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Bessel’s Equation
One of the most important of all variable-coefficient differential equations is
𝑥2𝑑2𝑦𝑑𝑥2
+ 𝑥𝑑𝑦𝑑𝑥
+ (𝛼2𝑥2 − 𝜈2)𝑦 = 0 (1)
Which is known as Bessel’s Equation of order ν with parameter α. A special case is when α = 0 so that
𝑥2𝑑2𝑦𝑑𝑥2
+ 𝑥𝑑𝑦𝑑𝑥
+ (𝑥2 − 𝜈2)𝑦 = 0 (1’)
Which is simply known as Bessel’s Equation of order ν.
Property #1
For all values of ν, a complete solution of Bessel’s equation (Eq 1’) of order ν can be written in the form
𝑦(𝑥) = 𝐶1𝐽𝜈(𝑥) + 𝐶2𝑌𝜈(𝑥)
If ν is not an integer, a complete solution can also be written as
𝑦(𝑥) = 𝐶1𝐽𝜈(𝑥) + 𝐶2𝐽−𝜈(𝑥)
𝑱𝝂(𝒙) is known as the Bessel function of the first kind of order ν.
Bessel values 𝒀𝝂(𝒙) is known as the Bessel function of the second kind of order ν.
are calculated from an infinite series and are tabulated, similar to the values of sin and cos. They can be looked up in a table, inserted as a function in Excel, or almost any other software that performs calculations. For the curious, the infinite series representing Bessel’s functions can be found online.
Modified Bessel’s Equation
Certain equations closely resembling Bessel’s equation occur so often that their solutions are also named and studied as functions in their own right. The most important of these is
𝑥2𝑑2𝑦𝑑𝑥2
+ 𝑥𝑑𝑦𝑑𝑥
− (𝑥2 − 𝜈2)𝑦 = 0 (2)
Which is known as the modified Bessel Equation of order ν. A closer look reveals that this is in fact simply Bessel’s Equation with the parameter α = i.
𝑥2𝑑2𝑦𝑑𝑥2
+ 𝑥𝑑𝑦𝑑𝑥
− (𝑖2𝑥2 − 𝜈2)𝑦 = 0 (2)
With solution
𝑦(𝑥) = 𝐶1𝐼𝜈(𝑥) + 𝐶2𝐼−𝜈(𝑥) (3)
𝐼𝜈 is known as the modified Bessel Function of the first kind of order ν. 𝐼−𝜈 is sometimes named 𝐾𝜈, which is known as the modified Bessel Function of the second kind of order ν. An equivalent form of equation (3) is therefore.
𝑦(𝑥) = 𝐶1𝐼𝜈(𝑥) + 𝐶2𝐾𝜈(𝑥) (3’)
Likewise, the modified Bessel Functions are also tabulated and available in software packages that perform calculations.
Equations Solvable In Terms of Bessel Functions
There are additional types of equations that can be solved in terms of Bessel Functions, in particular, the large and important family described in the following;
Property #2
If (1 − 𝑎2) ≥ 4𝑐 and if neither p, s nor q is zero, then the differential equation
𝑥2𝑑2𝑦𝑑𝑥2
+ 𝑥(𝑎 + 2𝑏𝑥𝑝)𝑑𝑦𝑑𝑥
+ [𝑐 + 𝑠𝑥2𝑞 + 𝑏(𝑎 + 𝑝 − 1)𝑥𝑝 + 𝑏2𝑥2𝑝]𝑦 = 0 (4)
Has a complete solution
𝑦(𝑥) = 𝑥𝛼𝑒−𝛽𝑥𝑝[𝐶1𝐽𝜈(𝜆𝑥𝑞) + 𝐶2𝑌𝜈(𝜆𝑥𝑞)]
Where 𝛼 = 1−𝑎2
; 𝛽 = 𝑏𝑝
; 𝜆 = �|𝑠|𝑞
; 𝜈 = �(1−𝑎)2−4𝑎𝑎2𝑞
If s < 0, 𝐽𝜈 and 𝑌𝜈are replaced by 𝐼𝜈and 𝐾𝜈, respectively. If ν is not an integer 𝑌𝜈 and 𝐾𝜈 can be replaced by 𝐽−𝜈 and 𝐼−𝜈 if desired.
