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Questions tagged [exponential-function]

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

5
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4answers
2k views

Prove that $a^x$ is continuous

I'm having trouble with proving the following: Let $a > 0$ be a positive real number. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) := a^x$ is continuous. I'm a ...
1
vote
2answers
45 views

Unintuitive behavior of x^y for x<0

I take advantage of mathematics a lot while programming, but in spite of that I'm very mathematically ignorant. Sorry if I'm asking something stupid. Often I need to exponentially amplify some values....
-1
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0answers
38 views

The matrix exponential [on hold]

Prove that $$\left(e^A\right)^{∗}=e^{A^∗} .$$ I've tried messing around with both sides, I just can't get the two to match up. Any ideas?
0
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1answer
27 views

maximizing entropy -> solving the dual problem analytically

I have to maximize entropy and therefore formulated the dual function, calculated its derivatives, set them equal to zero and now I have to solve the following system analytically: $$ e^{-\mu_1 - \...
0
votes
1answer
32 views

Why is this expression real valued and nonnegative?

$$ g(s)= \begin{cases} 1.4, &\text{for } s=0\\ (0.9)^{|s|},&\text{for } 1\leq |s|\leq 9\\ 0, &\text{else } \end{cases} $$ I want to show that $$ \sum_{s\in\mathbb{Z}}e^{-i\lambda s}g(s)\...
0
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1answer
56 views

Why is the solution to dy/dx = y exponential?

I am trying to understand how e was originally derived as a solution to the below differential: I understand how we reach: For my sake let's call the above solution to the differential f(x) instead ...
0
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1answer
39 views

For $a<b<c$, is it true that $c^n>a^n+b^n$ for large enough $n$?

Given three numbers $a$, $b$, $c$ with $a < b < c $, does the following hold? $$c^n > a^n+b^n \qquad \text{ for large enough }n$$
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0answers
42 views

Solving exponential and logarithm mixed together

Exp(-x)=Cosx. So i shifted the exponent to the right hand side. 0=exp (x)Cosx. Got stuck here. Don't know whether to solve them individually. Like Cosx =0 Exp (x)=0
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8answers
98 views

Show $\sin(\frac{\pi}{3})=\frac{1}{2}\sqrt{3}$

I have to show that $$\sin\left(\frac{\pi}{3}\right)=\frac{1}{2}\sqrt{3}$$ and $$\cos\left(\frac{\pi}{3}\right)=\frac{1}{2}$$ Should I use the exponential function?
79
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25answers
14k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
1
vote
1answer
39 views

Definite integral involving modified bessel function of first kind, exponential of higher order [on hold]

Is there a solution of integral of the following form: $$\int_0^\infty e^{-x^n} J_0(x) dx $$ I have tried searching a lot but cannot find anything. Any kind of help will be appreciated.
0
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1answer
696 views

Is exponential growth and decay faster than polynomial growth and decay?

I know the answer is yes for growth conditions, but I don't see how it's obvious that exponential decay is faster than polynomial decay, say for a polynomial $x^2$.
2
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1answer
24 views

For a triangular array, if $\max_{1\le k\le n}x_{n,k}\to 0$ and $\sum _{k=1} ^nx_{n,k}\to \lambda$ does $\prod_{k=1}^n(1+x_{n,k })\to e^{\lambda}$?

Consider the following: we have a triangular array of nonnegative numbers $$x_{1,1 } \\x_{2,1 } \ x_{2,2 } \\ x_{3,1 } \ x_{3,1 } \ x_{3,3 } \\... $$ The maximum on each row converges to zero: $\...
2
votes
4answers
51 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
0
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4answers
44 views

complex numbers. help in proof that $e^{-it}=1/e^{it}$

I need to show that $e^{-it}=\frac{1}{e^{it}}$. but I don't understand what needs to be proven, it seems trivial to me. If anyone could help me. Is the claim true even if t is not real? Thank you
0
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0answers
33 views

Is the zero truncated Poisson Distribution part of the Exponential Family?

This is the density of a truncated Poisson: $$P(X = x \mid X > 0) = \frac{\lambda ^ x e^{- \lambda} }{x ! \left ( 1 - e^{- \lambda} \right )}$$ To show that it's member of the Exponential ...
3
votes
6answers
80 views

Showing that $\lim_{n \rightarrow \infty}\left(1+\frac{1}{n}+\frac{1}{n^2}\right)^n = e$

$\lim_{n \rightarrow \infty}\left(1+\frac{1}{n}+\frac{1}{n^2}\right)^n = e$ I got this limit while solving another problem. Usually limits that look like this can be easily handled by factoring ...
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votes
1answer
71 views

Solve for $x$ (both 2 values as plotted on graph):

find a way to solve for x: $$ 2^x = x + 5$$ You can easily see one of the values is $3$. If you plot this in a graph, with $y=2^x$ and $y=x+5$, you'll see it has 2 values. If, in a calculator ...
3
votes
4answers
97 views

Can someone show me how to solve $2^x + x = 4$?

