Questions tagged [exponential-function]

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

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1answer
33 views

Why this equation of absolute value of exponential function,complex number holds or even is this equation correct?

I've written the below 2 claims. Claim1. $$ \left| \prod_{i=1}^{n} a_{i} \right|=\prod_{i=1}^{n} \left|a_{i} \right|$$ $$ \left\{ a \right\} :=\text{sequence which holds} ~a_{i} \in\mathbb{R} $$ $$...
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0answers
52 views

Proving $e^x < (2x + 1)\frac{x^x}{x!}$ to conclude $\lim_{x\to\infty}\frac{\sqrt[x]{x!}}{x}=\frac{1}{e}$

The question is to prove an inequality: $$e^x < (2x + 1)\frac{x^x}{x!}$$ Conclude from it that: $$\lim_{x\to\infty}\frac{\sqrt[x]{x!}}{x}=\frac{1}{e}$$ I've tried to play around derivative of left ...
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0answers
18 views

How can the following expressions be simplified using only the rules of logarithm (and natural logarithm) and exponents?

How can the two expressions above be simplified to A in both cases using just the basic exponential, or logarithmic rules? Thanks.
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1answer
56 views

Find this Question value using range method.

If the range of this function $f(x)$=$\frac{x-A}{x^2-5x+6}$ = R. Then, solve for A . How can I solve this question.
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20 views

Find residue of $f(z) = e^{\frac{1}{1+√z}}dz$ in the point $z = 1$ for all branches of $e^{\frac{1}{1+√z}}$.

As is said in the title I would like to evaluate the residue of $f(z) = e^{\frac{1}{1+√z}}dz$ in the point $z = 1$ for all branches of $e^{\frac{1}{1+√z}}$. This is new for me, but after reading some ...
2
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1answer
28 views

Total Variation on compact interval

Let $f(x)=e^{-x} \cos x$. Show that $f$ is of bounded variation on any compact interval. For each $k$ in $\mathbb{N}$, compute $V(f ; 0,2 k \pi)$ and show that $\{ V(f; 0, 2 k \pi): k$ in $\mathbb{N} \...
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1answer
46 views

Given a set $S$ with $n$ positive reals prove that for at most $n-2$ different integers $k$, $\exists a,b,c\in S: a+b+c=3^{k} $

I came across the claim that for any $n$-membered set of positive real numbers you can choose different $a,b,c $ from it such that $a+b+c=3^{k}$ only for at most $n-2$ choices of $k$. Sadly I could ...
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3answers
137 views

How to add (or subtract) two exponential components with same base?

I am to solve for $x$ using logs: $e^{2x}-e^x-110=0$ During my steps, I'm unsure how to combine $e^{2x}-e^x$ into one. My attempt: $$e^{2x}-e^x-110=0$$ $$e^{2x}-e^x=110$$ Here's where I get confused: $...
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1answer
29 views

How can you find the locus of a point that is the same distance from a exponential function?

Okay so this is a question that I came up with few weeks ago, and I still cannot find a answer to this problem. Abstract idea: If you draw a line that is perpendicular to the tangent line of a point $(...
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0answers
43 views

find the integral (X^r((exp(x)^2/(b^2)(exp(-((exp(x)^2/(2b^2) [closed]

Please can you help me find $$\int X^r\frac{e^{x^2}}{b^2}e^{\frac{e^{x^2}}{2b^2}}dx$$
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2answers
53 views

Proof that $\exp(x) - 1 - x \leq x^2, x \in [0, 1]$ [duplicate]

Is there a simple way to show that $$\exp(x)-1-x \leq x^2,~x~\in~[0, 1]$$
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2answers
64 views

Differentiate $x^{a^x}$ without logarithmic differentiation?

