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Bessel Functions of the First and Second Kind
Reading Problems
- Background Outline
- Definitions
- Theory
- Bessel Functions.
- Modified Bessel Functions
- Kelvin’s Functions
- Hankel Functions
- Orthogonality of Bessel Functions
- Assigned Problems
- References
Background
Bessel functions are named for Friedrich Wilhelm Bessel (1784 - 1846), however, Daniel Bernoulli is generally credited with being the first to introduce the concept of Bessels func- tions in 1732. He used the function of zero order as a solution to the problem of an oscillating chain suspended at one end. In 1764 Leonhard Euler employed Bessel functions of both zero and integral orders in an analysis of vibrations of a stretched membrane, an investigation which was further developed by Lord Rayleigh in 1878, where he demonstrated that Bessels functions are particular cases of Laplaces functions.
Bessel, while receiving named credit for these functions, did not incorporate them into his work as an astronomer until 1817. The Bessel function was the result of Bessels study of a problem of Kepler for determining the motion of three bodies moving under mutual gravita- tion. In 1824, he incorporated Bessel functions in a study of planetary perturbations where the Bessel functions appear as coefficients in a series expansion of the indirect perturbation of a planet, that is the motion of the Sun caused by the perturbing body. It was likely Lagrange’s work on elliptical orbits that first suggested to Bessel to work on the Bessel functions.
The notation Jz,n was first used by Hansen^9 (1843) and subsequently by Schlomilch^10 (1857) and later modified to Jn(2z) by Watson (1922).
Subsequent studies of Bessel functions included the works of Mathews^11 in 1895, “A treatise on Bessel functions and their applications to physics” written in collaboration with Andrew Gray. It was the first major treatise on Bessel functions in English and covered topics such as applications of Bessel functions to electricity, hydrodynamics and diffraction. In 1922, Watson first published his comprehensive examination of Bessel functions “A Treatise on the Theory of Bessel Functions” 12.
(^9) Hansen, P.A. “Ermittelung der absoluten Strungen in Ellipsen von beliebiger Excentricitt und Neigung, I.” Schriften der Sternwarte Seeberg. Gotha, 1843. 10
11 Schlmilch, O.X. “Ueber die Bessel’schen Function.” Z. fr Math. u. Phys. 2, 137-165, 1857. 12 George Ballard Mathews, “A Treatise on Bessel Functions and Their Applications to Physics,” 1895 G. N. Watson , “A Treatise on the Theory of Bessel Functions,” Cambridge University Press, 1922.
3. Modified Bessel Equation
By letting x = i x (where i =
− 1 ) in the Bessel equation we can obtain the modified Bessel equation of order ν, given as
x^2 d^2 y dx^2
- x dy dx − (x^2 + ν^2 )y = 0
The solution to the modified Bessel equation yields modified Bessel functions of the first and second kind as follows:
y = C Iν (x) + D Kν (x) x > 0
4. Modified Bessel Functions
a) First Kind: Iν (x) in the solution to the modified Bessel’s equation is referred to as a modified Bessel function of the first kind. b) Second Kind: Kν (x) in the solution to the modified Bessel’s equation is re- ferred to as a modified Bessel function of the second kind or sometimes the Weber function or the Neumann function.
5. Kelvin’s Functions
A more general form of Bessel’s modified equation can be written as
x^2 d^2 y dx^2
- x dy dx − (β^2 x^2 + ν^2 )y = 0
where β is an arbitrary constant and the solutions is now
y = C Iν (βx) + D Kν (βx)
If we let
β^2 = i where i =
and we note
Iν (x) = i−ν^ Jν (ix) = Jν (i^3 /^2 x)
then the solution is written as
y = C Jν (i^3 /^2 x) + DKν (i^1 /^2 x)
The Kelvin functions are obtained from the real and imaginary portions of this solution as follows:
berν = Re Jν (i^3 /^2 x) beiν = Im Jν (i^3 /^2 x) Jν (i^3 /^2 x) = berν x + i bei x kerν = Re i−ν^ Kν (i^1 /^2 x) keiν = Im i−ν^ Kν (i^1 /^2 x) i−ν^ Kν (i^1 /^2 x) = kerν x + i kei x
0 2 4 6 8 10 12 14 x
-0.
-0.
0
1
Jn
x
J J J
Figure 4.1: Plot of the Bessel Functions of the First Kind, Integer Order
or by noting that Γ(ν + k + 1) = (ν + k)!, we can write
Jν (x) =
∑^ ∞
k=
(−1)k(x/2)ν+2k k!(ν + k)!
Bessel Functions of the first kind of order 0 , 1 , 2 are shown in Fig. 4.1.
