Why Bessel functions of the second kind does not define from the second linearly independent solution of the Bessel equation that solved by Frobenius method?
For example about the Bessel function of the second kind of order $0$ , somebody had done a great job of solving Bessel equation of order $0$ by Frobenius method that in http://tw.knowledge.yahoo.com/question/article?qid=1712010200992. He find that the second group of linearly independent solutions of the Bessel equation of order $0$ that solved by Frobenius method is $y_2=C_2\left(\ln x+\sum\limits_{n=1}^\infty\left(\dfrac{(-1)^nx^{2n}}{4^n(n!)^2}\left(\ln x-\sum\limits_{k=1}^n\dfrac{1}{k}\right)\right)\right)$ .
But in fact the series form of the Bessel function of the second kind is $Y_0(x)\propto\ln\dfrac{x}{2}+\gamma+\sum\limits_{n=1}^\infty\left(\dfrac{(-1)^nx^{2n}}{4^n(n!)^2}\left(\ln\dfrac{x}{2}+\gamma-\sum\limits_{k=1}^n\dfrac{1}{k}\right)\right)$ .
How can you explain those issues?
