results in
(3.2)
x
d2 y
dy
+ (c − x) − a y = 0
2
dx
dx
which is called the confluent hypergeometric equation. But the equation has
a regular singular point at 0 and an irregular singular point at ∞.
Substituting z/b in (3.4) and letting b to infinity results in a series
(3.3)
+∞
X
(a)k k
Φ(a, c; z) := 1 F1 [a; c; z] =
z .
(c)k k!
k=0
This series is called a Kummer series. We note that the Gauss hypergeometric
series has radius of convergence 1, and after the above substitution of z/b results in
a series that has radius of convergence equal to |b|. So the Kummer series has radius
of convergence ∞ after taking the limit b → ∞. We call all solutions to the equation
(3.2) confluent hypergeometric functions because they are obtained from the
confluence of singularities of the hypergeometric equations. Further details can be
found in [4], better known as the Bateman project volumn one.
3.3
A Definition of Bessel Functions
We define the Bessel function of first kind of order ν to be the complex function
represented by the power series
(3.1)
+∞
+∞
X
X
(−1)k ( 21 z)ν+2k
(−1)k ( 21 )ν+2k 2k
ν
=z
z .
Jν (z) =
Γ(ν
+
k
+
1)
k!
Γ(ν
+
k
+
1)
k!
k=0
k=0
Here ν is an arbitrary complex constant and the notation Γ(ν) is the Euler Gamma
function defined by
Z ∞
(3.2)
Γ(z) =
tz−1 e−t dt, <(z) > 0.
0
24
CHAPTER 3. AN NEW PERSPECTIVE
It can be analytically continued onto the whole complex plane C, and has simple
poles at the negative integers including 0. So it is a meromorphic function. In
particular, Γ(k + 1) = k! = k(k − 1) · · · 2 · 1. So the Euler-Gamma function is a
generalization of the factorial notation k!.
Let us set a = ν + 12 , c = 2ν + 1 and replace z by 2iz in the Kummer series.
That is,
∞
X
(ν + 21 )k
Φ(ν + 1/2, 2ν + 1; 2iz) =
(2iz)k
(2ν + 1)k k!
k=0
ν + 12 2iz
= 1 F2
2ν + 1 z 2
− iz
= e 0 F1
−
ν + 1
2
∞
X (−1)k z 2k
= eiz
(ν + 1)k k! 22k
k=0
= eiz
∞
X
(−1)k Γ(ν + 1) z 2k
Γ(ν + k + 1) k! 2
k=0
∞
z −ν z ν X
(−1)k
z 2k
·
= e Γ(ν + 1)
2
2 k=0 Γ(ν + k + 1) k! 2
z −ν
= eiz Γ(ν + 1)
· Jν (z),
2
iz
where we have applied Kummer’s second transformation
a − 2
2x
x
(3.3)
4x = e 0 F1
1 F1
2a a + 21 in the third step above, and the identity (ν)k = Γ(ν + k)/Γ(ν) in the fourth step.
Since the confluent hypergeometric function Φ(ν + 1/2, 2ν + 1; 2iz) satisfies
the hypergeometric equation
(3.4)
z
d2 y
dy
+ (c − z) − a y = 0
2
dz
dz
with a = ν + 1/2 and c = 2ν + 1. Substituting eiz Γ(ν + 1)
and simplfying lead to the equation
(3.5)
z2
d2 y
dy
+ x + (z 2 − ν 2 )y = 0.
2
dz
dz
z −ν
2
· Jν (z) into the (3.4)
3.3. A DEFINITION OF BESSEL FUNCTIONS
25
The above derivation shows that the Bessel function of the first kind with order
ν is a special case of the confluent hypergeometric functions with specifically chosen
parameters.
26
CHAPTER 3. AN NEW PERSPECTIVE
Chapter 4
Physical Applications of Bessel
Functions
We consider radial vibration of circular membrane. We assume that an elastic
circular membrane (radius `) can vibrate and that the material has a uniform density.