Property #3
For differential equations of the special form (often observed in cylindrical coordinates) and (1 − 𝑟)2 ≥ 4𝑏 ; a ≠ 0 ; and either r – 2 < t or b = 0, then
𝑑𝑑𝑥
�𝑥𝑟𝑑𝑦𝑑𝑥� + (𝑎𝑥𝑡 + 𝑏𝑥𝑟−2)𝑦 = 0 (5)
There is a complete solution:
𝑦(𝑥) = 𝑥𝛼[𝐶1𝐽𝜈(𝜆𝑥𝛾) + 𝐶2𝑌𝜈(𝜆𝑥𝛾)] (6)
Where = 1−𝑟2
; 𝛾 = 2−𝑟+𝑡2
; 𝜆 = 2�|𝑎|2−𝑟+𝑡
; 𝜈 = �(1−𝑟)2−4𝑏2−𝑟+𝑡
If a < 0, 𝐽𝜈 and 𝑌𝜈are replaced by 𝐼𝜈and 𝐾𝜈, respectively. If ν is not an integer 𝑌𝜈 and 𝐾𝜈 can be replaced by 𝐽−𝜈 and 𝐼−𝜈 if desired.
Example:
𝑥2𝑑2𝑦𝑑𝑥2
+ 2𝑥𝑑𝑦𝑑𝑥
− 𝑘𝑥𝑦 = 0
𝑑 �𝑥2 𝑑𝑦𝑑𝑥�
𝑑𝑥− 𝑘𝑥𝑦 = 0
Following the form of eq. 5, r = 2, b = 0, t = 1, a = -k
Check conditions:
(1 − 𝑟)2 ≥ 4𝑏; (1 − 2)2 ≥ 4(0); 𝑻𝑻𝑻𝑻
a ≠ 0 ; a = -k ; 𝑻𝑻𝑻𝑻
b = 0 ; 𝑻𝑻𝑻𝑻
Conditions are met, so :
𝛼 = 1−𝑟2
= 1−22
= −12
𝛾 = 2−𝑟+𝑡2
= 2−2+12
=12
𝜆 = 2�|𝑎|2−𝑟+𝑡
= 2�|−𝑘|2−2+1
= 2√𝑘;
𝜈 = �(1−𝑟)2−4𝑏2−𝑟+𝑡
= �(1−2)2−4(0)2−2+1
= 1
Substitution into (6) gives
𝑦(𝑥) = 𝑥𝛼[𝐶1𝐽𝜈(𝜆𝑥𝛾) + 𝐶2𝑌𝜈(𝜆𝑥𝛾)]
𝑦(𝑥) = 𝑥−1/2�𝐶1𝐼1�2√𝑘𝑥1/2� + 𝐶2𝐾1�2√𝑘𝑥1/2��
Where modified Bessel Functions were used because a < 0.
Properties of Bessel Functions
Most important:
𝑱𝝂(𝟎) 𝒂𝒂𝒂 𝑰𝝂(𝟎) are finite, but 𝒀𝝂(𝟎) 𝒂𝒂𝒂 𝑲𝝂(𝟎) go to infinity.
