# Tagged Questions

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### Asymptotic for the integral involving exponential

The integrand seems extremely easy: $$I_n=\int_0^1\exp(x^n)dx$$ I want to determine the asymptotic behavior of $I_n$ as $n\to\infty$. It's not hard to show that $\lim_{n\to\infty}I_n=1$ follows from ...
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### Integral of smooth function

Another prelim problem: Suppose that $f(x,y)$ is a smooth function defined on $\mathbf{R}^2$. Prove that $$\int_{x^2+4y^2\leq r^2}f(x,y)\,dx\,dy = ar^2+br^4+O(r^5)$$ Express $a$, and $b$ in terms ...
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### Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]

I am reading about the Riemann hypothesis, and the article mentioned the Li function: $$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$ They said that this function can be approximated: ...
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### Laplace's method

I'm still having a little trouble applying Laplace's method to find the leading asymptotic behavior of an integral. Could someone help me understand this? How about with an example, like: ...
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### $f(x)=\int_{0}^{1}e^{ixz^2}dz$ as $x\rightarrow +\infty$.

Find the asymptotic behaviour as $f(x)=\int_{0}^{1}e^{ixz^2}dz$ as $x\rightarrow +\infty$. Could anyone show me how to do this with either the method of stationary phase or integration by parts? ...
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### Asymptotics of exponential integral

Hello I wonder if there is any asymptotics known for such integral: $$I(x) = \int_2^x \frac{e^t}{t} dt \qquad\text{when  x\to+\infty }.$$ Thank you very much.
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### Asymptotic expansion of integral

Consider the following integral: $$\int_{0}^{\infty} e^{-xt} \ln(1+\sqrt{t})dt$$ Calculate its asymptotic expansion to ALL orders as $x\rightarrow\infty$. It seems the natural thing to do is ...
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### Laplace integral and leading order behavior

Consider the integral: $$\int_0^{\pi/2}\sqrt{\sin t}e^{-x\sin^4 t} \, dt$$ I'm trying to use Laplace's method to find its leading asymptotic behavior as $x\rightarrow\infty$, but I'm running into ...
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### Asymptotic evaluation of integral

I am studying how to evaluate the integral $$\int_{0}^{\pi/4}{d\theta \over \epsilon^2+\sin^2\theta}$$ as $\epsilon \rightarrow 0$ with asymptotic methods. The book: perturbation methods by Hinch ...
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### Asymptotic analysis of the integral $\int_0^1 \exp\{n (t+\log t) + \sqrt{n} wt\}\,dt$

The integral I'm trying to study is $$F(n) = \int_0^1 \exp\left\{n(t+\log t)+\sqrt{n}wt\right\}\,dt, \tag{1}$$ where $w$ is a fixed complex number with $\Re(w) < 0$ and $\Im(w) > 0$. As ...
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### convergence of an oscillatory integral

Let $\alpha$ be real numbers and let $f\colon\mathbb{R}\to \mathbb{C}$ be a function in $L^2 (\mathbb{R})$ (actually smooth and compactly supported, but this doesn't seem to be relevant). I am ...
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### Help with the integral for the variance of the sample median of Laplace r.v.

When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.: ...
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### Estimate on an oscillatory integral

During my research I ran into the following type of an oscillatory integral, for some values of nonzero reals $a,b$: $f(R):=\int_{0}^{R} e^{2 \pi i (ar^2 + br)} dr$ and I am interested in finding a ...
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