Let u(x, y, t) denote the displacement of the membrane at time t from its equilibrium
position. We use polar coordinate in the xy−plane by the change of variables:
(4.1)
x = r cos θ,
y = r sin θ.
Then the corresponding equation
(4.2)
2
∂ 2u
∂ 2u 2 ∂ u
=
c
+
∂t2
∂x2 ∂y 2
can be transformed to the form:
(4.3)
2
∂ 2u
1 ∂ 2u
1 ∂ 2u 2 ∂ u
=
c
+
+
.
∂t2
∂r2
r ∂r
r2 ∂θ2
Since the membrane has uniform density, so u = u(r, t) that is, it is independent of
the θ. Thus we have
2
∂ 2u
1 ∂ 2u 2 ∂ u
(4.4)
.
=
c
+
∂t2
∂r2
r ∂r
The boundary condition is u(`, t) = 0, and the initial conditions take the form
(4.5)
∂u(r, 0)
= g(r).
∂t
u(r, t) = f (r),
Separation of variables method yields
(4.6)
u(r, t) = R(r)T (t),
27
28
CHAPTER 4. PHYSICAL APPLICATIONS OF BESSEL FUNCTIONS
which satisfies the boundary condition u(`, t) = 0. Thus
1
RT 00 = c2 R00 T + R0 T .
r
(4.7)
Hence
T 00
R00 + (1/r)R0
+ 2 ; = −λ2 ,
R
cT
(4.8)
where the λ is a constant. Thus
1
R00 + R0 + λ2 R = 0
r
T 00 + c2 λ2 T = 0.
(4.9)
(4.10)
We notice that the first equation is Bessel equation with ν = 0. Thus its general
solution is given by
(4.11)
R(r) = C1 J0 (λr) + C2 Y0 (λr).
. Since Y0 is unbounded when r = 0, so C2 = 0. Thus the boundary condition
implies that
(4.12)
J0 (λ`) = 0,
implying that µ = λ` is a zero of J0 (µ). Setting C1 = 1, we obtain for each integer
n = 1, 2, 3, · · · ,
(4.13)
Rn (r) = J0 (λn r) = J0 (
µn
r),
`
where µn = λn ` is the n−th zero of J0 (µ). Thus we have
un (r, t) = (An cos cλn t + Bn sin cλn t) · J0 (λn r),
(4.14)
for n = 1, 2, 3, · · · . Thus the general solution is given by
(4.15)
∞
X
n=1
un (r, t) =
∞
X
(An cos cλn t + Bn sin cλn t) · J0 (λn r),
n=1
and by
(4.16)
f (r) = u(r, 0) =
∞
X
n=1
An · J0 (λn r),
29
and
(4.17)
∞
X
∂u(r, 0) g(r) =
Bn cλn · J0 (λn r).
=
∂t
t=0
n=1
Fourier-Bessel Series theory implies that
Z `
2
(4.18)
An = 2 2
rf (r)J0 (λn r) dr,
` J1 (µn ) 0
(4.19)
2
Bn =
2
cλn ` J12 (µn )
Z
`
rg(r)J0 (λn r) dr.
0
30
CHAPTER 4. PHYSICAL APPLICATIONS OF BESSEL FUNCTIONS
Chapter 5
Separation of Variables of
Helmholtz Equations
The Helmholtz equation named after the German physicst Hermann von Helmholtz
refers to second order (elliptic) partial differential equations of the form:
(5.1)
(∆2 + k 2 )Φ = 0,
where k is a constant. If k = 0, then it reduces to the Laplace equations.
In this discussion, we shall restrict ourselves in the Euclidean space R3 . One
of the most powerful theories developed in solving linear PDEs is the the method
of separation of variables. For example, the wave equation
1 ∂2 (5.2)
∆2 − 2 2 Ψ(r, t) = 0,
c ∂t
can be solved by assuming Ψ(r, t) = Φ(r) · T (t) where T (t) = eiωt . This yields
ω2 (5.3)
∆2 − 2 Φ(r) = 0,
c
which is a Helmholtz equation. The questions now is under what 3−dimensional
coordinate system (u1 , u2 , u3 ) do we have a solution that is in the separation of
variables form
(5.4)
Φ(r) = Φ1 (u1 ) · Φ2 (u2 ) · Φ3 (u3 ) ?
