6
votes
2answers
87 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
1
vote
1answer
20 views

Find a power series expansion of $\frac{4x^2+2x}{1-3x-10x^2}$ about the point $x = \frac{1}{5}$

Find a power series expansion of $\frac{4x^2+2x}{1-3x-10x^2}$ Now I know that $\frac{1}{5}$ is a singularity of the $\frac{4x^2+2x}{1-3x-10x^2}$ and I know that $f(z) = ...
3
votes
3answers
236 views

Find a real entire function $f(z)$ asymptotic to $\ln(x^2+1)$ for real $x$.

Find a real entire function $f(z)$ asymptotic to $\ln(x^2 +1)$ for real $x$. More specific I want $f(0)=0$ and $\frac{1}{2} \ln(x^2+1) < f(x) < 2 \ln(x^2+1)$. Or prove it does not exist.
2
votes
0answers
43 views

Do asymptotically equivalent coefficients survive convolution at least in Θ?

This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that ...
2
votes
1answer
104 views

Why does convolution not maintain asymptotic equality of coefficients?

Assume I have four (generating) functions $f$, $f'$, $g$ and $g'$. If that is interesting, we can assume that they all share the same radius of convergence $\rho > 0$. In addition, we know that ...
5
votes
0answers
47 views

Singularities at roots of unity

I want to construct a function $f$ with the following properties: $f$ has a singularity at $z=1$, and for any $\zeta = e^{2\pi i\frac{a}{b}}$ with $(a,b)=1$, then $$\lim\limits_{x\to1^-}\frac{f(\zeta ...
3
votes
1answer
78 views

asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
0
votes
1answer
20 views

Asymptotic expansion of $z^{-x}$

Consider the function $z\mapsto z^{-x}$ for $x>1$ (real) and $z$ in the cut complex plane $\mathbb C\backslash\{z\leq 0, \text{ real}\}$. Does this function have an asymptotic expansion of the form ...
0
votes
1answer
28 views

What is the value of $p(z) \log(z)$ around a contour centered on the origin?

Given a polynomial $p(z)$, and a rectangle with vertices $2+iM, -3+iM, -3-iM, 2-iM$ what is the value of $f(z) = p(z) \log(z)$ around the contour? Or equivalently the change in argument? In ...
2
votes
1answer
83 views

Asymptotic evaluation of integral method of steepest descent

The question asks to show that the leading term of the integral $$ \int_{-\infty}^\infty (1+t^2)^{-1}\exp\left(ik(t^5/5+t)\right) dt $$ for large $k$ using the method of steepest descent is equal to ...
3
votes
0answers
65 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
1
vote
1answer
93 views

About asymptotic behaviour of a divergent integral.

I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. For the logarithm I am assuming a branch-cut along the positive imaginary axis starting at $x = ia$. ...
0
votes
0answers
45 views

How I can evalute numerically this improper complex integral?

I need a hand with the numerical evaluation, in Mathematica, for this integral: $$f(t)=\int_{-\infty}^\infty Exp\{it(\omega_H-\omega_l-\omega_k) - \sum _{j\neq(l,k)} S_j [1-e^{-it\omega_j}]\}\, dt$$ ...
1
vote
1answer
136 views

Expansion of lower incomplete gamma function $\gamma(s,x)$ for $s < 0$.

The lower incomplete gamma function for positive $s$ is defined by the integral $$ \gamma(s,x)=\int_0^{x} t^{s-1} e^{-t} dt. $$ Taylor expansion of the exponential function and term by term ...
0
votes
0answers
31 views

Question about picking value large enough so that an inequality holds for all values larger than said value

This question makes me wonder about more general inequalities, but I have a particular example. Let $C$ be a positive fixed constant, $0<\epsilon<1$ be given, and assume $\alpha,\beta\in ...
2
votes
1answer
60 views

Determining asymptotic behavior through generating functions

I need to determine the asymptotic behavior of $$a_n=\sum_{k=2}^{n-2}\frac1{\ln k\ln(n-k)}$$ as $n\to\infty$. I know some elementary methods that might help. For example, split the index $\lvert ...
1
vote
0answers
35 views

$\lim_{z \to x \pm i \infty} \Gamma(z) \zeta(z + \alpha) = 0$?

