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Questions tagged [exponential-function]

For question involving exponential functions and questions on exponential growth or decay.

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164 views

A “relative” of the exponential function

It is well known that $$\mathrm{e}=\sum_{n=0}^\infty \frac{1}{n!}.$$ Recently, I was reminded that the volume of an $n$-simplex in $n$-dimensional space, with vertices $v_0,v_1,\dots,v_n$ is $$\left| ...
12
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314 views

The most complex formula for the golden ratio $\varphi$ that I have ever seen. How was it achieved?

I am fascinated by the following formula for the golden ratio $\varphi$: $$\Large\varphi = \frac{\sqrt{5}}{1 + \left(5^{3/4}\left(\frac{\sqrt{5} - 1}{2}\right)^{5/2} - 1\right)^{1/5}} - \frac{1}{e^{2\...
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98 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of $\...
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267 views

On the conjecture that, for every $n$, $\lfloor e^{\frac{p_{n^2}\#}{p_{n^2 + 1}}}\rfloor $ is a square number.

I was playing around with numbers and I discovered that $$\left\lfloor e^{\frac{p_4\#}{p_5}}\right\rfloor=\left\lfloor e^{\frac{210}{11}}\right\rfloor =13981^2,$$ with floor function $\lfloor x \...
9
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530 views

Calculating growth rate of a population of Minecraft chickens

I have a rather strange question (for this Stack Exchange anyway). It felt too mathematical to ask elsewhere. If this is out of place here, please let me know. A chicken in Minecraft lays eggs; ...
6
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175 views

High dimensional integral of exponentials

I am attempting to marginalize a probability density function. But I got stuck on the following integral $$ \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \frac{\exp(\pmb x^T A\pmb z)} {|\exp(A\pmb ...
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176 views

Solving “ugly” equations

$k$, $c_1$ and $c_2$ are unkowns, while others are given. How can I solve the equations? $$ \begin{cases} k\left( {c_1 e^{\pi k} + \frac{c_2}{e^{\pi k}}} \right)^{(1 + 2\theta)/\theta} = - \theta^2\...
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234 views

Application of the Dominated convergence theorem for series

The following exercise is taken from a Calculus I course exam: Let $k\in \mathbb N$. Prove the existence of $$x = \lim_{k\to \infty}\sum_{n=1}^{\infty}\exp\left(-n+\frac{k}{n}e^{-\frac{k}{n}}\right)...
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387 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
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247 views

Closed-form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y+...
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97 views

Can you tell my proof of $\lim_{n\to\infty} (1 + \frac{1}{n})^{n} = \sum\limits_{n=0}^\infty \frac{1}{n!}$ is correct?

I am currently studying analysis with Rudin's PMA myself without looking at proofs for theorems stated in the book. I'm now at the stage where I should prove $e = \lim_{n\to\infty} (1 + \frac{1}{n})^{...
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143 views

A nontrivial solution for $ f(f(x)) = \exp(\exp(x)) $

Consider the equation $$ f(f(x)) = \exp(\exp(x)) $$ Valid for all real $x$, $f(x) \neq \exp(x)$ (not identically equal everywhere), $f(x)$ is analytic and $$ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 +...
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66 views

Say how many solutions an equation has

$16^x+81^x+625^x=60^x+90^x+150^x$ How many solutions does this equation has? I solved this but I am looking for another approach. I used the arithmetic-mean, geometric-mean property. This is how I ...
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64 views

Let $X$ be a left-invariant vector field. If $\exp(X)\in Z(G)$ then $X$ is right-invariant.

Let $G$ be a Lie Group and $X$ a left-invariant vector field. Show that if $\exp(X)$ is in the center of $G$, i.e $\exp(X)\in Z(G)$, then $X$ is right-invariant. I'm trying to use the fact that $X$ ...
5
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44 views

(School) How long does it take to send an e-mail to 25000 people

My math teacher gave us a task: There are 30 students each student sends an e-mail with a message to 2 other people. And everybody who recieved this e-mail sends it again to 2 other people and so ...
5
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113 views

function ${\displaystyle \varphi }$ such that ${\displaystyle \varphi (\varphi (u))=\exp(u)}$