Problem. Compute $\frac{d}{dx} x^{a^x}$. Method 1: Logarithmic Differentiation (Correct): \begin{align*} y&= x^{a^x} \\ \ln(y)&=a^x \cdot \ln(x) \\ \frac{1}{y} \frac{dy}{dx} &=a^x \frac{1}{...
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2answers
43 views

How to show a sequence of functions is increasing

How does one show that for all $x\geq 0$ that the following sequence of functions is increasing where $f_n$ is defined by $$f_n(x)= x\left(1+\frac{x^2}{n}\right)^n$$ using the fact that for all $y\geq ...
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1answer
21 views

Find the joint distribution and covariance with exponential density

Let $X_1, \dots , X_n$ be independently distributed with exponential density $$ f(x) = (2θ)^{−1}e^{−x/2θ}, x \geq 0 $$ and let the ordered $X$’s be denoted by $X_{(1)} \leq X_{(2)} \leq \cdots \leq ...
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2answers
112 views

Find the integral $\int_0^\infty \mu^x / \Gamma(x + 1) dx$ [duplicate]

Basically, I'm looking for advice on how I could find the value of $$\int_0^\infty \frac{\mu^x}{\Gamma(x + 1)}dx $$ where $\mu > 0$ is an arbitrary positive constant. Based on the infinite series, ...
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1answer
51 views

How to solve this system of equations with complex exponential in MATLAB?

Given $\Delta t$, I want to solve in MATLAB the following system of equations with complex exponentials in unknowns $\alpha_k$ and $\nu_k$ $$\lambda_k = \exp \left((-\alpha_k + j2\pi\nu_k)\Delta t)\...
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1answer
40 views

Error propagation on a 5 order polynomial

I have a 5 order polynomial equation which gives log of a chi factor required to convert equivalent widths to normalised H alpha luminosity for M dwarfs, (Reiners et al. 2008). I would like to get the ...
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0answers
32 views

Could someone explain the the steps mention in the description? [closed]

Why$$ \ln|y| = \ln|1+x| + c $$ can be written as $$ y = e^{\ln|1+x| + c} $$ Also, why $$ e^{\ln|1+x|} = 1+x$$
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2answers
33 views

How to simplify sum of exponential functions

I have been having some issues with simplifying the following equation: $p\Big[ \frac{e^{ay}+e^{a(10+y)}}{e^{a(10-y)}+e^{a(20-y)}} \Big]=1-p$ where $y$ is the variable, $a$ is a parameter and $p$ is a ...
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2answers
61 views

Problem manually evaluating definite integral

I seem to be quite stuck when trying to normalize a probability density given as $$p(x|\omega_i)\propto e^{-\frac{|x-a_i|}{b_i}}$$ with $a_i\in R$ and $b_i\in R^+$. Although I was able to "...
2
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1answer
36 views

Where does variable n gone of trignometric function and sum of infinite series?

I want to compute the Z-transform of the below signal function. $$x[n]=\cos(\omega n)u[n]$$ $$Z[x[n]]=\sum_{n=-\infty}^{+\infty}\cos(\omega n )u[n] \cdot z^{-n}$$ $$=\sum_{n=0}^{+\infty}\cos(\omega n )...
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4answers
68 views

Does $x^{0.5}$ have a negative solution?

A typical graph of $f(x) = n^x$ shows only positive solutions: But it seems like, for some values of $x$, there are negative solutions. For example, $2^{1/2}$ is $\sqrt{2}$, which has a negative ...
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1answer
17 views

The expectation including an indicator function

I stumbled upon this homework question and I simply cannot wrap my head around it: As far as I understood it, I would start off with the following: But I get completely stuck from here on out. ...
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0answers
17 views

One-sided decaying squared error

I'm trying to build a loss function to be used with a machine learning regression model that results in something that's almost a hybrid of classification and regression. The idea is to decrease the ...
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1answer
23 views

Multiple select question on finding square root of $-1$

This is a multiple select question. The values of $i$ where $i$ is square root is $-1$ is $a$ - $e^{\frac{\pi}{2}}$ $b$ - $e^{\frac{-\pi}{2}}$ $c$ - $e^{\frac{i\pi}{2}}$ $d$ - $e^{\frac{-i\pi}{2}}$ ...
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0answers
19 views

Basic Ordinary Differential Equation

Is this first linear ODE? I'm quite confused because the y is in the position of exponential of e.
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2answers
141 views

Elementary integral of exponential

I am keen to find the simplest expression for the following integral: $$ \int_{-1}^{1}(x^{2}-1)^{n}e^{i\sigma x}dx $$ where $n$ is a non-negative integer. The best I could come up with involves ...
4
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1answer
41 views

Let $X \in [a,b]$ satisfy $\mathbb{E}[\exp(-X)]=1$. Prove $\mathbb{E}[X] \le \frac18(b-a)^2$.

