Questions tagged [approximate-integration]

Use this tag for questions related to approximate integration, which constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

125 questions
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Approximation for the Marcum Q-function

I want to approximate the Marcum Q-function $Q_{m}(a,b)$ when $b$ goes to zero, where the Marcum Q-function $Q_{m}(a,b)$ is defined in an integral form, shown in https://en.wikipedia.org/wiki/Marcum_Q-...
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Approximate a double integral

I am struggling to approximate the following integral $$\sqrt n\int_0^\infty \int_0^\infty (1 + n x^2)^{-1}(1 + y^2)^{-1} \Phi\left(\frac{a}{\sqrt{1 + b + x^2y^2}}\right) \text{d}x \text{d}y,$$ where ...
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How do you determine a step function to approximate another function?

I have a question here in which I need to explicitly write down a sequence $f_n(x)$ that can approximate $e^x$. From reading, I known that I need to pick a partition sequence of $x_k$ so that I can ...
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Fourier partial sums of Sawtooth wave are not equal its convolution with the Dirichlet kernel!

Let $f$ be the $2\pi$-periodic function relating \begin{equation} f(x) = \frac{\pi-x}{2} \end{equation} on $(0, 2\pi)$. The coefficients of its Fourier series are easily calculated [see (*), ...
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Is it integration or not

I was solving a physics problem and I encountered to find the sum the function $xe^{-kx}$ where for all positive values of $x$ (i.e. from 0 to infinity) Is there any way to do that? I speculate that ...
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Trapezoidal rule

The trapezoidal rule is given by the following formula: $\int_{a}^{b}f(x) \approx \frac{b-a}{2}[f(a)+2\sum_{i=1}^{n-1}f(x_i)+f(b)]$ I have been given a question which wants me to evaluate the ...
Approximating $\int \frac{1}{1 + a p^4 + b p^6}$
I'm attempting to calculate an approximate "closed form" of the integral $$\int \frac{dp}{1 + a p^4 + b p^6}$$ as a function of $a$ and $b$, two small parameters (of the order of $10^{-2}$). I'm ...
I am trying to calculate the following integral : $$I(\lambda,\alpha)=\int_{\lambda}^1 \mathrm{d}\tau \frac{1-\tau^\alpha}{1-\tau}\exp(-k \tau)$$ where $\lambda<1$, $k$ is a positive constant ...