Difference between revisions of "Laguerre L"

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(Created page with "Laguerre's equation is $$x\dfrac{y^2x}{dx^2}+(1-x)\dfrac{dy}{dx}+ny=0.$$ One of the solutions of this differential equations are the Laguerre polynomials $$L_n(x) = \displayst...")
 
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One of the solutions of this differential equations are the Laguerre polynomials
 
One of the solutions of this differential equations are the Laguerre polynomials
 
$$L_n(x) = \displaystyle\sum_{k=0}^n (-1)^k \dfrac{n!}{(n-r)!(r!)^2}x^r.$$
 
$$L_n(x) = \displaystyle\sum_{k=0}^n (-1)^k \dfrac{n!}{(n-r)!(r!)^2}x^r.$$
 
+
The first few Laguerre polynomials are given by
 +
$$\begin{array}{ll}
 +
L_0(x) &= 1 \\
 +
L_1(x) &= -x+1 \\
 +
L_2(x) &= \dfrac{1}{2}(x^2-4x+2) \\
 +
L_3(x) &= \dfrac{1}{6}(-x^3+9x^2-18x+6) \\
 +
L_4(x) &= \dfrac{1}{24}(x^4-16x^3+72x^2-96x+24)\\
 +
\vdots
 +
\end{array}$$
 
=Properties=
 
=Properties=
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> The following formula holds:
 
<strong>Theorem:</strong> The following formula holds:
 
$$\dfrac{e^{\frac{-xt}{1-t}}}{1-t} = \displaystyle\sum_{k=0}^{\infty} L_k(x)t^k.$$
 
$$\dfrac{e^{\frac{-xt}{1-t}}}{1-t} = \displaystyle\sum_{k=0}^{\infty} L_k(x)t^k.$$
 +
<div class="mw-collapsible-content">
 +
<strong>Proof:</strong> █
 +
</div>
 +
</div>
 +
 +
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 +
<strong>Theorem:</strong> The following formula holds:
 +
$$L_n(x) = \dfrac{e^x}{n!} \dfrac{d^n}{dx^n} (x^n e^{-x}).$$
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █  
 
<strong>Proof:</strong> █  
 
</div>
 
</div>
 
</div>
 
</div>

Revision as of 19:16, 4 October 2014

Laguerre's equation is $$x\dfrac{y^2x}{dx^2}+(1-x)\dfrac{dy}{dx}+ny=0.$$ One of the solutions of this differential equations are the Laguerre polynomials $$L_n(x) = \displaystyle\sum_{k=0}^n (-1)^k \dfrac{n!}{(n-r)!(r!)^2}x^r.$$ The first few Laguerre polynomials are given by $$\begin{array}{ll} L_0(x) &= 1 \\ L_1(x) &= -x+1 \\ L_2(x) &= \dfrac{1}{2}(x^2-4x+2) \\ L_3(x) &= \dfrac{1}{6}(-x^3+9x^2-18x+6) \\ L_4(x) &= \dfrac{1}{24}(x^4-16x^3+72x^2-96x+24)\\ \vdots \end{array}$$

Properties

Theorem: The following formula holds: $$\dfrac{e^{\frac{-xt}{1-t}}}{1-t} = \displaystyle\sum_{k=0}^{\infty} L_k(x)t^k.$$

Proof:

Theorem: The following formula holds: $$L_n(x) = \dfrac{e^x}{n!} \dfrac{d^n}{dx^n} (x^n e^{-x}).$$

Proof: