0
votes
0answers
42 views

How to get the asymptotic formula of generalized Bessel function?

How to get the asymptotic formula of generalized Bessel function? $$J_{\nu}^{(\mu)}(z)=\frac{2}{\sqrt{\pi}\Gamma(\nu+1-1/\mu)}\Big(\frac{z}{2}\Big)^{\mu \nu/2} \int_{0}^{1} ...
5
votes
3answers
115 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
1
vote
1answer
29 views

Why does the asymptotic equation of the modified Bessel of the second kind (Iv) have an imaginary part?

This is a follow up to this question. How does one arrive at the asymptotic expressions for the bessel functions? After looking at: G. N. Watson, "A Treatise on the Theory of Bessel Functions", 2nd ...
1
vote
2answers
68 views

How does one arrive at the asymptotic expressions for the bessel functions?

It is known that Bessel functions for large arguments will behave as exp or cos/sin however I was wondering how does one arrive at those results. The motivation being that I would like to use these ...
0
votes
1answer
48 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 ...
0
votes
0answers
42 views

Limits of generalized hypergeometric functions

For a (quite fiddly) asymptotic matching, I would like to be able to write the solution to \begin{equation} \frac{\mathrm{d}^5}{\mathrm{d}x^5}f(x) + \frac{10}{15^{1/2}} \frac{\mathrm{d}}{\mathrm{d}x} ...
1
vote
0answers
41 views

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 $|y_1|+|y_2|\to \infty$ ...
4
votes
3answers
227 views

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 ...
2
votes
1answer
76 views

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 ...
1
vote
1answer
79 views

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.
4
votes
1answer
91 views

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.
3
votes
1answer
72 views

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 ...
2
votes
3answers
124 views

Bernoulli numbers: comparison to factorials

I am trying to understand the behaviour of the Bernoulli numbers with respect to factorials, specifically I'd like to know whether it is true that, for all $n \in N$ with $n \ge 2$ we have $$ ...
1
vote
0answers
61 views

function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
1
vote
0answers
215 views

Relation between the exponential function and the modified bessel function of second kind

I found the following sentence at the wikipedia page : Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Iα and Kα(this is the mod. bessel function of the ...
1
vote
1answer
341 views

Bounding the modified Bessel function of the first kind

i'm looking for an upper bound for the modified Bessel function of the first kind of a +ive real argument. It seems that it satisfies the inequality : $$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$ ...
3
votes
1answer
85 views

Asymptotics for infinite sum with erf

I'm interested in approximating the infinite sum $$ \sum_{i=1}^\infty Z\left(\frac{\alpha i\pm1}{\beta}\right) $$ where $\alpha,\beta$ are constant and $$ Z(a\pm ...
8
votes
2answers
214 views

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: ...
6
votes
1answer
286 views

Missing term in series expansion

I asked a similar question before, but now I can formulate it more concretely. I am trying to perform an expansion of the function $$f(x) = \sum_{n=1}^{\infty} \frac{K_2(nx)}{n^2 x^2},$$ for $x \ll ...
2
votes
1answer
82 views

Growth of $\Gamma(n+1,n)$ and $\operatorname{E}_{-n}(n)$

Quite often when I ask W|A to solve something it gives me an answer in terms of $\Gamma(n+1,n)$ or exponential integral $\operatorname{E}_{-n}(n)$. Looking up the definition of the incomplete gamma ...
2
votes
0answers
83 views

Series expansion of a series

I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
1
vote
2answers
63 views

Roots of the equation $I_1(b x) - x I_0(b x) = 0$

I'm interested in the roots of the equation: $I_1(bx) - x I_0(bx) = 0$ Where $I_n(x)$ is the modified Bessel function of the first kind and $b$ is real positive constant. More specifically, I'm ...
6
votes
0answers
177 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
1
vote
1answer
180 views

asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$

consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ : $$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$ And: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ ...
3
votes
0answers
145 views

Saddle point and stationary point approximation of the Airy equation

Happy New Year to you all. Let $$\tag 1 J(N)=\int_a^b e^{Nf(x)}dx$$ where $N\in\mathbb R$ and $N>>1$ and $f(x)$ has a global maximum at $x=x_0$ with Taylor expansion $$f(x) \approx ...
2
votes
0answers
69 views

asymptotics of $ J_{iu} (ia)$ for a Bessel function

Let $J_{iu}(ia)$ be the Bessel function of imaginary order. ($a$ is a real number (positive or negative) and $u$ is also real.) In the limit $u \to \infty $ how does the function $J_{iu} (ia)$ ...
0
votes
1answer
137 views

