36. $f(x) = \begin{cases} 0 & ; \ -1 \le x < 0 \\ x & ; \ 0 \le x < 1 \end{cases}$ fonksiyonuna ait Legendre serisinin ilk üç terimini bulunuz.
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Step 1: The Legendre series of a function $f(x)$ is given by: $$f(x) = \sum_{n=0}^{\infty} a_n P_n(x)$$ where $a_n$ is the coefficient of the $n$th Legendre polynomial $P_n(x)$. Show more…
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(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 $$y_{2}(x)=x+\frac{2}{3 !} x^{3}+\frac{(-3)(-1) \cdot 2 \cdot 4}{5 !} x^{2}+\cdots$$$$=x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\dots=\frac{1}{2} \ln \frac{1+x}{1-x}$$ Verify this by solving (I) with $n=0,$ setting $z=y^{\prime}$ and separating variables.
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Orthonormalize the powers of $x$, on the interval $-1 \leq x \leq 1$, to obtain the first four Legendre polynomials (Equation 3.91).
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