All Questions
Tagged with legendre-polynomials special-functions
81
questions
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Derivation of the associated Legendre Polynomials
I have been struggling to find a proper derivation of the associated Legendre Polynomials and a derivation of
$$P_l^{-m}(\mu)=(-1)^m\frac{(l-m)!}{(l+m)!}P_l^m(\mu)$$
Can someone point to a proper ...
1
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1
answer
69
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How to find an expression for the $n$th partial derivatives of $1/r$?
From Pirani's lectures on General Relativity I got the following identity which he asks the reader to prove by induction which is easy
$$\partial_{\alpha_1} \partial_{\alpha_2} ... \partial_{\alpha_n} ...
- 265
1
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0
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64
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Integral of product of Legendre polynomial and exponential function
Kindly help me with the following integral :
$
I_l(a) = \int_{-1}^{+1} dx\, e^{a x} P_l(x) \quad
$
($a$ is real and positive).
I thought to use the following relation given in Gradshteiyn and also ...
- 65
1
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2
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129
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Calculation for negative integer order Associated Legendre Function
I am currently engaging with the following hypergeometric function as a result of attempting to find a solution for this probability problem for $n$ number of dice:
$$_2F_1\left (\frac{n+k}{2}, \frac{...
2
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1
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80
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What is the combinatorics meaning of the generating function for Legendre polynomials?
I know the generating function has been a super useful tool when finding the Legendre polynomials (or other special functions), or even used to estimate the static electric potential. In the Physics ...
- 21
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0
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75
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Coefficient of $x^n$ in Legendre series expansion
Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely
$$
f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x)
$$
where $P_n(x)$ is the $n^{th}$ Legendre polynomial and
$...
- 31
1
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1
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566
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On the vanishing of the integral $\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}$ for $\ell \geq n+2$
I came across the following integral
\begin{equation}
\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}
\, ,
\end{equation}
where in the above $P_{\ell}(x)$ is the Legendre ...
user1211323
1
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0
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47
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partial alternating sum of legendre polynomial
My question concerns the (generalized) Legendre polynomials in the $d$-dimensional space (see, e.g. Mueller, Freeden & Gutting):
$$
P_n(d;t) = \frac{|S^{d-3}|}{|S^{d-2}|} \int_{-1}^1 (t + i \sqrt{...
- 1,179
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97
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Integration of a square of Conical (Mehler) function
I want to evaluate
$$\int_{\cos\theta}^1 \left( P_{-1/2+i\tau}(x) \right)^2 dx,$$
where $P$ is the Legendre function of the first kind, $i$ is the imaginary unit, and $\tau$ is a real number.
Are ...
- 33
1
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2
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289
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The $n$th derivative of the $n$th spherical Bessel function
I quote Problem 12.4.7 of the 5th edition of Mathematical Methods for Physicists by Arfken, Weber, and Harris:
A plane wave may be expanded in a series of spherical waves by the Rayleigh equation:
$$
...
- 347
3
votes
1
answer
414
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Proof of Bonnet's Recursion Formula for Legendre Functions of the Second Kind?
I'm doing some self-study on Legendre's Equation. I have seen and understand the proof of Bonnet's Recursion Formula for the Legendre Polynomials, $P_n(x)$.
$$(n+1)P_{n+1}(x) = (1+2n)xP_n(x) - nP_{n-...
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1
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74
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Could someone help prove some properties of Legendre Polynomials?
I have already proved other properties of the Legendre polynomials, like:
$$P_n(-x) = (-1)^n \, P_n(x)$$
$$P_{2n+1}(0) = 0$$
$$P_n(\pm1)= (\pm1)^n$$
But I can't get this one:
$$P_{2n}(0) = \frac{(-1)^...
- 29
2
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1
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148
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How to prove Legendre Polynomials' recurrence relation without using explicit formula?
Assume there is an inner product on linear space $V = \{ \text{polynormials}\}$:
$$\langle f, g\rangle = \int_{-1}^1 w(t) f(t)g(t) dt$$ with $w(t) \ge 0$ and not identically zero.
Then we can ...
- 1,231
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0
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80
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where these formulas for Legendre polynomials and series come from?
I am reading a textbook on differential equations.The chapter I am studying now is about solving differential equations using series.The book solves Legendre's differential equation$$(1-x^{2})y''-2xy'+...
- 411
3
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1
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125
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Evaluating an integral with derivatives of Associated Legendre polynomials
I came across the following integral
$$\int_{-1}^{+1} (1-x^{2}) \frac{\partial P_{lm}(x)}{\partial x} \frac{\partial P_{km}(x)}{\partial x} dx$$
where $P_{lm}(x)$ is an associated Legendre polynomial, ...
- 403