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3. Neumann Functions, Bessel Functions of the 2 nd Kind

3. Neumann Functions, Bessel Functions of the 2 nd Kind. Neumann Functions :. x << 1 . . . . Ex.14.3.8. agrees with. x << 1 . Mathematica. For x  , periodic with amp  x 1/2 /2 phase difference with J n. Integral Representation. Ex.14.3.7 Ex.14.4.8.

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3. Neumann Functions, Bessel Functions of the 2 nd Kind

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  1. 3. Neumann Functions, Bessel Functions of the 2nd Kind Neumann Functions : x << 1 

  2.  Ex.14.3.8 agrees with x << 1

  3. Mathematica For x  , periodic with amp  x 1/2 /2 phase difference with Jn

  4. Integral Representation Ex.14.3.7 Ex.14.4.8

  5. Recurrence Relations  

  6. Since Y satisfy the same RRs for J, they are also the solutions to the Bessel eq.  Caution: Since RR relates solutions to different ODEs (of different ), it depends on their normalizations.

  7. Wronskian Formulas For an ODE in self-adjoint form Ex.7.6.1 the Wronskian of any two solutions satisfies Bessel eq. in self-adjoint form :  For a noninteger  , the two independent solutions J& J satisfy

  8. Acan be determined at any point, such as x = 0.   

  9. More Recurrence Relations Combining the Wronskian with the previous recurrence relations, one gets many more recurrence relations

  10. Uses of Neumann Functions • Complete the general solutions. • Applicable to any region excluding the origin ( e.g., coaxial cable, quantum scattering ). • Build up the Hankel functions ( for propagating waves ).

  11. Example 14.3.1. Coaxial Wave Guides EM waves in region between 2 concentric cylindrical conductors of radii a & b. ( c.f., Eg.14.1.2 & Ex.14.1.26 ) For TM mode in cylindrical cavity (eg.14.1.2) : For TM mode in coaxial cable of radii a & b : with Note: No cut-off for TEM modes.

  12. 4. Hankel Functions, H(1)(x) & H(2)(x) Hankel functions of the 1st & 2nd kind : c.f. for xreal For x<< 1,  > 0 : 

  13. Recurrence Relations

  14. Contour Representations See Schlaefli integral  • The integral representation • is a solution of the Bessel eq. if at end points of C. 

  15. The integral representation is a solution of the Bessel eq. for any C with end points t = 0 and Re t = . Consider Mathematica  If one can prove then

  16. Proof of   

  17. QED i.e. are saddle points. (To be used in asymptotic expansions.)

  18. 5. Modified Bessel Functions, I(x) & K(x) Bessel equation :  oscillatory Modified Bessel equation :  Modified Bessel functions exponential  Bessel eq.  Modified Bessel eq.  are all solutions of the MBE.

  19. I(x) Modified Bessel functions of the 1st kind : I (x) is regular at x = 0 with  

  20. Mathematica

  21. Recurrence Relations for I(x)  

  22. 2nd Solution K(x) Modified Bessel functions of the 2nd kind ( Whitaker functions ) : Recurrence relations : For x 0 : Ex.14.5.9

  23. Integral Representations Ex.14.5.14

  24. Example 14.5.1. A Green’s Function Green function for the Laplace eq. in cylindrical coordinates : Let

  25. §10.1   Ex.14.5.11

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