The Bessel function of the second kind, Yν (x) is sometimes referred to as a Weber function or a Neumann function (which can be denoted as Nν (x)). It is related to the Bessel function of the first kind as follows:
Yν (x) = Jν (x) cos(νπ) − J−ν (x) sin(νπ)
where we take the limit ν → n for integer values of ν.
For integer order ν, Jν , J−ν are not linearly independent:
J−ν (x) = (−1)ν^ Jν (x) Y−ν (x) = (−1)ν^ Yν (x)
in which case Yν is needed to provide the second linearly independent solution of Bessel’s equation. In contrast, for non-integer orders, Jν and J−ν are linearly independent and Yν is redundant.
The Bessel function of the second kind of order ν can be expressed in terms of the Bessel function of the first kind as follows:
Yν (x) =
π Jν (x)
ln x 2
π
ν∑− 1
k=
(ν − k − 1)! k!
(x 2
) 2 k−ν
π
∑^ ∞
k=
(−1)k−^1
[(
k
k + ν
)]
k!(k + ν)!
(x 2
) 2 k+ν
Bessel Functions of the second kind of order 0 , 1 , 2 are shown in Fig. 4.2.
0 2 4 6 8 10 12 14 x
-1.
-0.
0
1
Yn
x
Y0 (^) Y Y
Figure 4.2: Plot of the Bessel Functions of the Second Kind, Integer Order
from Bowman, pg. 57
J 0 (x) =
π
∫ (^) π/ 2 0
cos(x sin θ) dθ
π
∫ (^) π/ 2 0
cos(x cos θ) dθ
Second Kind for Integer Orders n = 0, 1 , 2 , 3 ,...
Yn(x) = − 2(x/2)−n √ πΓ
− n
1
cos(xt) dt (t^2 − 1)n+1/^2 x > 0
Yn(x) =
π
∫ (^) π 0
sin(x sin θ − nθ) dθ −
π
∫ (^) π 0
[
ent^ + e−nt^ cos(nπ)
]
exp(−x sinh t) dt
x > 0
Y 0 (x) =
π^2
∫ (^) π/ 2 0
cos(x cos θ)
[
γ + ln(2x sin^2 θ)
]
dθ x > 0
Y 0 (x) = − 2 π
0
cos(x cosh t) dt x > 0
Approximations
Polynomial Approximation of Bessel Functions
For x ≥ 2 one can use the following approximation based upon asymptotic expansions:
Jn(x) =
πx
[Pn(x) cos u − Qn(x) sin u]
where u ≡ x − (2n + 1) π 4 and the polynomials Pn(x) and Qn(x) are given by
Pn(x) = 1 − (4n^2 − 12 )(4n^2 − 32 ) 2 · 1(8x)^2
(4n^2 − 52 )(4n^2 − 72 ) 4 · 3(8x)^2
(4n^2 − 92 )(4n^2 − 112 ) 6 · 5(8x)^2
and
Qn(x) = 4 n^2 − 12 1!(8x)
(4n^2 − 32 )(4n^2 − 52 ) 3 · 2(8x)^2
(4n^2 − 72 )(4n^2 − 92 ) 5 · 4(8x)^2
The general form of these terms can be written as
Pn(x) = (4n^2 − (4k − 3)^2 ) (4n^2 − (4k − 1)^2 ) 2 k(2k − 1)(8x)^2 k = 1, 2 , 3...
Qn(x) = (4n^2 − (4k − 1)^2 ) (4n^2 − (4k + 1)^2 ) 2 k(2k + 1)(8x)^2 k = 1, 2 , 3...
For n = 0
sin u =
(sin x − cos x)
cos u =
(sin x + cos x)
J 0 (x) =
√πx [P 0 (x)(sin^ x^ + cos^ x)^ −^ Q 0 (x)(sin^ x^ −^ cos^ x)]
Asymptotic Approximation of Bessel Functions
Large Values of x
Y 0 (x) =
πx
[P 0 (x) sin(x − π/4) + Q 0 (x) cos(x − π/4)]
Y 1 (x) =
πx
[P 1 (x) sin(x − 3 π/4) + Q 1 (x) cos(x − 3 π/4)]
where the polynomials have been defined earlier.
Power Series Expansion of Bessel Functions
First Kind, Positive Order
Jν (x) =
∑^ ∞
k=
(−1)k(x/2)ν+2k k!Γ(ν + k + 1)
Γ(1 + ν)
(x 2
)ν { 1 − (x/2)^2 1(1 + ν)
(x/2)^2 2(2 + ν)
(x/2)^2 3(3 + ν)
The General Term can be written as
Zk = −Y k(k + ν)
k = 1, 2 , 3 ,...