Upper bound of function including Pochhammer symbol

How can I find the upper bound of $$\left\vert\frac{(c+1/2+\lambda)_{n}}{\lambda^{n}}\right\vert,\quad\text{where}\quad(c+1/2+\lambda)_{n}=\frac{\Gamma(c+1/2+\lambda+n)}{\Gamma(c+1/2+\lambda)}$$ and ...
4
votes
1answer
738 views

About the asymptotic formula of Bessel function

For $ \nu \in \Bbb R$, I want to prove the well-known formula $$ J_\nu (x) \sim \sqrt{\frac{2}{\pi x}} \cos \left( x - \frac{2 \nu +1}{4} \pi \right) + O \left( \frac{1}{x^{3/2}} \right) \;\;\;\;(x ...
2
votes
1answer
296 views

Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?

Motivation: I was looking at the approximation of the truncated Prime $\zeta$ function $$ P_x(s)=\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O \left(\cdot \right) $$ (to be found here with or ...
5
votes
2answers
822 views

Asymptotic expansion of integral involving modified Bessel-function

I would like to obtain the asymptotic expression for $\alpha \to \infty$ of the following integral $$I(\alpha)=\int_0^\infty\!dx\,x (1 - \cos[2\alpha K_0(x)]) = \int_0^\infty\!dx\, 2x \sin^2[\alpha ...
1
vote
2answers
242 views

Bounds on integral $x^{-a} \int_{1}^x y^{a-1} \exp(-y a) dy$

Consider the function $$ I(a,x) = x^{-a} \int_{1}^x y^{a-1} \exp(-y a) dy $$ where $x \geq 1$, and $a \geq 0$. I am not really interested in the parameter $x$, so define $$ I(a) = \sup_{x \geq 1} ...
4
votes
2answers
136 views

'Error term' in zeta function [duplicate]

Possible Duplicate: What is the expression of $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$? Asymptotic formulas for the n-th harmonic number are well-known: $$ \sum_{k=1}^n\frac1n=\log ...
3
votes
1answer
147 views

Dilogarithm asymptotics for an exponential parameter.

So this question is about this dilogarithm function. Assume the argument $z$ is real then I want to show the formula $$\operatorname{Li}_2(e^{-z})=\frac{\pi^2}{6} + z\log z -z+O(z^2) $$ as $z$ ...
3
votes
1answer
73 views

Asymptotics of a solution

Let $x(n)$ be the solution to the following equation $$ x=-\frac{\log(x)}{n} \quad \quad \quad \quad (1) $$ as a function of $n,$ where $n \in \mathbb N.$ How would you find the asymptotic behaviour ...
7
votes
1answer
526 views

the limit of the ratio of two $\Gamma(x)$ functions

I am interested in the quantity $$ a_{n} = \sqrt{n/2} \frac{\Gamma((n-1)/2)}{\Gamma(n/2)}$$ (this is the geometric bias of the non-central t-distribution with $n$ d.f.) After some plotting, my hunch ...
4
votes
1answer
667 views

Variations on the Stirling's formula for $\Gamma(z)$

I am currently reading some material that makes heavy usage of Hypergeometric functions, and there is one particular point about applying Stirling's approximation to various terms consisting of ...
2
votes
1answer
209 views

Asymptotic order of $\frac{\mathrm{erfi}(\sqrt{x})}{\exp(x)\sqrt{x}}$

I need to approximate this expression in order to sum it. Asymptotically I obtain $\frac1{\sqrt{\pi}x}+\frac1{2\sqrt{\pi} x^2} + O\left(\frac1{x^3}\right)$. Although this looks fine there is the ...
1
vote
3answers
193 views

better understanding of incomplete gamma function $\Gamma(0,x)$

By definition incomplete Gamma function is:$$\Gamma(0,x)=\int_{x}^{\infty}t^{-1}e^{-t}dt $$ I have an expression which includes $$\Gamma(0,r(A)e^{i\phi(A)}),$$ where $A>0$ is a parameter, and ...
10
votes
3answers
537 views

How does Lambert's W behave near ∞?

How does $W$ behave near $+\infty$ compared to $\log$? In particular, I'm interested in the asymptotic expansion of $$\frac{W(x)}{\ln(x)}$$ near $\infty$ (but along the positive real line, if that ...