# 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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### Picking out a single term from a sum that depends on Associated Legendre Polynomials through integration

Although this question is rooted in physics, the question is a mathematical one, please read the entire question before voting. I have defined a state for some $m'$ \left|\Psi_{m'}\...
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### Probability of being in a circle, given normal

Let's assume a bivariate normal distribution with center $\mu$ and covariance matrix $\Sigma$. Let a circle $C$ be given as $C=\{x\in\mathbb{R}^2:||x-\mu||\leq R\}$. I would like to calculate the ...
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### Approximation of series using integral

In notes of statistical physics I found the following approximation $$\sum\limits_{n=0}^{\infty}F\left(n+\frac{1}{2}\right)\approx \int_{0}^{\infty}F(x)dx+\frac{1}{24}F'(0)$$ for $F$ such that the ...
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### Need some help understanding notation for composite gauss quadrature formula

Reading through some notes on 2-point gauss quadrature, I came across the following general formula. I'm currently doing an assignment with 3-point quadrature, and have gotten to a similar formula, ...
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### (Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
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### How can we use the Lindley's method to approximate the following expression?

The Lindley's(1980) approximation is one of the most popular methods that is used to obtain Bayes estimates. In this method we need to maximum likelihood estimators(MLEs) of the unknown parameters. ...
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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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### 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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### 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 ...