I guess $\lim_{z \to x \pm i \infty} \Gamma(z) \zeta(z + \alpha) = 0$ where $x$ and $\alpha$ are real numbers. The guessing is from numerical experiments and I know $\Gamma(z)$ vanishes exponentially ...
0
votes
1answer
35 views

A complex integration problem

The problem is: Let $\gamma$ be the circle of radius $R$ centered at $0$. Let $m$, $n$ be positive integers. Prove that, as $R$ goes to infinity, $\int _\gamma\frac{z^m}{z^n+1}dz=O(R^{m+1-n})$. And ...
3
votes
2answers
78 views

Asymptotics of coefficients

This is a question that asks the reader for a $strategy$ to solve a particular problem. I cannot solve this problem myself so I am looking around for general methods one might use to confront it with. ...
4
votes
0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
0
votes
0answers
36 views

Proving big-O asymptotics for inverted matrices

Suppose $A,B:\mathbb{C}\rightarrow\mathbb{C}^{n\times n}$ are non-constant and invertible as matrices everywhere, and satisfy that $B$ is an entire holomorphic mapping and $A(z)=B(z)+O(1/z)$ as ...
6
votes
1answer
154 views

How fast does the function $\displaystyle f(x)=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $ grow?

Let $x$ be a positive real number and $f(x):=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $. How fast does this function grow ? In other words can we find a good asymptote for ...
1
vote
0answers
24 views

Can an entire $f$ satisfy $x>k | f(x+yi)=\ln(x+yi+z)+o(1) $?

Let $z$ be a complex number. Let $i$ be the imaginary unit. Let $x,y,k$ be positive real numbers. Consider $$x>k | f(x+yi)=\ln(x+yi+z)+o(1) $$ true for all $x>k,y$ and some $k,z$. Is there ...
7
votes
1answer
161 views

Order and type of an entire function $f$ such that the numbers $f^{(n)}(0)$ are integers.

Let $f$ be an entire function with order $p=1$ and such that the numbers $f^{(n)}(0)$ are integers. Then show that the type $\sigma$ is at least $1$. I appreciate any suggestions.
1
vote
1answer
63 views

Find “singular expansion” of a function

I have the function $(1-z)^{-z}$, analytic except on $\mathbb{R}_{\geq 1}$ Now in the text, it says the "singular expansion" at $z=1$ is $\displaystyle \frac{1}{1-z} + \log(1-z)+O((1-z)^{1/2})$ I'm ...
1
vote
0answers
28 views

Steepest descent?

Here I would like to see the behavior of a function as an integral when its argument (which is a parameter in the integral) goes to zero. If I try to evaluate an integral ...
1
vote
1answer
23 views

Approximating the modulus of a Complex Function near a point.

Let $\Omega$ be a domain in $\mathbb{C}$, and let $z_0 \in \Omega$. Let $f$ be analytic on $\Omega$. Let $z=z_0+re^{i\theta}$ for $r$ small. Assume that $f(z_0) \neq 0$ and $f'(z_0) \neq 0$. I want ...
2
votes
1answer
59 views

I have a Big-O Problem

I want to show: $(1-3z)^{3/2}$ is O(1-3z) as $z\rightarrow 1/3$ where $z \in \mathbb{C}$ I would like to be able to write: $\displaystyle \frac{(1-3z)^{3/2}}{1-3z}=(1-3z)^{1/2}$, and then show that ...
1
vote
0answers
63 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
2
votes
1answer
57 views

Showing that $\frac{1}{(1-\frac{\pi}{\sqrt{6n}})^n} = O(e^{\pi\sqrt{n/6}})$

Here's a small introduction on what I am doing skip to the %%%%%%%%%% if you just want the question. Definition: We say that $f(z) << g(z)$ if $|f(z)| \leq |g(z)| \quad \forall z \in D$. In ...
1
vote
1answer
122 views

Asymptotic Expansion of a Two Variable Function

How is the double asymptotic expansion defined? I can't seem to find it anywhere. Suppose $$f(x)\sim \sum_{n=0}^\infty a_n\phi_n(x)$$ as described in the Wikipedia aritcle. How is then for ...
5
votes
0answers
175 views