Eisenhart, L. P. (”Separable Systems of Stäckel.” Ann. Math. 35, 284-305,
1934) determines via a certain Stäckel determinant is fulfilled (see e.g. Morse,
P. M. and Feshbach, H. “Methods of Theoretical Physics, Part I”. New York:
McGraw-Hill, pp. 125–126, 271, and 509–510, 1953).
31
32CHAPTER 5. SEPARATION OF VARIABLES OF HELMHOLTZ EQUATIONS
Theorem 5.0.1 (Eisenhart 1934). There are a total of eleven curvilinear coordinate systems in which the Helmholtz equation separates.
Each of the curvilinear coordinate is characterized by quadrics. That is,
surfaces defined by
(5.5)
Ax2 + By 2 + Cz 2 + Dxy + Exz + F yz + Gx + Hy + lz + J = 0.
One can visit http://en.wikipedia.org/wiki/Quadric for some of the quadric
surfaces. Curvilinear coordinate systems are formed by putting relevant orthogonal
quadric surfaces. Wikipedia contains quite a few of these pictures. We list the eleven
coordinate systems here:
33
Name of system
(1) Cartesian
Co-ordinate Systems
Transformation formulae
x = x, y = y, z = z
Degenerate Surfaces
(2) Cylindrical
x = ρ cos φ, y = ρ sin φ, z = z
ρ ≥ 0,
−π < φ ≤ π
(3) Spherical polar
x = r sin θ cos φ,
y = r sin θ sin φ,
z = r cos θ
r ≥ 0, 0 ≤ θ ≤ π, −π ≤ φ ≤ π
Parabolic
(4)
cylinder
x = u2 − v 2 , y = 2uv,
z=z
u ≥ 0,
−∞ < v < +∞
Elliptic
(5)
cylinder
x = f cosh ξ cos η, y = f sinh ξ sin η,
z=z
ξ ≥ 0, −∞ < η < +∞
Rotation
(6)
paraboloidal
x = 2uv cos φ, 2uv sin φ,
z = u2 − v 2
u, v ≥ 0,
−π < φ < π
Prolate
(7)
spheroidal
x = ` sinh u sin v cos φ,
y = ` sinh u sin v sin φ,
z = ` cosh u cos v,
u ≥ 0, 0 ≤ v ≤ π, −π < φ ≤ π
Finite line ;
segment
Two half-lines
Oblate
(8)
spheroidal
x = ` cosh u sin v cos φ,
y = ` cosh u sin v sin φ,
z = ` sinh u cos v,
u ≥ 0, 0 ≤ v ≤ π, −π < φ ≤ π
Circular plate (disc);
Plane with
circular aperture
(9) Paraboloidal
x = 12 `(cosh 2α + 2 cos 2β − cosh 2γ,
y = 2` cosh α cos β sinh γ,
z = 2` sinh α sin β cosh γ,
α, γ ≥ 0, −π < β ≤ π
Parabolic plate;
Plane with
parabolic aperture
(10) Elliptic conal
x = krsn αsn β;
y = (ik/k 0 )rcn αcn β,
z = (1/k 0 )rdn αdn β;
r ≥ 0, −2K < α ≤ 2K,
β = K + iu, 0 ≤ u ≤ 2K 0
Plane sector;
Including quarter
plane
2
Half-plane
Infinite strip;
Plane with
straight aperture
Half-line
34CHAPTER 5. SEPARATION OF VARIABLES OF HELMHOLTZ EQUATIONS
Coordinate system
Laplace & Helmholtz
Laplace Equation
Helmholtz equation
(1) Cartesian
(Trivial)
(Trivial)
(2) Cylinderical
(Trivial)
Bessel
(3) Spherical polar
Associated Legender
Associated Legender
(4) Parabolic cylinder
(Trivial)
Weber
(5) Elliptic cylinder
(Trivial)
Mathieu
(6) Rotation-paraboloidal
Bessel
Confluent hypergeometric
(7) Prolate spheroidal
Associated Legender
Spheroidal wave
(8) Prolate spheroidal
