math121b:02-26
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math121b:02-26 [2020/02/26 17:03] pzhou created |
math121b:02-26 [2026/02/21 14:41] (current) |
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| - | ===== 2020-02-26, | + | ===== 2020-02-26, |
| Today we begin Chapter 12, the series solution to ODE. | Today we begin Chapter 12, the series solution to ODE. | ||
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| We can get that, the series converges for $|x|<1$. | We can get that, the series converges for $|x|<1$. | ||
| - | If $l$ is an integer, then one of the series converges. If $l_1 + l_2 = 1$, then $l_1$ and $l_2$ gives the same solution. That is why we use $l(l+1)$ to label the different solutions. | + | If $l$ is an integer, then one of the series converges. If $l_1 + l_2 = -1$, then $l_1$ and $l_2$ gives the same solution. That is why we use $l(l+1)$ to label the different solutions. |
| ==== Eigenvalue problem ==== | ==== Eigenvalue problem ==== | ||
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| ===== Rodrigue Formula ===== | ===== Rodrigue Formula ===== | ||
| - | $$ P_l(x) = \frac{1}{2 l!} \d_x (x^2 -1)^l $$ | + | $$ P_l(x) = \frac{1}{2^l l!} (\d_x)^l (x^2 -1)^l $$ |
| Let's show that it satisfies the Legendre equation. | Let's show that it satisfies the Legendre equation. | ||
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math121b/02-26.1582736597.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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