An entire function of strict order 2

Here is a problem from Stein and Shakarchi Complex Analysis, can somebody help me to solve it? I guess we can use Phragmen-Lindelof theorem but I don't know the exact way. Suppose $f(z)$ is an entire ...
2
votes
1answer
51 views

Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
1
vote
1answer
107 views

Asymptotic formula for complex gamma function at $+\infty+i \times y$

I am currently looking for the behaviour of the complex gamma function at real infinity: $\lim_{x \to \infty}\Gamma\left(x+i\times y\right)$ and more particularly for asymptotic formulas for the ...
1
vote
1answer
125 views

Trying to find more information about “Darboux's method/theorem” on coefficients of an analytic function

My supervisor briefly showed me a statement of something she called "Darboux's theorem," but I am having trouble finding more information about it on the internet. Here is what I have written down ...
4
votes
0answers
150 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
1
vote
1answer
175 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
1
vote
0answers
48 views

What is the relationship between singularities for complex times and high frequency asymptotics?

As said in a paper I am reading on p 2677 in the text directly above FIG3, this should be a standard result about Fourier transforms of analytic functions. In the paper the authors use these methodes ...
1
vote
1answer
92 views

Can Cauchy theorem be applied to $\log{(z)}e^{ixz}$?

I'm reading about asymptotic analysis on the integral $I(x)=\int_0^1{\ln{t}e^{ixt}}dt$. The book tells me that I can use Cauchy theorem to deform the contour into a rectangular contour:0->iT, ...
1
vote
1answer
42 views

Limits of entire functions

Given an entire function $f \left(x \right)$, which entire function $g \left(x \right)$ is asymptotic to $f \left(x \right)$ as $x \rightarrow \infty$ and asymptotic to $1$ as $x \rightarrow 0$? When ...
2
votes
2answers
147 views

Asymptotic behavior of entire functions

Which entire function $f\left(x\right)$ goes asymptotically to $\dfrac{e^{-x}}{x}$ as $x$ goes to infinity with $x$ positive? That is, $\left(e^{-x}/x \right)/f \left(x \right) \rightarrow 1$.
1
vote
1answer
187 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: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ ...
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 $J_{iu} (ia)$ ...
7
votes
2answers
2k views

Value of Summation of $\log(n)$

Context: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: ...
18
votes
2answers
427 views

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 ...
-1
votes
1answer
180 views

How to find asymptotic entire functions?

I want to know how to find analytic functions $f(z)$ that are asymptotic and analytic on and near the real line of functions of the type $\ln(C +\exp(P(z^2)))$ where $C$ is a complex constant and $P$ ...
2
votes
1answer
80 views

Finding the asymptotic limit of an integral.

I'm having trouble finding the asymptotic of the integral $$ \int^{1}_{0} \ln^\lambda \frac{1}{x} dx$$ as $\lambda \rightarrow + \infty$. Can anyone help? Thank you!
2
votes
1answer
199 views

Coefficient growth in the power series $\sum u_n z^n = e^{1/(1-z)}$?

Let $\sum u_n z^n$ denote the power series of $e^{1/(1-z)}$. As our radius of convergence is $1$, it follows that $u_n$ exhibits sub-exponential growth. On the other hand, $\{u_n\}$ must grow ...
1
vote
1answer
126 views

Singularity analysis of integer power of logarithm ($\log^\beta (1-z)^{-1}$)

This is a theorem of Flajolet and Odlyzko (I think): Let $f(z)$ be a function analytic in a domain $$D = \{z : |z| \leq s_1, |\text{Arg}(z-s)| > \frac{\pi}{2} - \eta \},$$ where $s, s_1 > s,$ ...
8
votes
2answers
637 views

Approximation of Products of Truncated Prime $\zeta$ Functions

The problem arose, while I was looking at products of power prime zeta functions $$ P_x(ks)=\sum_{p\,\in\mathrm{\,primes}\leq x} p^{-ks}, $$ with $k\in \mathbb{N}$ and $s=it$ with real $t$. By using ...