Associated Legender
Spheroidal wave
(9) Paraboloidal
Mathieu
Whittaker-Hill
(10) Elliptic conal
Lame
Spherical Bessel, Lame
(11) Ellipsoidal
Lame
Ellipsoidal
35
1. Associated Legender:
(5.6)
d2 y
m2 o
dy n
(1 − x ) 2 − 2x + n(n + 1) −
y=0
dx
dx
(1 − x2 )
2
2. Bessel:
x2
(5.7)
d2 y
dy
+ (x2 − n2 )y = 0
+
x
2
dx
dx
3. Spherical Bessel:
x2
(5.8)
d2 y
dy
+ 2x + (x2 − n(n + 1))y = 0
2
dx
dx
4. Weber:
d2 y
1
+ (λ − x2 )y = 0
2
dx
4
(5.9)
5. Confluent hypergeometric:
(5.10)
x
dy
d2 y
+ (γ − x) − αy = 0
2
dx
dx
6. Mathieu:
d2 y
+ (λ − 2q cos 2x)y = 0
dx2
(5.11)
7. Spheroidal wave:
(5.12)
o
d2 y
dy n
µ2
2
2
(1 − x ) 2 − 2x + λ −
+ γ (1 − x ) y = 0
dx
dx
(1 − x2 )
2
8. Lame:
(5.13)
d2 y
+ (h − n(n + 1)k 2 sn 2 x)y = 0
2
dx
9. Whittaker-Hill:
(5.14)
d2 y
+ (a + b cos 2x + c cos 4x)y = 0
dx2
10. Ellipsoidal wave:
(5.15)
d2 y
+ (a + bk 2 sn 2 x + qk 4 sn 4 x)y = 0
dx2
Remark 5.0.1. The spheroidal wave and the Whittaker-Hill do not belong to the
hypergeometric equation regime, but to the Heun equation regime (which has four
regular singular points).
36CHAPTER 5. SEPARATION OF VARIABLES OF HELMHOLTZ EQUATIONS
5.1
What is not known?
It is generally regarded that the Bessel functions, Weber functions, Legendre functions are better understood, but the remaining equations/functions are not so well
understood.
1. Bessel functions. OK! Still some unknowns.
2. Confluent hypergeometric equations/functions. NOT OK.
3. Spheroidal wave, Mathieu, Lame, Whittaker-Hill, Ellipsoidal wave are poorly
understood. Some of them are relatd to the Heun equation which has four
regular singular points. Its research has barely started despite the fact that
it has been around since 1910.
4. Mathematicans are separating variables of Laplace/Helmholtz equations, but
in more complicated setting (such as in Riemannian spaces, etc)
Bibliography
[1] Milton Abramowitz and Irene A. Stegun, editors. Handbook of mathematical
functions, with formulas, graphs, and mathematical tables, volume 55 of National
Bureau of Standards Applied Mathematics Series. Superintendent of Documents,
US Government Printing Office, Washington, DC, 1965. Third printing, with
corrections.
[2] G. E. Andrews, R. A. Askey, and R. Roy. Special Functions. Cambridge University Press, Cambridge, 1999.
[3] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions, volume 2. McGraw-Hill, New York, 1953.
[4] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions, volume 1. McGraw-Hill, New York, 1953.
[5] T. J. Rivlin and M. W. Wilson. An optimal property of Chebyshev expansions.
J. Approximation Theory, 2:312–317, 1969.
[6] G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, second edition, 1944.
37