I tried every way imaginable to solve the function $2^x + x = 4$, but I can't figure it out. I know it's not an easy one. I spend my free time helping people out with their math questions online, and ...
10
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1answer
3k views

domain of $x^x$

What will be the domain of $f(x)=x^x$? I have asked this to some teachers, they say that the domain is set of all nonnegative real numbers. It is true that there are infinite negative numbers for ...
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votes
1answer
35 views

Solving logarithm with variable in base and exponent for x

$$ log_{1+\frac{0.0231}{x}}{5x} = 10/9$$ What is the solution to this equation and how would I approach the problem? For the purposes of the question, algebraic manipulation with log and exponential ...
2
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3answers
22 views

If an element $a$ in a Lie group commutes with another one, does this element also commutes with a one-parameter subgroup?

Let $G$ be a Lie group and $b=exp(X) \in G $, where $X \in Lie(G)$. Suppose that there exists an element $a \in G$ such that $a$ and $b$ commutes. Now I'm wondering if it is true that $a$ commutes ...
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2answers
18 views

Half life of a radioactive sample

The half of thallium-$201$ is $73$ hrs. How many hours will it take for an amount of thallium-$201$ to decay so that only $5%$ of the original amount remains ? So I decide to use this $$0.05P=P(0....
2
votes
1answer
59 views

Prove that $e^{\frac12i\,\text{gd}(\pi t)}\cos^\frac12(\text{gd}(\pi t))=\frac{1+i}{e^{\pi z}+i}e^{\frac{\pi z}2}$

This question was derived from this post about Gamma function. Juan said: $$ \Gamma\left(\frac12+it\right)=\sqrt{\frac{\pi}{\cosh\pi t}}\exp\left\{i\left(2\vartheta(t)+t\log(2\pi)+\arctan\tanh\...
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1answer
46 views

Definitions of sin and cos using the exponent

I follows these steps: Define $e^z:=\sum_{k=0}^{\infty}\frac{z^k}{k!}$. Show thatthe series is absolutely convergent. Define $\sin(z):=\frac{e^{iz}-e^{-iz}}{2i}$, and $\cos(z):=\frac{e^{iz}+e^{-iz}}{...
1
vote
1answer
67 views

Help proving inequality $\alpha^\alpha\left(1 - \alpha \right)^{1-\alpha} \geq \beta^\alpha \left(1 - \beta\right)^{1-\alpha}$

So I got a request from a friend to try and solve this, but I've stuck. $$\alpha^\alpha\left(1 - \alpha \right)^{1-\alpha} \geq \beta^\alpha \left(1 - \beta\right)^{1-\alpha}$$ for $\alpha, \beta \...
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0answers
15 views

Periodicity of $\exp(jm(2\pi / N)n)$.

Problem: Show that the fundamental period of the discrete-time signal ($n, m, N \in \mathbb{N}$) $$x[n] = e^{jm(2\pi / N)n}$$ is $$N_0 = \frac{N}{\gcd(m, N)}$$ Attempt: Let $N_x \in \mathbb{N}$, $$x[...
0
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1answer
58 views

Differentiation under Integral Sign involving an Exponential

An economics paper that I am reading has this expression: $Q_{t} = \int_{t}^{\infty}\delta e^{-(\bar{r}+\delta)(s-t)-\int_{t}^{s}\pi_{u}du}ds$. After taking derivative with respect to time, the above ...
2
votes
1answer
37 views

Bernstein-Chernoff inequality

I am studying the proof of the following result due to Bernstein and Chernoff: Let $\xi_1,\ldots,\xi_m$ be independent random variables with $|\xi_i|\leq 1$ for $i=1,\ldots,m$. Let $\beta = \sum_{i=1}...
0
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1answer
17 views

Expected Value of Exponential Function of a Single Variable

Given that I know: 1) $E(n)=2$ (or expected value of $n$ is $2$) 2) $n$ must be a positive integer ($n=1,2,3,\ldots$) I am just wondering is there any method to evaluate $E(2^{n-1})$ (the expected ...
0
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1answer
21 views

questions on solving exponential equations

I have these questions on exponentials and I wanted to know if I am correct.. 1. $2^{3-x} = 565$ $3-x\log 2 =\log 565$ $3-x =\frac{\log 565}{\log 2}$ $x = 3-\frac{\log 565}{\log 2}$ 2. $(1+\frac{...
21
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1answer
587 views

Continuous function with infinitely many zeros

Let $f:[0;1]\rightarrow\mathbb{R}$ a continuous function. Let $Z$ denote the set of zeros of $f$. If $Z$ is finite, it's easy to prove that $\lim\limits_{n\rightarrow+\infty}\left|\displaystyle\int_{...
1
vote
1answer
29 views

Find a Formula for an exponential function satisfying $f(1)=5$ and $f(3)=d$. [closed]

Can anyone help me find a formula for an exponential function satisfying $f(1)=5$ and $f(3)=d$ with the answer in terms of the parameter $d$.Is anyone able to help me out with this question? I don't ...
3
votes
3answers
80 views

Combinatorial Proof that the Logarithm of a Product is the Sum of the Logarithms

I've been strongly drawn recently to the matter of the fundamental definition of the exponential function, & how it connects with its properties such as the exponential of a sum being the product ...
3
votes
5answers
437 views

How to calculate $\lim_{x\to\infty}{\left(1+\frac{1}{3x}\right)}^{4x}$?

