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

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9
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0answers
129 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
5
votes
0answers
112 views

How prove this inequality $\frac{1}{n!}\sum\limits_{k=0}^{\infty}\frac{k^n}{k!}\ge e(C\ln{n})^{-n}$

Show that: $$\dfrac{e^n}{(\ln{n})^n}\ge \dfrac{1}{n!}\sum_{k=0}^{\infty}\dfrac{k^n}{k!}\ge e(C\ln{n})^{-n},\ n\ge 2$$ where $C>e$ is constant. My try: ...
4
votes
0answers
41 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
3
votes
0answers
42 views

Why is the base of an exponential function limited to the set of real numbers greater than zero?

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 339). W. H. Freeman. An exponential function is a function of the form $f (x) = b^x$, where $b > 0$ and $b \neq 1$. Why is $b$ ...
3
votes
0answers
59 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 $$ ...
3
votes
0answers
59 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 ...
3
votes
0answers
54 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
3
votes
0answers
48 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 ...
3
votes
0answers
96 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m ...
3
votes
0answers
160 views

Infinite series of matrices almost but not quite matrix exponential

I'm working on a problem that has brought up for me the need to address infinite series of the following form, $$ \sum_{i=k}^\infty \frac{1}{i!}A^{i-k+1} $$ where $A$ is an $n\times n$ matrix. If $k = ...
2
votes
0answers
30 views

Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density ...
2
votes
0answers
27 views

Parameterizing an implicit curve

I have to parameterize this curve: $$F(x,y)=y-x^2+x-e^{-yx^2}=0$$ But I don´t know how to do it. thanks
2
votes
0answers
46 views

Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
2
votes
0answers
38 views

Fractional derivative of exponential function

With the $n$th order derivative ($n$ as a positive integer) of $e^{ax}$ given by $$D^{n}e^{ax}=a^ne^{ax},$$ is the generalized (or fractional) derivative the same? Does it apply for any arbitrary ...
2
votes
0answers
27 views

Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
2
votes
0answers
21 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
2
votes
0answers
27 views

Area of intersection of polynomial and exponential functions

I was inspired to explore this by a recent post on the math subreddit, which to my knowledge went nowhere. Consider the families of functions $x^y$ and $y^x$. Given some $y \in \Bbb R$, the roots ...
2
votes
0answers
41 views

Lie group - exponential Diffeomorphism

Let $G$ be a nilpotent, connected simply connected Lie group and $\mathfrak{g}$ its Lie algebra. It is known that the exponential map $\exp$ is a diffeomorphism. Now let $\mathfrak{g}_0$ be a Lie ...
2
votes
0answers
31 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 + ...
2
votes
0answers
42 views

Exponential equations solving methods?

Do you have an idea or general method to solve the following equation?: $$a^{\alpha x}+b^{\beta x} = c^{\gamma x}+ d^{\delta x}$$ when $a,b,c,d$ aren't zero, and $\alpha, \beta, \gamma, \delta$ are ...
2
votes
0answers
54 views

Application of exponential distributions

The magnitudes of earthquakes in a region of North America can be modeled by an exponential distribution with mean 2.5 (measured on the Richter scale). If 3 earthquakes occur in a given month, what is ...
2
votes
0answers
54 views

Exponential Distribution and Possible Memoryless Property

My attempt : Do you guys think this is right? This might have something do the with the exponential's memoryless property?
2
votes
0answers
237 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 ...
2
votes
0answers
82 views

Inverse function of product of exponential matrices

I am looking for the value of $\mathbf{X}$ in a function of the type \begin{align} (\mathbf{X}-\mathbf{A})e^{\mathbf{X}}e^{-\mathbf{A}} = \mathbf{B} \end{align} where ...
2
votes
0answers
33 views

How to solve the equation $ (x-2)^{\log_{100}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2\log_{10}(x-2)}$?

If $\displaystyle (x-2)^{\log_{10^2}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2.\log_{10}(x-2)}$, then value of $x$ is ... My try:: Let $\log_{10}(x-2) = y\Leftrightarrow (x-2)=10^y$. Then ...
2
votes
0answers
289 views

How to calculate uncertainties from a natural exponent graph?

I conducted an experiment in which position of items were shifted on an object, either on the ends of wings of it, or on the base (I'd rather not get too much into what it's about), and the effect on ...
2
votes
0answers
57 views

differential operator

I've read journal "On the Comparison of Several Mean Values: An Alternative approach" (Welch, 1951). I don't understand this expression: $$E\left(\exp\left[ \sum_t ( w_t - \omega_t ) ...
2
votes
0answers
435 views

exp(ab) decomposition

How can one write $e^{a(x) \cdot b(x)} = c(x) e^{ b(x) }$ with $c(x)$ not implicitly depending on $b(x)$. I do not believe this is generally possible so alternatively one can use an infinite series or ...
1
vote
0answers
32 views

Approximations for finite n in limit-based definition of the exponential function

The exponential function can be defined via: $$ e^x = \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} = \lim_{n \rightarrow \infty} g(x; n) $$ In my problem, I actually have the right ...
1
vote
0answers
76 views

How to approximate large sum of exponential variables

Is there any way to approximate the following sum: $$ \sum_{i_1=1}^N \sum_{i_2=1}^N \cdots \sum_{i_k=1}^N \cdots \sum_{i_N=1}^N \exp(-r_{i_1} - r_{i_{k+1}} - r_{i_{2k+1}} - r_{i_{3k+1}} \cdots - ...
1
vote
0answers
28 views

Can this be solved analytically?

