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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Saddle Point / Steepest Descent for Bessel Functions
I am trying to understand how to approximate integrals with Bessel functions. In particular I have something like:
$$I_{\ell} = \int_{0}^{\infty} j_{\ell}(pr) dr = \frac{\sqrt{\pi} \Gamma[(1+\ell)/2] ...
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Sum of $\sum_{k=1}^{+\infty} \frac{(-1)^{k-1}}{k^2}$ correct to five decimal places [closed]
Correct to five decimal places would mean an error of less than 0ドル.000005$. So looking for an $n\in\mathbb{N}$ such that $a_{n+1}< 0.000005$ would imply that
$$
\frac{1}{(n+1)^2}<0.000005=\frac{...
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How this estimation $\frac{2A(n-1)^n}{n^{n}\sqrt{n-1} } $ was made?(in the absence of Stirling's formula, what method of estimation was available?)
This estimation is from the paper "A Method of approximating the Sum of the Terms of the Binomial
$(a+b)^n$ expanded to a Series"(1733) by Abraham de Moivre.(Full Text or Solutions of ...
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Is this 20ドル^\circ$ approximation construction using triangle, square, and pentagon a known method?
I recently found a simple straightedge-and-compass construction that approximates a 20ドル^\circ$ angle, and I wonder if it has been known or studied before.
Construction:
Draw a segment $AB$.
Construct ...
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Why is the small strain hypothesis considered sufficient for finite strain linearization?
In continuum mechanics, the infinitesimal strain tensor $ \varepsilon $ is introduced as:
$$ \varepsilon_{ij} = \frac{1}{2} \left( u_{i,j} + u_{j,i} \right), \quad u_{i,j} = \frac{\partial u_i}{\...
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Approximating pumpkin pi (the ratio of an idealized pumpkin's circumference to its diameter, appropriately defined)
As everyone knows, a circle has the property that:
$$\frac{\text{circumference}}{\text{diameter}} = \pi$$
Now, let's consider a more complicated shape, the cross-section of a pumpkin through its ...
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How do we find an asymptotic approximation of the function $\mathbf{P}(r)=\left(t_1+\prod\limits_{k=1}^{r}(t_2+k^{s})\right)^n$?
Suppose we have the following function, where $s\in\mathbb{R}$ and $t_1,t_2,n\in\mathbb{N}\cup \{0\}$ are constants:
$$\mathbf{P}(r)=\left(t_1+\prod_{k=1}^{r}(t_2+k^{s})\right)^n$$
Question: What is ...
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How can I approximate this function to be able to integrate it?
Suppose I have the following real function
$$f(x) = \frac{\left[ (2 + b)^2 - x \right]^{1/2} (b^2 - x)^{3/2} \left[ (2 + b)^2 + 2x \right] (x + 2a^2) (x - 4a^2)^{1/2}}{x^{3/2} (c^2 - x)^2}$$
defined ...