𝑑𝑑𝑥
[𝑥𝜈𝜉𝜈(𝛼𝑥)] = 𝛼𝑥𝜈𝜉𝜈−1(𝛼𝑥) 𝜉 = 𝐽,𝑌, 𝐼
𝑑𝑑𝑥
[𝑥𝜈𝜉𝜈(𝛼𝑥)] = −𝛼𝑥𝜈𝜉𝜈−1(𝛼𝑥) 𝜉 = 𝐾
𝑑𝑑𝑥
[𝑥−𝜈𝜉𝜈(𝛼𝑥)] = −𝛼𝑥−𝜈𝜉𝜈+1(𝛼𝑥) 𝜉 = 𝐽,𝑌, 𝐼
𝑑𝑑𝑥
[𝑥−𝜈𝜉𝜈(𝛼𝑥)] = 𝛼𝑥−𝜈𝜉𝜈+1(𝛼𝑥) 𝜉 = 𝐾
𝑑𝑑𝑥
[𝜉𝜈(𝛼𝑥)] = 𝛼𝜉𝜈−1(𝛼𝑥) −𝜈𝑥𝜉𝜈(𝛼𝑥) 𝜉 = 𝐽,𝑌, 𝐼
𝑑𝑑𝑥
[𝜉𝜈(𝛼𝑥)] = −𝛼𝜉𝜈−1(𝛼𝑥) −𝜈𝑥𝜉𝜈(𝛼𝑥) 𝜉 = 𝐾
𝑑𝑑𝑥
[𝜉𝜈(𝛼𝑥)] = −𝛼𝜉𝜈+1(𝛼𝑥) +𝜈𝑥𝜉𝜈(𝛼𝑥) 𝜉 = 𝐽,𝑌, 𝐼
𝑑𝑑𝑥
[𝜉𝜈(𝛼𝑥)] = 𝛼𝜉𝜈+1(𝛼𝑥) +𝜈𝑥𝜉𝜈(𝛼𝑥) 𝜉 = 𝐾
2𝑑𝑑𝑥
[𝐼𝜈(𝛼𝑥)] = 𝛼[𝐼𝜈−1(𝛼𝑥) + 𝐼𝜈+1(𝛼𝑥)]
2𝑑𝑑𝑥
[𝐾𝜈(𝛼𝑥)] = 𝛼[𝐾𝜈−1(𝛼𝑥) + 𝐾𝜈+1(𝛼𝑥)]
𝜉𝜈(𝛼𝑥) =𝛼𝑥2𝜈
[𝜉𝜈+1(𝛼𝑥) + 𝜉𝜈−1(𝛼𝑥)] 𝜉 = 𝐽,𝑌
𝐼𝜈(𝛼𝑥) = −𝛼𝑥2𝜈
[𝐼𝜈+1(𝛼𝑥) − 𝐼𝜈−1(𝛼𝑥)]
𝐾𝜈(𝛼𝑥) = −𝛼𝑥2𝜈
[𝐾𝜈+1(𝛼𝑥) − 𝐾𝜈−1(𝛼𝑥)]
Ways to simplify Bessel functions
𝐽1/2(𝑥) = � 2𝜋𝑥
sin (𝑥) 𝐽−1/2(𝑥) = � 2𝜋𝑥
cos (𝑥)
𝐼1/2(𝑥) = � 2𝜋𝑥
sinh (𝑥) 𝐼−1/2(𝑥) = � 2𝜋𝑥
cosh (𝑥)
𝐽−𝑛(𝛼𝑥) = (−1)𝑛𝐽𝑛(𝛼𝑥) n = 0 or integer
𝐼−𝑛(𝛼𝑥) = 𝐼𝑛(𝛼𝑥) n = 0 or integer
𝐾−𝑛(𝛼𝑥) = 𝐾𝑛(𝛼𝑥) n = 0 or integer
As 𝒙 → ∞
𝐽𝜈(𝑥) = � 2𝜋𝑥
cos (𝑥 −𝜋4−𝜈𝜋2
)
𝑌𝑛(𝑥) = � 2𝜋𝑥
sin (𝑥 −𝜋4−𝑛𝜋2
) n = integer
𝐼𝜈(𝑥) =𝑒𝑥
√2𝜋𝑥
𝐾𝑛(𝑥) =𝑒−𝑥
√2𝜋𝑥
n = integer
As 𝒙 → 𝟎;𝒀 → −∞ ;𝑲 → ∞
𝐽𝜈(𝑥) = 𝐼𝜈(𝑥) ≈1
2𝜈𝜈!𝑥𝜈
𝐽−𝜈(𝑥) = 𝐼−𝜈(𝑥) ≈2𝜈
(−𝜈)!𝑥−𝜈
Where ν is integer not equal to zero
𝑌𝑛(𝑥) ≈−2𝑛(𝑛 − 1)!
𝜋𝑥−𝑛
𝑌0(𝑥) ≈2𝜋
ln (𝑥) n = integer
𝐾𝑛(𝑥) ≈ 2𝑛−1(𝑛 − 1)! 𝑥−𝑛 n is an integer not equal to zero
𝐾0(𝑥) ≈ −ln (𝑥)
What some Bessel Functions look like
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10
J0
J1
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
0 5 10
Y0
Y1
0
2
4
6
8
10
0 2 4 6
I0
I1
0
0.5
1
1.5
2
0 5 10 15
K0
K1