I have a sum of two Gaussian type functions, $g_1(x) = C_1 Exp[-\alpha (X_1-x)^2]$ and $g_2(x) = C_2 Exp[-\beta (X_2-x)^2]$ and have found that the derivative w.r.t. $x$ is $f(x) = 2 C_1 (X_1 - x) ...
1
vote
0answers
71 views

Only one positive solution

When $(a+b)^2=(x-3a)(x-b)e^x$ has only one positive solution, find the relationship between a and b. Here, a and b are constants and satisfy $a>b>0$. Hint: consider the graph of ...
1
vote
0answers
72 views

How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
1
vote
0answers
40 views

What is the product of bessel functions of first and second kind when their arguments are same and tends to zero?

As we know, $\lim_{x \to 0} J_m(x)=0$ where $m\geq 1$ and $\lim_{x \to 0} Y_m(x)=\infty$ then what would be $\lim_{x \to 0}J_m(x)Y_m(x)$. Matlab shows the product is finite and $< 1$. What should I ...
1
vote
0answers
33 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???
1
vote
0answers
41 views

Sigmoid Function Question

Ive been trying for well over a week to try to understand how to use a simple sigmoid or logistic function works. Specifically I'm trying to understand how to build proper polynomia parameters for ...
1
vote
0answers
21 views

The variance of an arrival process with shifted exponential interval

Here we have a arrival process. The inter-arrival time follows a shifted negative exponential distribution as: $f(t)=e^{−\lambda(t−\theta)}$ How to derive the variance of the number of arrivals in ...
1
vote
0answers
54 views

Inverse Fourier transform of two variable function $F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}$

I am trying to find the inverse Fourier transform of: $$ F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}, $$ where $k^2 = k_x^2 +k_y^2 +k_z^2 = k_\rho ^2 +k_z^2 $ is a constant. I am getting confused as to ...
1
vote
0answers
93 views

Maximum Likelihood Estimator of the exponential function parameter based on Order Statistics

Let $X_1, \ldots, X_n$ be a random sample from the exponential distribution $\exp(\lambda)$. Let $M_n=\max\{X_1, \ldots, X_n\}$ with probability density function $$g_{M_n}(x)=n\lambda e^{-\lambda ...
1
vote
0answers
22 views

Decay Function with some additional features

I'm writing a computer function to model data which seems to slope down exponentially until it gets to an optimal point. At that point it steadily grows and then stabilizes. What type of function ...
1
vote
0answers
55 views

Hopelessly Basic Question Regarding Cancelling Exponents

This is actually from my physics class, but it's the algebra that's the problem. It's been driving me insane. So we're doing torque problems. $\Sigma\tau = I \alpha $ We're routinely replacing ...
1
vote
0answers
31 views

How many distinct fixpoints does $\sum_{i=1}^n 2 \sinh(a_i z)$ have?

Let $n$ be a positive integer. Let $z$ be a complex number. Let $a_1,a_2,...,a_n$ be a given sequence of distinct reals larger than $\ln(2)$. How many distinct solutions are there to $\displaystyle ...
1
vote
0answers
60 views

Negative exponential/ exponential power distribution between 0 .0 and 1.0?

Note: I'm not very familiar with distribution and higher level math Heyho, I'm currently looking for a way to generate random values between 0.0 and 1.0 with an exponential power or negative ...
1
vote
0answers
325 views

Exponential integral approximation

I have an equation that contain exponential integral of the form: $$ \begin{equation} E_k\left(\frac{a+b ~x}{c}\right) \end{equation} $$ Where $k\geq 0$ ($k=0,1,2,...$), $a$, $b$, and $c$ are ...
1
vote
0answers
62 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 ...
1
vote
0answers
30 views

Explanations of the Euler's continued fractions to compute exponential

After looking for explanations of the Euler's continued fractions to compute exponential on internet and after reading Euler's explanations about, I still don't understand how Euler find this ...
1
vote
0answers
744 views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
1
vote
0answers
128 views

Stable solution of nonlinear equation

I'm trying to numerically solve equation for $t\in \mathbb{R}$ $$ n \cdot e^{At}x_0= c $$ $A \in \mathbb{R^{3\times 3}},n,x_0 \in \mathbb{R^3}, c \in \mathbb{R}$. I'm looking for smallest positive ...
1
vote
0answers
139 views

$n$-th derivative of exponential integral

I would like to adapt Abramowitz and Stegun 5.1.27 formula to my problem. Equation 5.1.27 states: ...
1
vote
0answers
78 views

Long tail distributions

I have just started to study sub-exponential distributions. I wonder how to prove this statement : Let $X_1,\dots,X_n$ are independent random variables with distributions $F_1,\dots,F_n$ respectively. ...