Tuesday, 1 October 2013

Solution of a Modified Bessel Differential Equation with Complex Coefficient



Consider the following ordinary differential equation (ODE)




[d2dr2+1rddr(ri+μmr2)]R(r)=0



where r is a real variable, μm is a real constant and ri is a complex constant. I already know that if ri is a negative or positive real number then the solution of this ODE is given by



R(r)=C1Iμm(rir)+C2Kμm(rir),ri>0R(r)=C1Jμm(rir)+C2Yμm(rir),ri<0



where C1,C2 are real constants and Iμm and Kμm are modified Bessel functions of order μm. Also, Jμm and Yμm are the Bessel functions of order μm.




What will be the solution when ri is a complex constant?


Answer



For a purely imaginary ri, the solutions are linear combinations of Kelvin functions. Thus, for a general complex ri, solutions are superposition of Bessel or modified Bessel functions (depending on the real part of the parameter) and Kelvin functions.


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