The limit $$\lim_{x\to\infty}{\displaystyle \left(1+\frac{1}{3x}\right)}^{4x}$$ apparently has a value of $e^{4/3}$. I can't see why this would be the case.
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1answer
13 views

Check if the given function is degenerate

A function $f:\mathbb{R^2}->\mathbb{R}$ is called degenerate on $x_i$ if $f(x_1,x_2)$ remains constant when $x_i$ varies ($i=1,2$). Define $$f(x_1,x_2)=\mid 2^{\pi i / x_1}\mid^{x_2}\text{ for }...
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1answer
1k views

Proof That Exponential Function Is Convex

There are several closed questions on this site with inadequate answers to the question about how to prove that an exponential function, $b^x$, encloses a convex set. I encountered this question in ...
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1answer
19 views

Exponential decay to a line with slope

I have a question about exponential decay. Suppose I have an exponential decay at point $(x,y)$. Instead of decaying to $x-$axis, I want to decay it to a line with slope $(s')$. I also want the decay ...
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0answers
55 views

Prove $f$ is positive and has no roots

I need help to prove the following function is positive for all $0<x<1$, $0<y<1$, $x\neq y$ and an integer $a\ge 2$, $$f(x,y,a)=(y-1) x^a \left((a-2) x^2-(a-1) x (y+1)+a y\right)+(x-1)^2 ...
28
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9answers
14k views

About $\lim \left(1+\frac {x}{n}\right)^n$

I was wondering if it is possible to get a link to a rigorous proof that $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
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0answers
12 views

problem on translation in quantum mechanics [duplicate]

$$\lim_{N\to \infty} \left(1-\frac{x}{N}\right)^N=\exp(-x)$$ Can anyone give a proof for it, I see a similar result in quantum mechanics ( Momentum as a generator of translation ). The textbook does ...
0
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1answer
48 views

Finding the $m^\text{th}$ term of an expression?

How to find the $m^\text{th}$ term for the following expression: $$ \left.\frac{\partial^m}{\partial s^m}e^{a s^2}\right|_{s=0}$$ Is there any analytical approach? I computed first few terms ...
0
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3answers
98 views

Evaluating $\lim_{x\to 0} \left( \frac{e^x - 1 }{x} \right) $ without L'Hopital's rule

I have to solve this limit: $$\lim\limits_{x\to 0} \left( \dfrac{e^x - 1 }{x} \right) $$ I can easily do it by using hospital's rule, numerator becomes $e^0$ and denominator becomes $1$. How ...
2
votes
4answers
160 views

Why $(-2)^{2.5}$ isn't equal to $((-2)^{25})^{1/10}$?

I've tried both calculations on Wolfram Alpha and it returns different results, but I can't get a grasp of why it is like that. From my point of view, both calculations should be the same, as $2.5=25/...
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0answers
38 views

Exponential of Matrix Rewritten as Hyperbolic Functions

Working through a paper and cannot seem to confirm the following equality: $\exp(-\beta\hbar\omega_i\hat{S_{iz}}) = \cosh(\frac{\beta\hbar\omega_i}{2})-2\hat{S_{iz}}\sinh(\frac{\beta\hbar\omega_i}{2})...
1
vote
3answers
131 views

Why the $e$ in $e^z$?

I am studying complex analysis, and I am learning about the complex-valued function $e^z$. But why not use another value other than $e^z$? In other words, why not use $2^z$, or $10^z$, for example? ...
3
votes
2answers
691 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
votes
1answer
90 views

General solution to complex number to complex power in complex form

Given the form: $(a+bi)^{(c+di)}$ Does there exist a generalized solution for the principle branch where: $(e+fi) = (a+bi)^{(c+di)}$ I ask this because addition and multiplication (with ...
0
votes
1answer
32 views

Solve the system of equations with exponential term.

I am trying to solve this system of equations. I know the answer but I am struggling with the working. I need to solve for $m$ and $s^2$ in terms of all the other parameters. The system of equations ...
0
votes
0answers
20 views

transformation of the Gaussian function

consider a gaussian function: $$ \frac{1}{\sqrt{\pi}} \exp \left\{-\frac{1}{2}\left(x^{2}+y^{2}\right)\right\} $$ And I have to prove that transformation of this function: $$ \exp \left\{-\eta\left(...