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Questions tagged [exponential-function]

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

1
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1answer
33 views

Showing that the limit of contour integrals is zero

I want to prove that $$\lim\limits_{N \to \infty}{\oint_{C_N}{\frac{z}{exp(z)-1}}\cdot\frac{dz}{z^{2\cdot k+1}}}=0,$$ where $C_N=\{z\in \mathbb C : |z|=2\pi(N+\frac{1}{2}) \}$ and $k\in \mathbb N$...
0
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0answers
46 views

Is there any hope with this integral?

I can't figure out how to take this integral. Looks pretty standard to me, but somehow can't find anything helpful in the literature: $\int_0^{2\pi} d\phi\; (1-2 a \cos \phi + a^2)^{k+ (\kappa/2)} e^{...
-1
votes
0answers
15 views

Upper bound of complex function

i search for an upper bound of $\frac{1}{|exp(z)-1|}$ on circles of the form $C_N = \{z \in \mathbb C | |z|=2\pi(N+1/2) \} $. The denominator is non zero on such circles so i guess this function can ...
7
votes
3answers
237 views

Proving that $f(z;\sigma)=\sum_{k\in\Bbb Z}\frac{1}{\sqrt{2\pi}\, \sigma}{\rm e}^{-\frac{(z-k)^2}{2\sigma^2}}$ converges to $1$ as $\sigma\to\infty$

I have an application where I get following function as a result: $$f(z;\sigma) = \sum_{k \in \mathbb{Z}} \frac{1}{\sqrt{2 \pi} \, \sigma} \textrm{e}^{-\frac{(z - k)^2}{2 {\sigma}^{2}}}$$ It appears ...
0
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1answer
20 views

exponential growth saving function

I'm figuring out some kind of money saving function where the amount, in the end, should be 100.000 value with a time period of 10 years. I thought of an exponential growth function since people ...
3
votes
3answers
19 views

Need help with some basic math/exponential rules I don't understand. How do I get from a to be here?

See image. I've looked up just about every rule I can find and I can't figure out how I am supposed to arrive at that answer. Can someone explain to me what has been done step by step here?
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1answer
54 views

Units of Function [on hold]

What will be the unit of a function $f(V)=e^V$, where $V$ is in Volts? I mean what will be the unit when the function is written in an implicit format and what will be the units when function has a ...
0
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0answers
21 views

Integral involving power, exponential and Confluent Hypergeometric function from 1 to Infinity

I am trying to integrate the following $\int_{1}^\infty e^{-a\,y}\,y^{b-2}\,W_{\kappa,\nu}(c\,y)\,dy$ where $a,b,\nu\,\in\Bbb R$, with $a,b,\nu>0$ and $c,\kappa\in\Bbb C$, with $Re(c)=0$, $Im(c)&...
1
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0answers
16 views

Questioning the commutative property in complex powers [duplicate]

We have:$$e^{2\pi\cdot i(1+\frac{1}{x})}$$ Power properties state that: $a^{b\cdot c}=(a^{b})^{c}=(a^{c})^{b}$. Thus we could re-write the above power as: $$(e^{2\pi\cdot i})^{1+\frac{1}{x}}=1$$ ...
0
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0answers
17 views

$g_t = \frac{e^{f_t}}{\int e^{f_t}}$ is differentiable

Let $\Omega$ be a metric space and $f_0, f_1: \Omega \rightarrow \mathbb{R}$ be any two differentiable functions. For each $t \in [0,1]$, define the function $f_t = tf_1 + (1-t)f_0$. I need to show ...
0
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1answer
26 views

Simplifying equation with exponentials

I am trying to move the $x$ on the right over to the left side of this equation to solve for $x$: $$x = \large e^\frac{{{{ \Large(z / x - 1 - 0.2029)}}}} {{\large {-0.022}}}$$ I am basically trying ...
1
vote
1answer
21 views

Find an equation of the tangent line of the exponential function at the point (0,1)

So I differentiated the expression $$ y=e^{2x}\cos(\pi x) $$ And I got $$ y'=2e^{2x}\cos(\pi x)-\pi e^{2x}\sin(\pi x)$$ But when from the given point $(0,1)$ when I plug in zero I get 2 I looked the ...
0
votes
2answers
43 views

Differentiate the exponential function $f(x)= \frac{x^2e^x}{x^2+e^x}$

$$f(x)= \frac{x^2e^x}{x^2+e^x}$$ Using product rule and quotient rule I computed $$f'(x)=\frac{(x^2+e^x)e^x(x^2 + 2 x ) - x^2e^x(2x+e^x)}{(x^2+e^x)^2}$$ Is my computation correct so far?
-4
votes
1answer
52 views

Determining $\lim_{x\to 1} (\frac{1}{x-1}) e^{-1/(1-x^2)}$ [on hold]

Show that $$\lim_{x\to 1^-} \frac{e^{-1/(1-x^2)}}{x-1}=0.$$
0
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1answer
47 views

Is this simple equivalence on Euler's identity true?

