# Tagged Questions

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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### Approximating $\chi_0$ pointwise with holomorphic functions

Define $\chi_0:\mathbb{C}\to \mathbb{C}$ as $$\chi_0(z)=\left\{ \begin{gathered} 1 \quad z=0\hfill \\ 0 \quad z\ne 0 \hfill \\ \end{gathered} \right.$$ Does there exist a sequence of ...
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### (Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
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### How to show this formula to get a square root of a number in “just few seconds” is true?

I don't remember in which topic I found it but I know it was there. And I still have not find a proof of this nice approximation. Let $x$ be a non perfect square number. If $y$ is the closer ...
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### Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
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### Partial Derivatives Approximation

By definition we know the following: $$\frac{\partial f(x,y)}{\partial x} \approx \frac {f(x+ \delta x,y)-f(x,y)}{\delta x}$$ \frac{\partial f(x,y)}{\...
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### Formal approximation for second-order ODE with varying coefficients

I have a differential equation of the form $$0=a+by(x)+cf(x)+z(x)f''(x)$$where the functions $y$ and $z$ are known and we want to find $f$. If $z$ is constant, i.e. $z(x)=Z$, it is straightforward to ...
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### Finding an approximation of a function's root

I have the polynomial function $f (x) = x^5+2x^2+1$. I am trying to find an approximation to its root in $[-2,-1]$, with the precision of $0.1$, and with a minimal number of steps. The answer I was ...
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### Behaviour of the Spectral Weight Function $\frac{\sin^2{(\pi f t)}}{(\pi f)^2}$

I'm looking into the properties of the so called spectral weight function $W_0 = \frac{\sin^2{(\pi f t)}}{(\pi f)^2}$. While not important for the question, this function is is encountered in the ...
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### Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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### Estimate the drift and diffusion function numerically

I have a 1D problem as following $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x} \Big[ \frac{1}{2} \frac{\partial (g(x) f)}{\partial x} -\mu(x)f \Big]$$ I have a time-series of ...
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### Fitting a continuous curve over a piecewise constant data

I have some measurements that are piecewise constant over a certain variable. For example, in the following image, the vertical axis represents the measurement data and the variable is on the ...
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### Normal Approximation - how many bookings so probability for “overbooking” stays under certain value

I need some help with the following: A hotel has $r$ rooms. The probability that a guest who booked a room also appears (which means: no cancellation) is $p = 0.9$. I'd like to know how many rooms ...
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### What is the value of $e^{-10000}$?

What is the value of $e^{-10000}$? We know that the function $e$ does not attain value $0$ anymore. But in R and Matlab the value of $e^{-10000}$ is given as $0$ which is not correct anymore. I ...
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### Perturbation: compute an approximation to the solution of the equation $y+\epsilon\sin y=x^2$

Compute approximation to the solution of the equation $y+\epsilon \sin y=x^2$ using perturbation method. Assume that terms involving powers of $\epsilon$ of order 3 or more can be ignored. So far I ...