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Questions tagged [exponential-function]

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

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1answer
21 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.
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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: $\...
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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
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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
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6answers
78 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 ...
3
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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 ...
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0answers
31 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 ...
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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
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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....
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1answer
34 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 ...
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0answers
14 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[...
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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}}{...
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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}...
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1answer
49 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 ...
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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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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{...
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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 ...
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1answer
28 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 ...
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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
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
54 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 ...
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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 ...
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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 ...
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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})...
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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? ...
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4answers
135 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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1answer
580 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_{...
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1answer
31 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 ...
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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(...
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2answers
55 views

Maclaurin expansion of $\exp(x)/(1-\exp(x))^2$

I am trying to expand following function $f(x)=\frac{e^x}{(1-e^x)^2}$ using the Maclaurin series. According to wolframalpha it should be: $\frac{1}{x^2}-\frac{1}{12}+\frac{x^2}{240}-\frac{x^4}{6048}+...
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2answers
99 views

How can I decipher which graph belongs to which equation?

Aside from plotting points, how else can I tell which graph is which?
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1answer
32 views

Find the limit $\lim_{z\to2k\pi i}\frac{z}{e^z-1}$ where $k\in\Bbb{Z}$

If $k=0$, then ${e^z-1\over z}=\sum_{n\ge0}{z^n\over (n+1)!}$. So. $\lim_{z\to0}\frac{z}{e^z-1}=1$. Now if $k\ne0$, then $\lim_{z\to2k\pi i}{e^z-1\over z}=\sum_{n\ge0}{(2k\pi i)^n\over (n+1)!}$, now ...
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3answers
40 views

Why is the domain of x raised to x (0,infinity)? [duplicate]

My math teacher was explaining how to draw graphs of given functions. For $f(x)=x^x$ he put the domain as $(0,\infty)$. Why is this true (if it is)? For $x= -2$, $(-2)^{-2}$ is $1/4$, and so the ...
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0answers
39 views

Integrating an exponential with a power of a sine function

I need some urgent help! I want to compute this integral $$ \int e^{\left(\mu + r\right)t + \frac{r}{\alpha}\text{sin}\left(\alpha t\right)} dt $$ can anyone offer any advice?
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1answer
29 views

Does $C(n)$ grow exponential versus $n$?

Let $λ(m)=n$ be the Carmichael Function of $m$. For each (even) number $n$, there is a largest number $m$ such that $λ(m)=n$. Let $C(n)$ denote the largest integer $m$ such that $λ(m)=n$. For instance,...
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0answers
61 views

Proof for the bound of a complex exponential function

I am carrying out sum proof of a particular calculation and I am stuck at the following step. Let there be two functions of variable $\delta$ given by $$f(\delta) = \left|\sum_{i=1}^N\frac{e^{j\pi i(\...
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0answers
32 views

Exponential touch Logarithm

When I write graph for fun I found that there is some value a that make $y=a^x$ and $y=\log_a (x)$ don't touch each other and there is some value a that make $y=a^x$ and $y=\log_a (x)$ intersect each ...
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0answers
16 views

3-dimensional integral of exponential square and power

Is there any analytical expression for the 3-dimensional integral below? $$ \int_{-\infty}^{\infty} ||\mathbf{x}||^{-2} \exp\left[-(\mathbf{x-a})^\mathbf{T}\mathbf{K^T K}(\mathbf{x-a})\right]\ \mathrm{...
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1answer
49 views

why -1 to the power of e is undefined

I'm currently just playing with the number e, I have tried $$ (-1)^e $$ But my calculator return undefined I am a bit confused by why undefined is returned. What is the logic/theory behind it? Can ...
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1answer
43 views

Exponential decay: continuous vs discrete [closed]

Why do people differentiate between discrete and continuous decay? If a block of whatever is decaying to half it’s amount ever $x$ years, it is perfectly natural, and continuous to say: $m_0(1/2)^{t/...
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1answer
77 views

The limit of $((1+x)^{1/x}-e)/x\;$ as $\;x$ tends to $ 0$

So, I have to solve the limit problem using a derivative. The problem is as follows: $$\lim_{x\to0}\frac{(1+x)^{1/x}-e}{x}$$ I really don't know what to do. Can someone help?
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1answer
21 views

Strategies for approximating $e$ with tangent planes

The exercise says we have to find an expression of a tangent plane to a function that gives us a good approximation of $\alpha =(0.99e^{0.02})^8$. I defined $f(x,y) = ((1-x)e^{2x})^y$, $0 \leq x < ...
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3answers
47 views

Graphically, the limit is 0, but algebraically, why isn't it 1?

$$\lim_{x\to\infty}e^{x-x^2} = 0$$ I'm a bit lost on this one. Graphically, it tends to 0, hence the limit is 0. But, how could I prove it algebraically?
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1answer
47 views

$\lim_n ((\log(n+1)/(n+2))/(\log(n*(n+2)/(n+1)^2))- n$

im kinda new in the field of mathematics, and i would really appreciate some help on the limit $$ \lim_{n\to\infty} \frac{\log\frac{n+1}{n+2}}{\log\frac{n(n+2)}{(n+1)^2}} - n. $$ Thank you all for the ...
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0answers
29 views

An integration involving exponential and trigonometric terms

Assuming $ d, r, f,c> 0 $, I am wondering how one could get the below integral analytically: $$\int_0^{\infty} \exp(-d \: b^2) \frac{-b \: c \:\cos(b\: c) + (1 + b^2 f)\sin(b\: c)}{b + b^3 r^2} \,\...
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1answer
30 views

make an exponential curve tend to a specific value

I am trying to make the following curve to exhibit an exponential decline from a given value to zero. So far, I only managed to make it go from infinite to zero.. Thus I still need to make the curve ...
0
votes
2answers
46 views

elementary inequality involving exp and ln

Suppose $x$ and $y$ are real numbers satisfying $x\geq 0$ and $y\geq 1$. Is the following inequality true? $xy \leq e^x + y \ln (y)$ If so, is there a reference or proof?
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1answer
48 views

How to prove that $f'(x) = e^{-x} g(x)$?

Hello guys can someone help me with that? I tried all the ways but it always leads me far away from the wanted result. Given $g(x)= e^{x}-x-1$ and $f(x)=x-1+(x+2)e^{-x}$, prove that $f'(x) = e^{...
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0answers
53 views

Explicit calculation of the given integral

Can we calculate this integral explicitly? $$\int\frac{dx}{1+e^{-k\left(\sin^2\left(\frac{\Gamma(x)+p}{x+q}\right) - 1\right)}}$$
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votes
1answer
62 views

Inequality involving $e$: $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$

$\forall n,k\in \mathbb{Z^+}$, prove that $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$ where $e$ is the [Euler's number](https://en.wikipedia.org/wiki/E_(mathematical_constant). I tried my ...