Y = (x/2)^2
where
B 0 = 1 B + 1 = Z 1 · B 0 B 2 = Z 2 · B 1 ...
Bk = Zk · Bk− 1
The approximation can be written as
Jν (x) = (x/2)ν Γ(1 + ν
∑^ U
k=
Bk
First Kind, Negative Order
J−ν (x) =
∑^ ∞
k=
(−1)k(x/2)^2 k−ν k!Γ(k + 1 − ν)
Γ(1 − ν)
(x 2
)−ν { 1 − (x/2)^2 1(1 − ν)
(x/2)^2 2(2 − ν)
(x/2)^2 3(3 − ν)
The General Term can be written as
Zk =
−Y
k(k − ν) k = 1, 2 , 3 ,...
Y = (x/2)^2
where
B 0 = 1 B + 1 = Z 1 · B 0 B 2 = Z 2 · B 1 ...
Bk = Zk · Bk− 1
The approximation can be written as
J−ν (x) = (x/2)−ν Γ(1 − ν)
∑^ U
k=
Bk
where U is some arbitrary value for the upper limit of the summation.
Note: xn+1 − xn → π as n → ∞
These roots can also be computed using Stoke’s approximation which was developed for large n.
xn = β 4
[
β^2
β^4
5 β^6
35 β^8
35 β^10
]
with β = π(4n + 1).
An approximation for small n is
xn = β 4
[
β^2
β^4
10 β^6
]
The roots of the transcendental equation
xnJ 1 (xn) − BJ 0 (xn) = 0
with 0 ≤ B < ∞ are infinite in number and they can be computed accurately and efficiently using the Newton-Raphson method. Thus the (i + 1)th iteration is given by
xi n+1 = xin − x
inJ 1 (xin) − BJ 0 (xin) xinJ 0 (xin) + BJ 1 (xin)
Accurate polynomial approximations of the Bessel functions J 0 (·) and J 1 (·) may be em- ployed. To accelerate the convergence of the Newton-Raphson method, the first value for the (n + 1)th root can be related to the converged value of the nth root plus π.
Aside:
Fisher-Yovanovich modified the Stoke’s approximation for roots of J 0 (x) = 0 and J 1 (x) =
- It is based on taking the arithmetic average of the first three and four term expressions
For Bi → ∞ roots are solutions of J 0 (x) = 0
δn,∞ = α 4
α^2
2 3 α^4
+^15116
30 α^6
with α = π(4n − 1).
For Bi → 0 roots are solutions of J 1 (x) = 0
δn, 0 = β 4
β^2
β^4
10 β^6
with β = π(4n + 1).
Potential Applications
- problems involving electric fields, vibrations, heat conduction, optical diffraction plus others involving cylindrical or spherical symmetry
- transient heat conduction in a thin wall
- steady heat flow in a circular cylinder of finite length
0 0.5 1 1.5 2 2.5 3 x
1
2
I
x
I
I
I
Figure 4.3: Plot of the Modified Bessel Functions of the First Kind, Integer Order
0 0.5 1 1.5 2 2.5 3 x
1
2
K
x
K
K
K
Figure 4.4: Plot of the Modified Bessel Functions of the Second Kind, Integer Order
Relations Satisfied by the Modified Bessel Function
Recurrence Formulas
Bessel functions of higher order can be expressed by Bessel functions of lower orders for all real values of ν.
Iν+1(x) = Iν− 1 (x) − 2 ν x Iν (x) Kν+1(x) = Kν− 1 (x) +^2 ν x Kν (x)
I ν′ (x) =
[Iν− 1 (x) + Iν+1(x)] K ν′ (x) = −
[Kν− 1 (x) + Kν+1(x)]
I ν′ (x) = Iν− 1 (x) − ν x Iν (x) K ν′ (x) = −Kν− 1 (x) − ν x Kν (x)
I ν′ (x) = ν x Iν (x) + Iν+1(x) K ν′ (x) = ν x Kν (x) − Kν+1(x)
d dx [xν^ Iν (x)] = xν^ Iν− 1 (x) d dx [xν^ Kν (x)] = −xν^ Kν− 1 (x)
d dx
[
x−ν^ Iν (x)
]
= x−ν^ Iν+1(x) d dx
[
x−ν^ Kν (x)
]
= −x−ν^ Kν+1(x)
Integral Forms of Modified Bessel Functions for Integer Orders n = 0, 1 , 2 , 3 ,...
First Kind
In(x) =
π
∫ (^) π 0
cos(nθ) exp(x cos θ) dθ
I 0 (x) =
π
∫ (^) π 0
exp(x cos θ) dθ
I 1 (x) =
π
∫ (^) π 0
cos(θ) exp(x cos θ) dθ
