# Questions tagged [approximation]

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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### Why is $-cN/z^\alpha$ a good approximation for $\ln(1-F(z))^N$?

Bertin's Statistical Physics of Complex Systems, 3rd ed. p. 65 defines a "complementary cumulative distribution $\tilde{F}(z)$" equal to $\int_z^\infty p(x)\,dx$, where the density $p(z)$ is ...
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### Approximation of PDF with summation to infinity, cos(x) and exp(x)

I would like to implement this probability density function in C++. However, on this current form, the algorithm takes a lot of time to return a result (especially because it include a summation). Do ...
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### For what values of $p>0$ is $\lim_{n\rightarrow\infty}\int_0^n\frac{(1-\frac{x}{n})^ne^x}{n^p}dx=0$?

For what values of $p>0$ is $\lim_{n\rightarrow\infty}\int_0^n\frac{(1-\frac{x}{n})^ne^x}{n^p}dx=0$? My thoughts: We know that $(1-\frac{x}{n})^n\leq e^x$, so the numerator is $\leq e^{2x}$. So, ...
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### Let $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x) = xm, m>1$ . Write Newton’s method for approximating root $x^∗ = 0$ of $f$ starting with $x_0= 0$. [closed]

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x) = xm, m > 1$ Write down Newton’s method for approximating the root $x^∗ = 0$ of $f$ starting with initial guess $x_0= 0$. Express the $n^{th}$ ...
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### In TSP where a modified triangle inequality holds, what is the approximation ratio of Christofides' algorithm?

In the metric TSP, we can use Christofides' algorithm to get a $\frac{3}{2}$-approximate solution. This is a consequence of the triangle inequality where $d_{ij} + d_{jl} \geq d_{il}$, that enables us ...
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### Gamma functions approaches solely n to a negative power

I have read $[x^n] (1-x)^\alpha\sim \frac{n^{-\alpha-1}}{\Gamma(-\alpha)}$ as $n\rightarrow\infty$ where $[x^n]$ means coefficient of $x^n$ in what follows. But I have never seen a proof of this ...
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1 vote
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### How the green function for the relativistic heat equation converges to the green function of the heat equation?

The relativistic heat equation or telegraphers equation is: $$(\alpha\partial_t^2 + \beta\partial_t - \omega\,\nabla^2_{\text{3D}})G_R = \delta$$ if $\alpha \rightarrow 0$ the solution must ...
1 vote
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### (Polynomial approximation) Why does the nth derivative of both the function and the polynomial has to be equal at center?

I've watched many videos about Taylor/Maclaurin polynomial but no tutor ever explained why it has to be f(n)(c) = p(n)(c) at center x = c. I've seen the behavior of the graph of p(x) and f(x) when ...
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1 vote
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### Linear approximation of expected value

I am reading a paper where there is an approximation of an expected value. I am not sure what sort of approximation method they are using. Reminds me a bit of Taylors Theorem, but I am just not ...
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