Is the following equivalence true? $$\cos(x\cdot a)+i\cdot \sin(x\cdot a)=e^{ix\cdot a}=(e^{ix})^a=(\cos(x)+i\cdot \sin(x))^{a}$$ why?
0
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0answers
29 views

Solve for parameters of a system of non linear equations that have forms of $0 = e^x + A_1 e^x + A_2 e^{2x} + A_3$

I'm self studying math and came across a problem that I have to solve for parameters of the following equations, $$ 0 = e^{-x} - A_1 - A_2 e^{-x} - A_3e^{-2x} \\ 0 = e^{-2x} - A_1e^{-x} - A_2 - A_3e^{-...
1
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2answers
46 views

Convert $(1+i) ^ {1+i}$ to polar form

Can someone please help me understand the exponent/logarithm relationships to get through this problem? Thank you.
0
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1answer
18 views

How to know if a graph is exponential just by looking at the data values

If I were given the points (1,3) (2,5) (3,7) and assumed this pattern continued forever, I know that it is linear as there is a constant the y value for an increase of one for the x value. If I were ...
0
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0answers
6 views

Growth Factor in an Exponential Word Problem

I was doing an exponential function word problem Question: Assume that the population of the United States is increasing exponentially with time. The 1970 census showed that the population was about ...
0
votes
1answer
31 views

Proof or relation between a Uniform and Exponential

Given $X\sim U(0,1)$, i have to determine the density of $Y=-\frac{1}{\lambda}lnx$. I can't apply the law of transformation of random variables because $g(X)$ is not a monotonic function. So, i ...
0
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0answers
47 views

Can you solve this riddle? [closed]

I have two parents. My parents have two parents, and I have four grandparents. Their parents each have two parents, and I have 8 great-grandparents. And so forth. Thus, it would seem that I ...
1
vote
4answers
59 views

Can I prove there is no real solution except $x=0, x=1$, without using the function $W(x)$?

Can I prove there is no real solution except $x=0, x=1$, without using the function $W(x)$? And is it possible to do it without using calculus? $$2^x=x+1.$$ Here is my attempts: $2^x>0 \...
0
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0answers
35 views

Studying the continuity of the function ln and exponential : [closed]

I didn't find the right way to find if ln is continuous starting from the fact that exponential is continuous "knowing that they are bijective".
100
votes
14answers
18k views

The math behind Warren Buffet's famous rule – never lose money

This is a question about a mathematical concept, but I think I will be able to ask the question better with a little bit of background first. Warren Buffet famously provided 2 rules to investing: ...
1
vote
4answers
67 views

What is the coefficient of $x^{11}$ in $(3x-9)^{19}$?

I am currently studying for finals, and I do not know how to do this problem from my study guide. I have tried to watch a few YouTube videos and I know that I will end up with $3x^{11} \times (-9)^8$, ...
1
vote
2answers
24 views

Discretization of an exponential variable

Given $X=Exp(\lambda)$, i have to define $Y=ceil(X)$ in order to prove the link between exponential and geometric variables. By definition of ceiling $\forall x\in \mathbb{R},\exists n\in \mathbb{N}:...
1
vote
1answer
63 views

Proof of exponential property

Given $f(x) = \exp(ax)$, then the derivative, by definition is given by $\frac{df}{dx} = a\exp(ax)$. Also, it is known that $f(0) = 1$ I want to prove that $\exp((a+b)x)$ is equal to $\exp(ax)\cdot\...
1
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1answer
31 views

Let $T$ be exponential with parameter $λ.$ Let $X$ be discrete defined by $X=k,$ if $k≤T<k+1,$ $k=0,1,2,\dots$. Find the pdf of $X.$

I am aware that this question has been asked already here, however there is no accepted answer to it yet. I have no idea where to start. we know that $$ f(t)= \lambda \times exp (-\lambda t)$$ ...
1
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8answers
125 views

Limit $ \lim_{n \rightarrow \infty}\sqrt{(1+\frac{1}{n}+\frac{1}{n^3})^{5n+1}} $

What's the limit of this? $$ \lim_{n \rightarrow \infty}\sqrt{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)^{5n+1}} $$ Can't find a way to solve it, I tried to use something looks like $(1+1/n)^n$ but ...
0
votes
1answer
29 views

The conditions under which the Taylor polynomial $P_n(x)$ will converge to $f(x)$

Background It is well-known that for the following function: $$f(x) = \begin{cases} e^{\frac{-1}{x^2}} & \text{if} & x \neq 0 \\ 0 & \text{if} & x = 0 \end{cases} $$ the Taylor ...
0
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0answers
50 views

Generated space by the harmonic oscillators $e^{iw}$ in $[\pi , \pi)$ .

Let $\mathbb{T}$ represent the $[-\pi,)$ interval and for each $w\in\mathbb{Z}$ denote the function $$e_w:\mathbb{T}\to\mathbb{C}$$ such that $$\forall x\in\mathbb{T}, \ e_w(x)=e^{2\pi iwx}.$$ It is ...
1
vote
1answer
64 views

How to compute Euler constant $(e^x)$ to its any power.