Here it's cited: the existence of the holomorphic function ${\displaystyle \varphi }$ such that ${\displaystyle \varphi (\varphi (u))=\exp(u)}$ had been demonstrated in 1950 by Hellmuth Kneser. ...
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71 views

The fractal dimension of the Kolakoski sequence is $2-1/e$

The Kolakoski sequence, which is defined as the infinite sequence of symbols {1,2} that is its own run-length encoding (Wikipedia), has been suggested to be self-similar$^{1}$. The fractral ...
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89 views

Solutions to $x \exp(x) = a x + b$

Are there any closed form solutions to $$ x \exp(x) = a x + b $$ for real-valued $x$, $a$, and $b$ (we wish to solve for $x$, and $a$ and $b$ are simple constants)? We can assume that everything is ...
5
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102 views

Invertibility of infinite order matrix

how the matrix $[e^{-(x_j-x_k)^2}]$ is invertible where $\{x_j\}$ be any real sequence such that $(x_{j+1}-x_j) >0$ for all $j \in Z$ where $Z$ denotes the set of integers.
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77 views

Why do perfect square values to $ax^2 +ax +1$ form an exponential function?

While playing around with numbers using Python, I found that the set of values of x which fulfilled $$ax^2 + ax +1 = p^2$$ Where p is an integer form an exponential function. For example, $$3x^2 + ...
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118 views

A method to evaluate functional roots of $e^x$

I've an idea to find exact function $f(x)$ such that $f(f(x))=e^x$. But it involves solving complicated systems of non-linear equations, the skills for which I don't have. Here's how I intend to do ...
4
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461 views

Expansion of Gaussian function in a series of exponential functions

Trying to solve a complicated integral, I got interested as a side question in the following problem. Suppose that $f(x)=\exp(-x^2/2)$ is a standard Gaussian function. Is it ever possible to represent ...
4
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90 views

integrating of exponential of exponential function

I need the analytic answer of this integral, if possible. I can calculate it numerically but I was wondering if an analythical solution exist. $$\large \int_0^{+\infty}\ \frac{\text{d}x}{\large e^...
4
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76 views

An asymptotic expansion in the form of nested exponents

Suppose I have a function $f(x)$ with a linear asymptotic behavior for $x\to\infty$: $$f(x)=a_0\,x+b_0+o(1).\tag1$$ Suppose, the last term decays roughly exponentially, so if we take its logarithm, we ...
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94 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! \...
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161 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x =\...
4
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122 views

Integral, possibly of Bessel or Exponential form.

I'm working with a hierarchical statistical model, whereby the output of a log-normal distribution affects the argument of an exponential distribution. I need to marginalize, obtaining the following ...
4
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217 views

Is Every transcendental entire function $f(z) = C + \exp a + \exp b + \exp c $?

Let $z$ be a complex number and let $f(z)$ be any transcendental entire function. Is it true that $f(z) = C + \exp a + \exp b + \exp c $ where $ a,b,c $ are entire functions of $z$ and $C$ is a ...
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603 views

exponentially decaying Fourier transform

Assume you have a real-valued function $f(x)$ defined over whole $\mathbb{R}$ and $f \in L^2(\mathbb{R})\cap L^1(\mathbb{R})$. What additional characteristics should this function have in order that ...
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97 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ e_q(z)=\prod_{k=0}^...
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361 views

Showing the exponential and logarithmic functions are unique in satisfying their properties

The question asks to prove that there exists a unique function defined on $\Bbb R$ and satisfying the following conditions: 1) $f(1) = a$ $(a>0, a \neq 0)$ 2) $f(x_1) \cdot f(x_2) = f(x_1 + x_2)$...
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91 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum $$\...
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57 views

How quickly can one compare exp(m/n) to a given rational?

For positive integers $\hspace{.06 in}m_{\hspace{.02 in}0}\hspace{.02 in},n_0\hspace{.02 in},m_1,n_1\:$, $\;$ how difficult is it to decide whether $$\exp\left(\hspace{-0.03 in}\frac{m_{\hspace{.02 in}...
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1k views

How to prove $x^y$ is jointly continuous?

