1. Modified Bessel Functions
ECE 6382
Fall 2017
David R. Jackson
Notes 20
Modified Bessel Functions
and
Kelvin Functions
Notes are from D. R. Wilton, Dept. of ECE

2. Modified Bessel Functions
Modified Bessel differential equation: or
(self-adjoint form)
Motivation: Bessel functions of imaginary argument:

3. Modified Bessel Function of the First Kind
Definition:
(I is a real function of x.)
Use the Frobenius solution for J :
Frobenius series solution for I :

4. Second Solution of Modified Bessel Equation
For   n, the modified Bessel function of the 2nd kind is defined as:
For  = n (an integer):

5. Relations Between Bessel and Modified Bessel Functions
The modified Bessel functions are related to the regular Bessel functions as
Note: The added factors in front ensure that the functions are real.

6. Small Argument Approximations
For small arguments we have:

7. Large Argument Approximations
For large arguments we have:

8. Plots of Modified Bessel Functions for Real Argumentsx I0 I1 I2
The In functions increase exponentially.
The are finite at x = 0.

9. Plots of Modified Bessel Functions for Real Argumentsx K0 K1 K2
The Kn functions decrease exponentially.
The are infinite at x = 0.

10. Recurrence Relations
Some recurrence relations are:

11. Wronskians
A Wronskian identity is:

12. Kelvin Functions The Kelvin functions are defined as
Note: These are important for studying the fields inside of a conducting wire.

13. Kelvin Functions (cont.)
The Ber functions increase exponentially.
They are finite at x = 0.
Ber0
Ber1
Ber2 x

14. Kelvin Functions (cont.)
The Bei functions increase exponentially.
They are finite at x = 0.
Bei1
Bei2
Bei0 x

15. Kelvin Functions (cont.)
Normalizing makes it more obvious that the Ber and Bei functions
increase exponentially and also oscillate. x x

16. Kelvin Functions (cont.)
Normalizing makes it more obvious that the Ber and Bei functions
increase exponentially and also oscillate. x x

17. Kelvin Functions (cont.)
The Ker functions decay exponentially.
They are infinite at x = 0.
Ker0
Ker1 x

18. Kelvin Functions (cont.)
The Kei functions decay exponentially.
They are infinite at x = 0.
Kei0
Kei1 x