How to compute $e^x$ ($2.71218...$) to its any power with any shortcut or a method. I want to know a method to calculate in big powers like $e^{50}$ not small powers, For eg-$0.02$ (using Taylor ...
0
votes
0answers
28 views

How can I solve this equation by analytical method? $2^x=x+1$

How can I solve this equation by analytical method only for $x\in\mathbb{R}$ $$2^x=x+1$$ I used the graphical method. I find $x=0, x=1$
3
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0answers
45 views

an inverse of the Artin-Hasse exponential?

In the p-adic world the Artin-Hasse exponential is the sollowing power series: $$ E_p(x)= exp \left( \sum_{n=0}^{\infty}\frac{x^{p^n}}{p^n} \right) $$ where $E_p(x)\in 1+x\mathbb{Z}_{(p)}[[x]]$ with ...
0
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1answer
14 views

transforming from absolute sign to plus minus sign

I have recently encountered the following algebra transformation from an absolute sign to plus minus sign. I am unable to get my head around on how it really works? What is the underlying principle ...
-1
votes
3answers
45 views

Compute series with exponential function [closed]

My aim is to compute $$ \sum_{k=1}^{\infty}\frac{e^{-k}}{k}$$ and to have an explicit formula. How can I do that?
3
votes
1answer
60 views

Asymptotic behavior of roots of an equation involving exponential and logarithm

Prelude This Post is a continuation of this Original Post. The original problem asked is: How many solutions does the following equation have: $$ a^x = \log_a(x) \,,\quad a \in (0,1) \wedge x \in\...
2
votes
0answers
23 views

Formula for the application of a linear differential operator to the product of exponential and polynomial functions

In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients: $$ P\left(\frac{d}{dx}\...
0
votes
1answer
25 views

How to find the sequence given the exponential generating function [closed]

I am trying to learn combinatorics by myself and I am currently stuck on this question. Let $ω$ denote a root(complex ) of the equation $z^2 + z + 1 = 0$. What sequence is $e^x + e^{\omega ...
6
votes
1answer
74 views

Why does calculating $\exp z$ using $\ln z$ via newton-raphson method fail to converge?

I am trying to calculate $\exp z$ using $\ln z$ via Newton-Raphson method $$x_{n+1} = x_n-\frac{f(x_n)}{f^{'}(x_n)}$$and got the formula $$x_{n+1}=x_n-\frac{\ln x_n-z}{\frac{1}{x_n}}$$ where $z = a + ...
2
votes
1answer
33 views

How to solve equations where the power of $x$ is a function of $x$?

I have been trying to find a solution for equations of the type $x^{px-c} = a$. I know how to use Lambert W. function to find solutions for $x^x = a$, but the function of $x$ at the exponent is making ...
3
votes
1answer
100 views

Why is it called “$e$”?

So I'm sure you have all heard of the number $e$ which is approximately $2.71828...$ . But why is it called $e$? It isn't due to Leonhard Euler, since he didn't name the number after himself, the ...
0
votes
0answers
2 views

How can I find the mean value of an exponential profile?

assume a surface density profile \Sigma(y)=\Sigma_p e^{-2y}. How can I find a (fixed) mean value to approximate the mentioned profile? Is it true fo find the value ...
3
votes
1answer
43 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 ...
0
votes
2answers
29 views

How to show that all the zeros of complex function are on unit circle.

Prove that if $be^{a+1}<1$ where $a$ and $b$ are positive and real, then the function $$z^ne^{-a}-be^z$$ has $n$ zeroes in the unit circle. I have no idea how to start this problem?
2
votes
1answer
61 views

Is there a function that satisfies $f(\log \frac{p}{1-p}) = p(1-p)$?

Let $p\in (0,1)$ I am wondering there exists a function $f:\mathbb{R}\to[0,{1\over 2}]$ that satisfies $$ f\left(\log \frac{p}{1-p}\right) = p(1-p) $$ I've tried messing around with exponentials ...
0
votes
1answer
58 views

Is $\lim_{x \to \infty }\sum_{n=1}^\infty \frac{n!}{e^{n^x}}=e^{-1}$? [closed]

Is it the case that: $$ \lim_{x \to \infty }\sum_{n=1}^\infty \frac{n!}{e^{n^x}}=e^{-1}$$ And why?
0
votes
1answer
21 views

Whether a Pathological Set Might Exist that could Foil a Theorem about the Exponential Function

There has been some discussion recently about the theorem $$\lim_{n \rightarrow \infty} \prod_{i = 1}^n \left( 1 + \frac{x_i}{n} \right) = \exp \left( \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i =...
0
votes
0answers
58 views

Approximate a solution for $\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx 0$

Is it possible to approximate (or even find) a solution for the following equation: $$\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx0,$$ where $x\ge 0$ and integer, and the ...
0
votes
0answers
30 views

The diophantine equation $a! = b! \times c!$ [duplicate]

One infinite family of solutions to the equation $a! = b! \times c!$ has $a = s!$ so we have $(s!)! = (s!-1)! \times s!$ but I'm hard pressed to find manually another type of solution apart from $10! =...