I would like to prove the following conjecture. Conjecture. Let $X$ be a measurable random variable supported on the interval $[a,b]$. If $\mathbb{E}[\exp(-X)]=1$, then $\mathbb{E}[X] \le \frac18 (b-...
7
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2answers
294 views

Squaring a complex exponential that represents a real number

Often, complex exponential functions are used to represent trigonometric functions, since $$ e^{i\theta} \equiv \cos\theta + i\sin\theta . $$ Thus, if for example I want to express the quantity $\cos ...
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0answers
37 views

Intuition for exponential derivatives

We have $d/dx(b^x) = b^x\log(b)$, with $b>0$. If $b < e$, then $b^x > b^x\log(b)$. If $b = e$, then $b^x = b^x\log(b)$. If $b > e$, then $b^x < b^x\log(b)$. In fact, if you take the $...
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2answers
44 views

Probability that one value from one exponential is smaller than the value from another

I have two exponential (Y1 time between incoming data, Y2 representing the time of cooldown) with $λ_1 = 0.05$ and $λ_2 = 0.5$. I'm trying to find the percentage of data that will be ignored because ...
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1answer
50 views

If for two complex-valued continuous functions $g_i$ it holds $g_i(0)=1$ and $g_1^n=g_2^n$, can we infer that $g_1=g_2$?

Remember that if $a\in\mathbb C\setminus\{0\}$, then $\phi\in\mathbb R$ is called argument of $a$ if $$a=|a|e^{{\rm i}\phi}\tag1.$$ Let $$\operatorname{Arg}a:=\left\{\phi\in\mathbb R:\phi\text{ is an ...
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1answer
53 views

Expansion of $e^x$ - correct form

I have come across in a textbook to an expansion of e to the x in the following form: $$ 1+ \frac1x + \frac1{x^2} + \frac1{x^3} + \ldots $$ Is the above correct or is it a typo? I am familiar with ...
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1answer
19 views

differential equations, exponential population growth

If p is population and t is time. Does that mean that when you do dp/dt = 0 you can find the maximum and minimum population
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2answers
57 views

Do $a^{1/n}$ and $n$th root of a mean the same thing?

I encountered a question in which I had to find $x$ and the question had $3^{1/x}$ and I got $x=1/2$ and $1/4$ as its solution. But my textbook said that "$x$th root of $3$" is valid for $x\...
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1answer
47 views

Where does this equation come from and the reason behind it to a model exponential function?

To model data based on the exponential equation $$ y=Ae^{Bx} $$ On the website https://mathworld.wolfram.com/LeastSquaresFittingExponential.html In equation (5) $$ \sum_{i=1}^{n}{y_i\left(\ln{\left(...
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1answer
73 views

Why is there more weight on smaller y values in transformed linear regression as compared to least squares regression for exponential models?

I was doing regression for an exponential model that follows the general formula $$ y=Ae^{Bx} $$ And found that using linear regression for the linearized data would model larger values of y poorly ...
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1answer
47 views

2 to the power of a matrix

I was watching 3b1b https://www.youtube.com/watch?v=O85OWBJ2ayo and I was wondering does there exist a definition for matrix exponentiation? Or is this concept of "matrix exponentiation" ...
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2answers
68 views

Proof that $\frac{2}{e}<\int_{-1}^1e^{-x^2}\, dx<2$

I have to prove that $\frac{2}{e}<\int_{-1}^1e^{-x^2}\, dx<2$. To prove this I have thought to use two methods:\ In $[-1,1]$ I have $\frac{1}{e}=e^{-1}\leq e^{-x^2}\leq e^{0}=1\implies \frac{2}{...
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1answer
37 views

Exponents, versus Exponentials, versus Exponential Functions. Splitting hairs on definitions.

