/sci/ - Science & Math

[OP No.1365410]

Hi guys, not a homework thread. Just doing some self teaching and I'm stuck.

Can anyone verify or correct the following steps?

<div class="math">f(n_i)=\ln{ \left( \prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!} \right)}+\alpha left( N-\sum n_i \right)+\beta \left( E-\sum n_i \epsilon_i \right)</div>
assume <span class="math">g_i >> 1 [/spoiler], and use Stirling's approximation ln(x!) = x(ln(x)-1)
<div class="math">f(n_i)=\sum_i \left( (n_i + g_i) \ln(n_i + g_i) - n_i \ln(n_i) - g_i \ln (g_i) \right) +\alpha\left(N-\sum n_i\right)+\beta\left(E-\sum n_i \epsilon_i\right)</div>

Take the derivative w.r.t <span class="math">n_i[/spoiler]
<div class="math"> \sum_i \left( \ln {\left( \frac{n_i + g_i}{n_i} \right)} \right) - \alpha \sum 1 - \beta \sum \epsilon_i </div>
<div class="math"> = \sum_i \left( \ln{ \left( \frac{n_i+g_i}{n_i} \right) } - \alpha - \beta \epsilon_i \right) = 0 </div>
<div class="math">\frac{n_i+g_i}{n_i} = \exp{\alpha + \beta \epsilon_i} </div>
<div class="math">n_i = \frac{g_i}{\exp{\alpha + \beta \epsilon_i} -1} </div>

I'm not sure if my differentiation was correct, or if the step were I moved from showing the summation vanishes to the step where the summand vanishes is legit.

It gives me the answer I was looking for though.

Any help would be much appreciated.
I pray to the God of Latex that this all works.

[OP No.1365419]

Extra open brace error should have read

<div class="math">f(n_i)= \ln{ \left( \prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!} \right)}+\alpha \left( N-\sum n_i \right)+ \beta \left( E-\sum n_i \epsilon_i \right)</div>

[Anonymous No.1365429]

Is this that Undetermined Multipliers thing I keep hearing about?

[OP No.1365428]

Ugghh
Also, the last and second last line should read

<div class="math"> exp{(\alpha + \beta \epsilon_i)} </div>
<div class="math">frac{g_i}{\exp{(\alpha + \beta \epsilon_i)} -1}</div>

[OP No.1365442]

Uggghh

Also, the last and second last lines should read

<div class="math">\exp{(\alpha + \beta \epsilon_i)}</div>
<div class="math">\frac{g_i}{\exp{(\alpha + \beta \epsilon_i)} -1}</div>

[Anonymous No.1365496]

pbfbump

[Anonymous No.1365548]

anti-front-page-religion-thread-pbfbump

[Anonymous No.1366332]

>>1365410

[Anonymous No.1366948]

I would be able to help you more if I knew what the fuck you were asking. What are all these terms you haven't explained or introduced or defined? Are you going to give us any goddamn context? What are you trying to do in the first place? Why did you put ni on the left and the run ni over all indices on the right? How come you still run the right side over all indices i when you differentiated wrt a specific index (shouldn't all but one vanish)? And generally speaking, you can't differentiate an approximation in order to achieve an approximation of a derivative. if that's what you're trying to do.

OP is a disaster area in terms of coherence and clarity. Also, use codecogs latex editor to check your latex before you post it (it parses the code to an image in front of your eyes).

[Anonymous No.1366959]

>>1365429 here

I don't know anything about this method you're using, but there is another way to derive Stirling's formula that I know of.

[Anonymous No.1367070]

>>1366959
Since threads never die on /sci/ I might as well tell you that you need to apply the method of steepest descent to the beta function, and make use of asymptotic expansion of the exponential integral.