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Questions tagged [exponential-function]

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

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138 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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88 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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3answers
270 views

Understanding solution to $y' = y$ and exponential distribution

My Understanding: I would derive the exponential random variable as follows: I consider an experiment which consists of a continuum of trials on an interval $[0,t)$. The result of the experiment ...
3
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0answers
65 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(...
3
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0answers
57 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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2answers
96 views

Prove $\sum_{k=0}^{n} \frac{x^k}{k!} < e^x, \forall x > 0$.

Claim: $$\sum_{k=0}^{n} \frac{x^k}{k!} < e^x, \qquad \forall x > 0$$ Proof: Let $f(x)=e^x$. Then the Taylor series gives $$f(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!}$$ $$= \sum_{k=0}^{n} \...
3
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393 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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63 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\{...
3
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1answer
83 views

summation of $e^{(f(x))}$

$$\sum _{x=0}^4e^{-\left[0.06\left(\frac{2x+1}{2}\right)+0.001\left(\frac{2x+1}{2}\right)^2\right]}$$ Hi, I am new, I am not sure if the above code works here, but I want to find $$\sum_{x={0}}^ 4 [ ...
3
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63 views

Local inverse/injectivity of a function involving products of $\exp$

Applying the inverse function theorem, we can quite easily show that the function $$ f \colon \mathbb{R}^2 \rightarrow \mathbb{R}^2\quad (x,y)\mapsto \begin{pmatrix}x \exp(y) \\ y \exp(x) \end{pmatrix}...
3
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155 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
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194 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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134 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
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0answers
433 views

Formula for exponential backoff/rate-limiting

It's been quite some time since I've done actual math and I'm hoping you guys can help me out. I'm consuming a rate-limited API service that provides up to $N$ calls within a a sliding window of $M$ ...
3
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1answer
381 views

When are Logistic Growth Models equivalent to exponential models?

When is the logistic growth model and exponential model equivalent? Logistic Growth Model: $P(1+t)=(P(t)*(-\frac b N )+1+b)*P(t)$ Where $b$ is the birth rate and $N$ is the Carrying Capacity. ...
3
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2answers
137 views

At what point does exponential growth dominate linear growth?

It's well-known that exponential growth eventually overtakes linear (and indeed polynomial) growth (see e.g. here). Given $a,b,c>0$, I would like to find bounds $t^*=t^*(a,b,c)$ such that \begin{...
3
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249 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
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163 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
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1answer
111 views

Find monotonic functions going from $0$ to $+\infty$ for $x \in (-\infty,+\infty)$ (similar to $e^x$)

How can we find functions on $\mathbb{R}$ with exponential-like properties, namely: $f(x)$ is infinitely differentiable; $f(x)$ and all its derivatives are monotonic; $f(x)$ and all its derivatives ...
3
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523 views

Expectation of exponential of lognormal distributed random variable

Why do we have that $$\mathbb{E}[\exp(\exp(Y))]=\infty$$ where $Y$ is Gaussian distributed? This is stated in a book, but what is a proof? Thanks
3
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3answers
117 views

Random Walk And Stochastic-Processes

Assume that $P(X_i = 1) =1/2, P(X_i =-1)= 1/4,\text{ and }P(X_i = 0)=1/4$. Consider the random walk starting at 1 given by $$S_n = 1 + X_1 + X_2 + \cdots + X_n$$ where $X_1,X_2, ...$ are i.i.d. ...
3
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76 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 ...
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168 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
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302 views

Decisions on the order of integration with double integrals (when Deriving PDF via CDF) (Bank Problem)

Consider the following problem: Gandalf, Saruman and Radagast go to a bank together. There are two open counters which Gandalf and Saruman immediately go to get their service. Radagast goes to the ...
3
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1answer
300 views

How to invert a simple exponential growth formula

I think this is simple but my math skills are limited. I have a basic exponential growth formula: $$y=x \cdot (1-p)^n$$ and I have $y$ and $x$ and $n$ values and I need value of $p$. Then when I solve ...
3
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0answers
358 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
3
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0answers
495 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 ...
3
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599 views

