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copied!<p>It's a placeholder for a variable. </p> <p>If you want to define a y(x)=x^2 function, you just could do:</p> <pre><code> f = #^2 & </code></pre> <p>The & "pumps in" the variable into the # sign. That is important for pairing & and # when you have nested functions. </p> <pre><code> In: f[2] Out: 4 </code></pre> <p>If you have a function operating on two vars, you could do:</p> <pre><code> f = #1 + #2 & </code></pre> <p>So </p> <pre><code> In: f[3,4] Out: 7 </code></pre> <p>Or you may have a function operating in a list, so:</p> <pre><code> f = #[[1]] + #[[2]] & </code></pre> <p>So: </p> <pre><code> In: f[{3,4}] Out: 7 </code></pre> <p><b> About <code>Root[]</code></b> </p> <p>According to Mathematica help: </p> <pre><code>Root[f,k] represents the exact kth root of the polynomial equation f[x]==0 . </code></pre> <p>So, if your poly is <code>x^2 - 1</code>, using what we saw above: </p> <pre><code> f = #^2 - 1 & In[4]:= Root[f, 1] Out[4]= -1 (* as we expected ! *) </code></pre> <p>And</p> <pre><code>In[5]:= Root[f, 2] Out[5]= 1 (* Thanks God ! *) </code></pre> <p>But if we try with a higher order polynomial:</p> <pre><code> f = -1 - 2 #1 - #1^2 + 2 #1^3 + #1^4 & In[6]:= Root[f, 1] Out[6]= Root[-1 - 2 #1 - #1^2 + 2 #1^3 + #1^4 &, 1] </code></pre> <p>That means Mathematica doesn't know how to caculate a symbolic result. It's just the first root of the polynomial. But it does know what is its numerical value: </p> <pre><code>In[7]:= N@Root[-1 - 2 #1 - #1^2 + 2 #1^3 + #1^4 &, 1] Out[7]= -2.13224 </code></pre> <p>So, <code>Root[f,k]</code> is a kind of stenographic writing for roots of polynomials with order > 3. I save you from an explanation about radicals and finding polynomial roots ... for the better, I think</p>

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