# Tagged Questions

37 views

### Asymptotic Expansion of Bessel $\frac{1}{\pi}\int_0^{\pi}e^{x\cos t}dt$

My question is how to find the asmyptotic expansion of $I(x)=\frac{1}{\pi}\int_0^{\pi}e^{x\cos t}dt$ as $x\rightarrow\infty$. I already got the expansion of $\int_0^{\pi/2}e^{-x\sin^2t} dt$ by using ...
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### Limits of generalized hypergeometric functions

For a (quite fiddly) asymptotic matching, I would like to be able to write the solution to \frac{\mathrm{d}^5}{\mathrm{d}x^5}f(x) + \frac{10}{15^{1/2}} \frac{\mathrm{d}}{\mathrm{d}x} ...
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### Asymptotic behavior of the Beta function

Let $B(z_1,z_2)$ be the Beta function, $z_1 = x_1 + iy_1$, $z_2 = x_2 + i y_2$. Suppose that $x_1$, $x_2 > 0$. I want to estimate the behavior of $|B(x_1+iy_1,x_2+iy_2)|$ as $|y_1|+|y_2|\to \infty$ ...
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### Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
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### Lommel function

I need to do this integral: $$\int_0^\infty dx\cdot x \sqrt{x^2+1}K_0(ax)$$ where K is the modified Bessel of second kind. I have seen that in Gradhsteyn 7th edition in 6.565.7 says that this ...
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### An integral relating to Bernoulli polynomials

Show that $$\int_{0}^{1}B_{2n+1}(x)(\cot({\pi}x)-2\sin(2{\pi}x))dx{\sim}0$$ where $B_{2n+1}(x)$ is the Bernoulli polynomials.
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### Bernoulli number type asymptotics

I find an interesting formula but I can not prove it. Show that $$I_n=(-1)^{n+1}\int_0^1 B_{2n+1}(x)\cot(\pi x) \, dx\sim\frac{2(2n+1)!}{(2\pi)^{2n+1}}$$ where $B_n(x)$ is the Bernoulli Polynomials.
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### Asymptotics of sequence depending on Tricomi's function

I'm dealing with the following sequence $$p_n = U(a, a - n, 1)$$ where $a > 0$ and $U$ is Tricomi's function. I suspect that asymptotically when $n \to \infty$ its behaviour is a power law ...
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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: ...