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

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0
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1answer
30 views

exponential regression fit error problem

I have the following data and im trying to get an exponential fit. Ive tried a variety of different tools for this, which all seem to give quite a large error margin at the top of the curve. Plotting ...
0
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1answer
18 views

finding limits as function approaches zero

Set $n=1+\epsilon$ and let $\epsilon$ tend to zero. $$ \begin{align} c_1 &= \frac1{2\pi} \left[\frac{e^{i\pi (1+\epsilon)}-e^{-i\pi(1+\epsilon)}}{(1+\epsilon)^2-1}\right]\\ &= ...
1
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2answers
47 views

Fourier Transform the following exponential and cosine function: $f(x) = e^{-a^{2}x^{2}}cos(bx)$

I have a previous exam here for my course (Provided by the professor) that requires me to do a Fourier Transform of the following equation. Here is the function: $f(x) = e^{-a^{2}x^{2}}cos{(bx)}$ ...
1
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1answer
35 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
2
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0answers
16 views

Area of intersection of polynomial and exponential functions

I was inspired to explore this by a recent post on the math subreddit, which to my knowledge went nowhere. Consider the families of functions $x^y$ and $y^x$. Given some $y \in \Bbb R$, the roots ...
3
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1answer
62 views

Integral definition of e

I know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...
10
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1answer
159 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
2
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4answers
46 views

Limit of a rational function to the power of x

Ok so I have been trying for days already to find a solution to this all around the web and in math books but to no success. The problem is to evaluate a limit of a function composed by polynomial ...
17
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2answers
478 views

Is this function a constant?

I am a french guest and I hope that my english isn't too bad... So here is my issue : I consider an entire function $f$ which satisfies the following property for all complex number $z\in \mathbb{C}$ ...
1
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3answers
68 views

On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?

I was seeing the proof that $\exp(A)\exp(B)=\exp(A+B)$ on link Show that $ e^{A+B}=e^A e^B$ where uses the hypothesis $AB=BA$? Thanks!
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2answers
76 views

Is it possible to convert $\sigma = \int_0^\infty e^{-x^2} dx$ to an integral problem over $(0,1)$? [on hold]

Is it possible obtain a transformation to convert $\theta=\displaystyle\int_0^\infty e^{-x^2}\, dx$ to an integral problem over $(0,1)$?
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0answers
18 views

How to integrate this function and decide lambda

I want to decide lambda for I also want to integrate the same function. Please help!
1
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1answer
25 views

division by sum of exponentials of large negative numbers

I need to evaluate the following numerically: $$ f = \frac{\exp(a)}{\exp(a)+\exp(b)+\exp(c) + \exp(d)} $$ $a,b,c$ and $d$ are large negative numbers, they are smaller than -1000. Numerically ...
5
votes
9answers
1k views

Proving $\lim \limits_{n\to +\infty } \left(1+\frac{x}{n}\right)^n=\text{e}^x$.

I knew that $e^x=\lim \limits_{n\to+\infty }{\left(1+\frac{x}{n}\right)^n}$. But I've never seen its proof. So I tried to prove it using $\exp(\ln x)=\ln(\exp(x))=x$. Here is what I've tried so far : ...
0
votes
2answers
28 views

What is the 'growth constant'?

I'm looking into the formula of growth, namely $$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'. What is the growth constant and how do I find it? I'm looking at a bug that has on average 1,67 ...
0
votes
1answer
248 views

Is a Distribution Exponential if its Mean equals its Standard Deviation

Can someone clarify if it is safe to declare that a distribution is not exponential if the mean and standard deviation are not equal, for example coefficient of variance, c < 1 and that it is ...
1
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3answers
52 views

Proof that $b^{\log_b(x)} = x$

I understand that the exponential functions are inverses, and would therefore map $x$ when formed as a composition, but I cannot find any formal mathmatical proofs. My thought process is: ...
0
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2answers
22 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
2
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0answers
21 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 ...
0
votes
1answer
22 views

Proof of simple interest formula

Can someone please prove to me that $I = PRT$, where $P$ is the principal, $R$ is the interest rate, and $T$ is the number of years/time. I have seen $I = P(1+TR) = P+PTR$ which does not equal $PRT$, ...
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3answers
98 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
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3answers
26 views

Programming Help - Solving for e(n)

I've been wrestling with this issue for a week and I just need some guidance on the math part of it. If I could just understand the math behind it I could piece together the functions to make it ...
1
vote
1answer
275 views

Equation containing modified bessel functions and exponential function

I'm trying to find a approximation solution for the following equation: ${e^{ - x}}\left[ {{I_o}\left( x \right) + {I_1}\left( x \right)} \right] = C$ where $I_0$ and $I_1$ is the modified Bessel ...
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0answers
16 views

What's the MLE of $\frac1\lambda$ for $f(x)=\frac1\lambda\exp − \frac x\lambda$?

What's the MLE of $\frac{1}{\lambda}$ for $f(x)=\frac{1}{\lambda}\exp \left(− \frac{x}{\lambda}\right)$? Is it equal to the mean of $x$ or the inverse of mean of $x$? Thanks,
-3
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1answer
28 views

Can I have an exponential function such that if x = infinity, y = 100?

