(Legendre function -Ql(X) for II = 1) Show that (7) with 11 = I gives )"2(X) = PI(x) = x
Chapter 5, Problem 5.1.49(choose chapter or problem)
(Legendre function \(-Q_{1}(x)\) for n = 1 ) Show that (7) with n = 1 gives \(y_{2}(x)=P_{1}(x)=x\) and (6) gives \(y_{1}(x)=-Q_{1}(x)\) (the minus sign in the notation being conventional),
\(y_{1}(x) =1-\frac{x^{2}}{1}-\frac{x^{4}}{3}-\frac{x^{6}}{5}-\cdots\)
\(=1-x\left(x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdots\right)\)
\(=1-\frac{1}{2} x \ln \frac{1+x}{1-x}\)
Text Transcription:
-Q_1 (x)
y_2 (x) = P_1(x) = x
y_1(x) = -Q_1(x)
y_1(x) = 1- x^2 / 1 - x^4 / 3 - x^6 / 5 - cdots
= 1 - x (x + x^3/ 3 + x^5 / 5 + cdots)
= 1 - 1 / 2 x ln 1 + x / 1 - x
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