I stumbled upon some tutorial videos on various topic in math by Lorenzo Stadun. I'm sharing the link because a cursory glance indicates that the content is pretty good. So the first thing I ever do ...
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3answers
42 views

Find an exponential equation that passes through the points $(2, 2.25)$ and $(5,60.75)$

I am asked to find an exponential equation that passes through $(2, 2.25)$ and $(5, 60.75)$. My textbook says the solution it's $y=0.25(3)^x$ whereas I got $y=0.028(9)^x$. Here is my working: Express ...
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1answer
61 views

What is the motivation for defining $f(x)=e^x$ as “ the unique function EXP differentiable on R such that EXP'$(x)=$EXP$(x)$ and EXP$(0)=1$ ”?

In some french books, I can find the above definition of the exponential function with base $e$. This defnition does not seem to be so common in books written in English. What is the point of defining ...
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0answers
16 views

Coefficients of taylor series of an analytic function about different centers.

To be concrete, I would like to limit my question to the exponential function and real numbers. The power series centered at any point converges to the exponential because its radius of convergence is ...
0
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1answer
30 views

Show that $e^{\alpha t} \ast (\frac{t^k}{k!}e^{\alpha t}) = \frac{t^{k+1}}{(k + 1)!}e^{\alpha t}$ for the convolution

As it says in the title, I would like to show that $e^{\alpha t} \ast (\frac{t^k}{k!}e^{\alpha t}) = \frac{t^{k+1}}{(k + 1)!}e^{\alpha t}$ for the convolution. However the only thing I know for the ...
0
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3answers
55 views

Solving $e^{2x} + 2e^x = 8$

it's been a while since I've done Analysis and I'm currently trying to solve for $x$. I'll just show what I did to solve to solve : $e^{2x} + 2e^x = 8$ $\leftrightarrow e^x \cdot e^x + 2e^x -8 = 0$ $\...
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0answers
21 views

Proving $\frac{b_{n}}{b_{n+1}}=1+\frac{1}{n}+\frac{p}{n \log n}+o\left(\frac{1}{n \log n}\right)$

If we define $b_{n}:=\dfrac{1}{n(\log n)^{p}},\:n \geq 2, p>0$, prove that $$\frac{b_{n}}{b_{n+1}}=1+\frac{1}{n}+\frac{p}{n \log n}+o\left(\frac{1}{n \log n}\right)$$ What I tried: We have $$\frac{...
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2answers
23 views

Derivation of exponential function (minor help)

This is basic maths but I'm just not seeing the connection here. How is 2-3 equation derived? $$e^{-\gamma(c_{t-1}+K_{t-1}+\nu_{t})}=E_te^{-\gamma(c_{t-1}+K_{t-1}+\nu_{t}+K_t+\nu_{t+1})}$$ $$e^{\gamma ...
0
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2answers
98 views

How to isolate $x$ in $a^x + b^x = c$? (For use in medical statistics)

I've broken a somewhat complex calculation down to the following mathematical equation to be solved: $a^x + b^x = c$ How do I find $x$ when $a, b$ and $c$ are given as parameters? I.e., if $a=3$, $b=4$...
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0answers
28 views

Simplify the expression $\log_2({x_1}^{y_1} + {x_2}^{y_2})$ for large values of $x,y$

Following this thread: Simplify the expression $\log(2^x+2^y)$ for large values of $x,y$ Is there a way to simplify (or get a good approximation) for the $\log_2$ of a sum of powers with different ...
1
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1answer
69 views

$1 - \lim_{x \to \infty} \left( 1 - \frac{2}{x^2} \left( 1 - \exp \left( -\frac{x^2}{2} \right) \right) p \right)^{kx^2}$

I'm currently working on my master thesis and I need to solve this limit. I forgot almost everything about limits since last time a I saw them was basically in high school. Anyway, I solved this limit ...

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