It's known that real exponentiation $x^y$ is continuous in each variable, but is real exponentiation jointly continuous in both the exponent and the base? I considering the function $(0,\infty)\times\...
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67 views

When is the exponential map between matrices injective?

Consider a complex matrix $A$. When does $e^A=e^B$ imply $A=B$? Is there any general statement that can be made as to when this holds? It is clearly not true in general, a trivial example being when ...
3
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0answers
50 views

Expectation normal distribution

I am trying to solve the following exercise: $$\mathbb E[e^Y\textbf 1_{X<T}]$$ where, X and Y $\sim \mathcal N(0,\sigma)$, $\sigma = 0.5$ $\operatorname{cor}(X,Y) = \rho =-0.98$ $T=-3$ I solved ...
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45 views

Find the preimage of a square in the complex plane.

Define $$f_n(z)=\left(1+\frac{z}{n}\right)^n$$ Find the preimage of the closure of a square centered at $0$ with side length 1 under $f_n$. And explain in a geometric way that why $\lim_{n \to \infty}...
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46 views

write the inverse of: $y= 3(4)^{2x+1} + 1$

write the inverse of: $y= 3(4)^{2x+1} + 1$ This is what I did: $ \begin{align} x=& 3(4)^{2y+1} + 1\\ x-1=& 3(4)^{2y+1} \\ \frac {(x-1)} 3=& (4)^{2y+1}\\ \log((x-1)/3)=& (2y+1)\log(...
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0answers
47 views

Simple form for improper integral $\int_0^{\infty}{x^{1/x-x}dx}$

I've been struggling a while now trying to find an alternate form for $$G=\int_0^{\infty}{x^{1/x-x}dx}$$ in function of known constants, (like $e$,$\pi$,etc.) Having looked at the problem, I don't ...
3
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276 views

Continued fraction of exponential function

There is this $$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$ and there is $$\exp(1)=1+\cfrac{1}{0+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{...
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62 views

How is $a^x$ defined (without using Logarithm) for $a<0$?

let $a\in \mathbb{R},x\in \mathbb{R}\backslash \mathbb{Q}$ . now : how is defined $a^x$ ?(without using Logarithm) i know that : if : $a>1$ then: $$a^x:=\sup\{a^r:r\in \mathbb{Q},r<x\}=\inf\{...
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107 views

Exponential system of equations

Find $x,y \in \mathbb{R}$ such that \begin{align*} 2^x+3^y=7, \\ 2^y+3^x=11 \end{align*} Obviously, $(x,y)=(2,1)$ is a solution, but I can't prove that it's the only one. I tried using contradiction ...
3
votes
0answers
143 views

analytic expression for integral weighted average of exponential function

Based on some numerical calculations I've done it appears that the following is true: $\frac{\int w(x) \exp(-f(x) z) dx}{\int w(x) dx} = a \exp(-b z) + c$ where $w(x)$ is a function with a single ...
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40 views

Math question - specifying the range of ln

The original problem is this: $$\lim_{x\to 0}\frac{1}{2x}\ln \frac{1}{n}\sum_1^n {e^{kx}} = 20$$ Find out the value of $n$, which is a natural number. But I'm having a bit of trouble solving this ...
3
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110 views

Can Pascal's Triangle be expressed as a exponential equation?

Pascal's triangle seems to follow a pattern. 11row-1 outputs PascalTriangle(row) while ...
3
votes
0answers
194 views

Does this equation have no solutions?

The question is this : The source from where I got this question was devoid of any answers to it, so I came here, this is how I proceeded : LHS : $((((({(x)^x})^{2x})^{3x})^{....x^2})^2 = (((((x)^{...
3
votes
0answers
147 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
3
votes
0answers
72 views

What is $\int \frac{e^{a x}}{1+x^2} dx $?

In my answer to another question (here: Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$), I came up with this integral: $\int \frac{e^{a x}}{1+x^2} dx $. I ...
3
votes
0answers
142 views

A property of exponential of operators 2

Let $X$ be a Banach space. The other day I asked if all bounded operators $A:X\to X$ satisfy the following property: (P): All bounded nonzero trajectories $t\mapsto e^{tA}x$ satisfy $$\inf_{t\in \...
3
votes
0answers
226 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...