 If we set

which converges for formula_17, and analytically continue it, we obtain a second linearly independent solution to the differential equation.The factor of 1/2 is inserted in order to make formula_18 correspond to the Bessel functions of the second kind.We have

and

In terms of "K", we have

Hence just as the Bessel function and modified Bessel function of the first kind can both be expressed in terms of formula_9, those of the second kind can both be expressed in terms of formula_18.