Hidden Markov Model, transition probabilities which are modeled with an exponential distribution

I'm looking at implementing an algorithm described in a paper, but I'm having trouble understanding how the transition probabilities for a Hidden Markov Model are defined. In the first sections, I ...
3
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0answers
114 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
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1answer
349 views

Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
3
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0answers
404 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 ...
3
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1answer
655 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 ...
3
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1answer
109 views

A question on Exponential Equation

I came across the following question a few week ago (Exponential equation+derivative): Solve $3^x+28^x=8^x+27^x$. The answer for the above question is 0 and 2. I generalized the question, as ...
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0answers
72 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 power-...
3
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2answers
770 views

The Lie exponential map and commuting elements

By Baker–Campbell–Hausdorff formula, if $[v,w]=0$ for $v,w$ in the Lie algebra $\mathfrak g$ of a Lie group $G$ then $\exp(v)$ and $\exp(w)$ commute in $G$. Does anyone know a reference or a method ...
3
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0answers
109 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 $(10)^{y.\frac{1}{2}...
3
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0answers
142 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 c_it^{n_i}e^{\...
3
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0answers
322 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
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1answer
47 views

Hard integral fraction on exponent and fraction multiplying

How does one integrate this, just came across this in a project and couldn't do it $$\int^1_{-1}\frac{a}{x-a}e^{-b/x}dx$$ I was thinking first substitution but it didn't work, then my next idea was ...
2
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0answers
31 views

Solving this limit without employing l'Hôpital's rule to define the exponential derivative.

Reading this page to understand where does Euler's Theorem come from, the proof gets to a point where it tries to define the derivative of the exponent functions $f(x) = a^x$ as follows: First it ...
2
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0answers
65 views

Help with expressing Partition Numbers as powers of two

Grateful for some help with expressing Partition Numbers as powers of two. I'm not great at maths, but this is as far as I have got: $P(n)=n/2+2^{n-1}-(2^{n-2})(2^x)$ where $x$ is an exponential ...
2
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0answers
73 views

integration of polynomial * exponential of polynomial

I have the following problem: $$\int_{\alpha}^{\beta} (ax^4 + bx^3 + cx^2 + dx + e)\exp{(fx^4 + gx^3 + hx^2 + ix + j)}\ dx$$ $b,c,d,e,g,h,i,j\in\mathbb{R}$. One can assume the final value of the ...
2
votes
1answer
30 views

Simple equation that will recreate this graph shape?

I am working on an audio synthesis project for a guitar, and one of the most challenging parts has been finding an equation so the decays of the partials (modes) can roughly match the behavior of real ...
2
votes
4answers
56 views

Why does $y=x^a$ flicker around the x-axis as you adjust $a$?

I know raising to an even power makes things positive and raising to an odd power preserves sign, but if you graph fractional powers they seem to alternate. Why is this? GIF demonstrating
2
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0answers
32 views

How to limit steepness of inv exp curve?

I am trying to implement a motorised system that will move an object whose velocity has the shape of an inverse exponential curve. In other words it will begin very fast and then slowly come to a halt....
2
votes
0answers
100 views

Solving $f(x/2)^2=f(x)$

Does $\left[f(\frac{x}{2})\right]^2=f(x)$ imply $f(x)=\exp(Ax)$? How can I go about finding all the solutions to this equation?
2
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0answers
26 views

Under which conditions can we algebraically manipulate around an $\approx$ sign?

I noticed the following, and have a more general question that stems from it. The natural logarithm can be defined as follows. $$\ln x := \lim_{\epsilon \rightarrow 0} \frac{x^\epsilon - 1}{\...
2
votes
1answer
51 views

how to handle a “Stiff” algebraic equation numerically?

I have a question of great practical importance for me, but I would like to ask it on a bit more of a theoretical mode, because I feel I lack the basic knowledge on it. I would also like to mention ...