Legendre function Qo(x) for n = 0) Show that (6)with 11 = 0 gives Yl(X) = Po(x) = I and

Chapter 5, Problem 5.1.48

(choose chapter or problem)

(Legendre function \(Q_{0}(x)\) for n = 0 ) Show that (6) with n = 0 gives \(y_{1}(x)=P_{0}(x)=1\) and (7) gives

\(y_{2}(x) =x+\frac{2}{3 !} x^{3}+\frac{(-3)(-1) \cdot 2 \cdot 4}{5 !} x^{5}+\cdots\)

\(=x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdots=\frac{1}{2} \ln \frac{1+x}{1-x}\)

Verify this by solving (1) with n = 0, setting z = y ‘, and separating variables.

Text Transcription:

Q_0(x)

y_1 (x) = P_0(x) = 1

y_2 (x) = x +2 / 3! x^3 + (-3)(-1) \cdot 2 \cdot 4 / 5! x^5 + cdots

= x + x^3 / 3 + x^5 / 5 + cdots = 1 / 2 ln 1 + x / 1 - x