All Questions
Tagged with legendre-polynomials derivatives
28
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Getting the recurrence relation for Legendre polynomials by Leibnitz rule
Exercise:
For each natural number $n$ define
$$\phi_n(x)=\frac{d^n}{d x^n}\left(x^2-1\right)^n$$
Derive the formulas
$$\phi_{n+1}^{\prime}(x)=2(n+1) x \phi_n^{\prime}(x)+2(n+1)^2 \phi_n(x)$$
$$\phi_{n+...
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What will be the explicit formula for Shifted Legendre's polynomial in interval $x\in[a,b]$
If I defined shifted Legendre polynomial $\tilde P_{n}(x)=P_{n}(\frac{2x-b-a}{b-a}) for\;all x\in[a,b]$ Then what will be the explicit formula $P_{n}(x)=\sum_{k=0}^{n} (-1)^{n+k} \frac{(n+k)!}{(n-k)! ...
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Prove for general form of function at -x containing derivatives of order n
I have stumbled across multiple casses of functions (explicitly Hermit and Legendre polynomials) for which I wanted to prove the symmetry. While doing so I always ended up with the following equations:...
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Formula for the derivative of finite power series in reversed order of terms.
I wanted to solve the polar part in Schrödinger's wave equation for the H-atom by direct substitution of functions of form:-
$$
\Theta_{lm}(\theta) = a_{lm} \sin^{|m|}\theta \sum_{r≥0}^{r≤(l-|m|)/2}(-...
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2
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1
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Behavior of Legendre polynomials $P_\ell(\cos\theta)$ under $\theta\to\pi-\theta$
I'm studying some notes on the hydrogen atom which define the Legendre polynomials via Rodrigues's formula, $$P_\ell(z)=\frac{1}{2^\ell \ell!}\frac{d^\ell}{dz^\ell}(z^2-1)^\ell,\quad z=\cos\theta.$$ I'...
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Summation $\sum_{n=0}^{\infty} \frac{P_n(\cos \theta)}{n+1} = \log\frac{\sin(\theta/2) +1}{\sin(\theta/2)}$
$P_n$ is legendres polynomial of $n$ degree terms. I tried to use the formula summation of $C_n P_n(x) = f(x)$ but not able to find $f(x)$. Also I tried using generating functions but that too didn't ...
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4
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$2n$th derivative of $(x^2-1)^n$
How is the derivative $\frac{d^{2n}}{dx^{2n}}(x^2-1)^n=(2n)!$. I have tried to use Leibniz's formula but couldn't reach the solution.
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Why swapping between the derivative operator and this infinite sum leads to different results?
While working on a mathematical physical problem, i came across seemingly contradictory results.
Notations
Let's consider $\mathbf{x}_1$ to be the origin of a spherical coordinate system and $\...
1
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1
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Legendre polynomial recurrence relation proof using the generation function
I want to prove the following recurrence relation for Legendre polynomials:
$$P'_{n+1}(x) − P'_{n−1}(x) = (2n + 1)P_n(x)$$
Using the generating function for the Legendre polynomials which is,
$$(1-...
- 519
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1
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192
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Trouble proving identity with Legendre polynomials
I am trying to prove the following identity:
$$\int_0^\pi P_l(\cos\theta)P_{l'}(\cos\theta)\sin\theta d\theta = \biggl\lbrace \begin{matrix}0, \ if \ l' \neq l \\ \frac2{(2l+1)}, \ if \ l'=l \end{...
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$\left|\frac{d^n}{dx^n}(x^2-1)^n\right|\le\sqrt\frac2\pi\cdot\frac{2^nn!}{\sqrt n}\cdot\frac1{\sqrt[4]{1-x^2}}$, a bound for Legendre Polynomial
Question
Show that
i)$$\left|\frac{d^n}{dx^n}(x^2-1)^n\right|\le\sqrt\frac2\pi\cdot\frac{2^nn!}{\sqrt n}\cdot\frac1{\sqrt[4]{1-x^2}},$$ or equivalently $$\left|P_n(x)\right|\le\sqrt\frac2{\pi n}\...
- 8,679
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Differentiate Legendre’s equation $m$ times using Leibniz' rule for differentiating products
$$(1-x^2)u''(x)-2xu'(x) + \ell(\ell+1)u(x)=0\tag{1}$$
Assume that $m$ is non-negative, differentiate $(1)$ (Legendre’s equation) $m$ times using Leibniz' theorem for differentiation to show that
...
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2n-th derivation of Legendre Polynomial
Let $u_n(x)=(x^2-1)^n$
Show that $\frac{(d^{2n}u_n(x)} {dx^{2n}} = 2n!$
$(x^2-1)^n = (x-1)^n(x+1)^n $ and then a should use Leibnitz formula. I thought if I write Leibnitz formula as a binomial I ...
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7
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How to show that $\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!$ for $x=1$
I am trying to show that
$$
\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!,
$$ for $x = 1$. I tried to prove it by induction but I failed because I lack axioms and rules for this type of derivatives.
...
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Differential Equation from a generating function related to legendre polynomial
Consider the generating function $$\frac1{1-2tx+t^2}=\sum_{n=0}^{\infty}y_n(x)t^n$$. I wish to find a second order differential equation of the form $$p(x)y_n''(x)+q(x)y_n'(x)+\lambda_ny_n(x)=0$$ and ...
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