1
vote
1answer
82 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
39 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
52 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
26 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
38 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
27 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
32 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
76 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
86 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
34 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
136 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
149 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
41 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
23 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
22 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
53 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
54 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
82 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
163 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
103 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
95 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
141 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
169 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
45 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
87 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
39 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
137 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
157 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
65 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 $J_{iu} (ia)$ ...
5
votes
2answers
2k views

Value of Summation of log(i)

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
400 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
166 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
75 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
184 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
625 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 ...
4
votes
0answers
87 views

Asymptotic Analysis of Complex Integrals

Let $\gamma\subset B(0,2)\subset \mathbb{C}$ a smooth Jordan curve envolving the origin and $\phi:\mathbb{C}\setminus (-\infty,0)\to \mathbb{C}$ an analytic function having finite boundary values, ...
20
votes
2answers
727 views

Proof $\sum\limits_{k=1}^n \binom{n}{k}(-1)^k \log k = \log \log n + \gamma +\frac{\gamma}{\log n} +O\left(\frac1{\log^2 n}\right)$

More precisely, $$\sum_{k=1}^n \binom{n}{k}(-1)^k \log k = \log \log n + \gamma +\frac{\gamma}{\log n} -\frac{\pi^2 + 6 \gamma^2}{12 \log^2 n} +O\left(\frac1{\log ^3 n}\right).$$ This is Theorem 4 ...
4
votes
1answer
614 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
2answers
458 views

On the growth order of an entire function $\sum \frac{z^n}{(n!)^a}$

Here $a$ is a real positive number. The result is that $f(z)=\sum_{n=1}^{+\infty} \frac{z^n}{(n!)^a}$ has a growth order $1/a$ (i.e. $\exists A,B\in \mathbb{R}$ such that $|f(z)|\leq ...
3
votes
0answers
182 views

proving the saddle point method in a specific case

$1:$ the problem Let $f : U \to \Bbb{C}$ be analytic on some open set $U$ that includes the closed unit ball. Define a path $\gamma$ by: $\gamma(t) = e^{i t}$, -$\pi < t \leq \pi$ I want to ...