I tried the most basic y = 100*constant^(1/x) assuming that 1/x = 0 when x is infinity, but it doesn't seem to work. This gives me a function that starts with a higher value of y and goes down till ...
1
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3answers
67 views

Prove that $e^x \gt 0$ for $x \in \mathbb{R}$ [duplicate]

This is a consequence of the exponential rule, but how do I actually prove it to be true?
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0answers
29 views

Continuous exponential functions

In my book, it makes it appear that any continuous exponential function, such as those regarding money, do not follow the traditional formula of $$\text{growth} = (1+\text{return})^x $$ Rather, it ...
14
votes
3answers
287 views

Is there a simple proof that $e^2$ is irrational using a positional numeral system?

My favorite proof that $e$ is irrational goes something like this. Observe that we can write any real number $r$ as $$ a \,+\, \frac{b_2}{2} \,+\, \frac{b_3}{3!} \,+\, \frac{b_4}{4!} \,+\, ...
1
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1answer
44 views

What's the MLE of lambda for $f(x)= \frac{1}{λ}\exp{\frac{−x}{λ}}$?

What's the MLE of $\lambda$ for $$f(x)= \frac{1}{λ}\exp\left({\frac{−x}{λ}}\right)$$ Values of x are 5,7,9,3,6,8 Is it just the mean of $x$? Thanks.
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0answers
43 views

Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
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0answers
19 views

What is the product of bessel functions of first and second kind when their arguments are same and tends to zero?

As we know, $\lim_{x \to 0} J_m(x)=0$ where $m\geq 1$ and $\lim_{x \to 0} Y_m(x)=\infty$ then what would be $\lim_{x \to 0}J_m(x)Y_m(x)$. Matlab shows the product is finite and $< 1$. What should I ...
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1answer
234 views
0
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2answers
65 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
1
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1answer
26 views

problem about population growth

At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of ...
2
votes
6answers
258 views

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\ldots+a_n<\frac{1}{2}$, prove that $(1+a_1)(1+a_2)\ldots(1+a_n)<2$.

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\cdots+a_n<\frac{1}{2}$, prove that $$(1+a_1)(1+a_2)\cdots(1+a_n)<2$$ I've tried using Holder's inequality (the same result can easily be derived using ...
0
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1answer
32 views

Find $t$ in $N = b \times g^t$.

The problem is the following: Find the value of $t$ in $N = b × g^t$. So for example "$512.000 = 2000 × 2^t$" I'm not really a mathematician so their may be a simple way or it could be hard.
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0answers
18 views

Generic Exponential curve base derivation

Alrighty so I am working on a computer program that forms ADSR envelopes including exponential curves for the attack, decay, and release segments. It uses the following equation for the exponential ...
0
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0answers
66 views

A question about exponential matrices

So here is my question, I would like to prove, If $R,S\in \mathcal M_{n\times n}(\mathbb R)$ are matrices such that, $$e^{t(R+S)}=e^{tR}e^{tS},\;\forall t\in\mathbb R$$ Then, $$RS=SR$$ And here is ...
3
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1answer
476 views

Is this algorithm an example of exponential interpolation?

We have an algorithm I'm trying to get my head around. The original author is gone and away and the whole thing seems to generally work, but I'd like to verify that it's working correctly. (And ...
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2answers
18 views

How do I find the inverse of this exponential function?

$x=-3(3^{-x})+9$ I know the steps up until a certain point. $x=-3(3^{-y})+9$ $x-9=-3(3^{-y})$ $\frac{(x-9)}{-3} = 3^y$ $ln (\frac{x-9}{-3}) = -y * ln 3$ Not sure what to do from here. I know I ...
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votes
2answers
21 views

Limit of n * ln(1+x/n)

How can you compute with the most primitive tools that: $$ \lim_{\stackrel{n \to \infty}{n > -x}}n \:\ln (1+\frac{x}{n})=x $$ Using l'hospital verifies this. However we hadn't proofed ...
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3answers
119 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
1
vote
4answers
175 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...
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2answers
43 views

An exponential/polynomial inequality

Prove that there is at least $1$ real number $a>0$ with the property $$a^x\ge x^a $$ for any $x>0$.
0
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1answer
36 views

Solution of an equation with polynomial and exponential terms

Can anyone solve for $t$ the equation: $$ e^t=\frac{1-nt}{1-t} $$ with $n \in \mathbb N$ (known) and $t>0$. Online solvers give an answer only for specific values of $n$, but I need a general ...
0
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1answer
29 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
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2answers
40 views

why is $\lim_{t \to \infty}t^m e^{-\alpha t} = 0$ for every $m \in \mathbb{N}$ fixed and $\alpha \in \mathbb{C}$ with $Re(\alpha) \gt 0$?

I'm having trouble trying proving this fact: $\lim_{t \to \infty}t^m e^{-\alpha t} = 0$ for every $m \in \mathbb{N}$ fixed and $\alpha \in \mathbb{C}$ with $Re(\alpha) \gt 0$ I tried to use ...
0
votes
1answer
39 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
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1answer
15 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
1
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0answers
29 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???