Modified Bessel Functions

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Presentation transcript:

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

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

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 :

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

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.

Small Argument Approximations For small arguments we have:

Large Argument Approximations For large arguments we have:

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

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

Recurrence Relations Some recurrence relations are:

Wronskians A Wronskian identity is:

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

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

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

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

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

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

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