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

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1
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1answer
110 views

Can I solve this with a Lambert Function?

New to W-Functions and do not understand it properly. How do I solve this equation? I know about numerical solutions (or graph solution), but I'm interested in pure algebraic solution if it exists. ...
1
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3answers
36 views

How to solve for f?

The question asks to solve for the variable: $$2=6(3^{4f-2})$$ I am not quite sure how to solve for $f$ because the bases on either side cannot be made equal. Here is an example of a similar ...
3
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1answer
32 views

Justification of exponent laws

How is the following operation justified? Specifically, what happens to the $4^n$ in the numerator. I tried looking online but could not determine how this was justified. Thanks. ...
4
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3answers
106 views

Proving positivity of the exponential function

Question. Without using the semigroup property ($\mathrm{e}^{x}\mathrm{e}^{y}=\mathrm{e}^{x+y}$), how can we show that $\mathrm{e}^{x}>0$ for all $x\in\mathbb{R}$ only by using the series ...
5
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4answers
96 views

Inequality $|e^z -1| \le 2 |z|$ for complex $z$ with $|z|\le1$

I am trying to prove that for $z \in \mathbb C, |z|\le 1$: $$|e^z -1| \le 2 |z|$$ But I'm stuck and I need help. I showed that for all $z$: $|e^z -1| \le |z|e^{|z|}$ but it does not seem useful. ...
0
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0answers
28 views

Exponential Decay of Advertising Effect

For the last couple of days i've tried to come up with a formula for an exponential effect of advertising on car sales. (The one where as sales increase it takes more and more spending to achieve ...
0
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1answer
40 views

Solution for $\mathrm{e}^{aX}=b I$ besides $X \propto I$?

Is there any solution for $e^{aX}=b I$, where $a$ and $b$ are numbers, $X$ is a matrix and $I$ is the identity matrix. One solution is of course $X=cI$, with some number $c$, because $$ ...
0
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1answer
51 views

Can I simplify $ \ln(A/B)+C$ any more?

This should be a rather simple problem however I am having difficulty getting this simplified. If I need to simplify the expression $$ \ln(A/B)+C$$ My first step is $$ A/B + e^c$$ However MATLAB and ...
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1answer
61 views

solve initial value problem using exponential matrix

$x'' = 2 x' +6y +3$ $y' = -x' -2y$ subject the the initial condition $x(0) = 0; x'(0) = 0; y(0) = 1$ The first part of the question is about finding $e^{At}$ of this matrix $A = \begin{bmatrix} ...
1
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1answer
38 views

Prove that $e^{\lambda A}Be^{-\lambda A}=B$

Prove that $$e^{\lambda A}Be^{-\lambda A}=B$$ if $[A,B]=0$. $A$ and $B$ are operators and $\lambda$ is a complex number. Can anyone explain how I should go about this question? How do I calculate ...
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0answers
19 views

Exponential curve with hyperbolic sine behavior on the tails

I have a dataset that I've fitted an exponential curve to that looks like a great fit at midrange values of the domain but is not such a good fit at low and high end domain values. Instead, at these ...
3
votes
1answer
34 views

Rational Exponents

I'm just checking to see if have the correct answer because my teacher didn't give us an answer key and i like to know that I have done one question properly before doing the rest. Evaluate. ...
1
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1answer
54 views

How to remodel sigmoid function so as to move stretch/enlarge it?

I have a question similar to this. I want the sigmoid to have asymptotes to $+1$ and $0$ in specific points $\frac{1}{A}$ and $-\frac{1}{A}$, as in the Figure (where $\frac{1}{A}=2$ and ...
1
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1answer
54 views

Closed form solution of $\int \exp(-a (b-x)^{3/2}-cx)\text dx$

Does following integral have a closed form solution (a, b, and c are constants) $$ \int \exp(-a (b-x)^{3/2}-cx)\text dx $$ If not possible, what about a function with close behavior. $$ \int \exp(-a ...
1
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1answer
63 views

Intuition for the exponential of a matrix

I'm trying to understand an algorithm that tries to map points from a lie group to its lie algebra using the exponential map. The background is the representation of 3d coordinate transformations as a ...
0
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0answers
45 views

Each year for $10$ years, the population of a city increased by $5\%$ of its value in the previous year

Each year for $10$ years, the population of a city increased by $5%$ of its value in the previous year. If the initial population is $200,000$, what was the population after $10$ years? My solution: ...
0
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1answer
48 views

Proof that the series expansion for exp(1) is a Cauchy sequence

Consider the series expansion for the exponential function at x = 1: $$a_n := \sum\limits_{i = 0}^n{\frac{1}{i!}}$$ I want to prove that this is a Cauchy sequence, using the remainder formula for ...
0
votes
1answer
35 views

What function can I use to fit through the points (0,1), (13,0) and (p,q)?

I'm trying to find a function that will go through $(0,1)$, $(13,0)$, with another point in between, say at $(6,p)$ where $0\le p\le 1$, that I can vary up and down to make the curve similar to a ...
0
votes
2answers
42 views

exponential distribution with probability about texts

It is 9:00 p.m. The time until Joe receives his next text message has an exponential distribution with mean 5 minutes. A text has not arrived for 5 minutes. Find the probability that none will arrive ...
0
votes
1answer
31 views

simultaneous equations-exponential and linear

I am trying to find a general formula for x and y given that $y=mx+c$ and $y=Ae^{kx}$, with m, c, A and k as constants (and e is Euler's number). essencially, find the point(s) where an exponential ...
2
votes
1answer
58 views

How to find $s(\exp(d(x)))$ ~ $ x + 2 $?

Let $x$ be a positive real. I want to find a pair of analytic functions $s(x),d(x)$ such that $s(d(x)) = x$ and $ s(\exp(d(x)))$ ~ $ x + 2 $ More presicely I Also want : $$ \lim_{x \to \infty} ...
1
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1answer
44 views

Integral involving $\operatorname{sinc}$ and exponential

Is there a closed form for the following integral: $$\int_{0}^a\exp\left[\frac{i\pi x^2}{b}\right]\operatorname{sinc}\left(\frac{\pi ax}{b}\right)dx$$ where $i=\sqrt{-1}$ and ...
2
votes
2answers
196 views

What is the equation for figuring out the change in pitch from changes in tempo?

I have various audio loops that need to change pitch when I change the tempo. The relationship is not linear, so it must be exponential, but I don't know what the equation would be. There is an ...
0
votes
1answer
38 views

find the inverse of $\frac{1-e^t}{1+e^t}$

Hi I am trying to prove that the inverse of $f(t) = \frac{1-e^t}{1+e^t}$ is $F^{-1}(t) = \ln\left(\frac{1-t}{1+t} \right )$ But I don't quite know where to start? Do I just sub ...
0
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2answers
16 views

Gamma Family Density Function of Y

I have tried to think of how to prove this but at a loss. I keep getting it squared so not sure what I'm doing wrong.
0
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1answer
32 views

Prove $x^n$ + x < ($x^n$)x using PMI

I need to prove $x^n$ + x < $x^n\cdot$ x, n $\in$ N, x $\in$ R>2 using induction. I started by $x^n$ + x + (x^(n+1)+x) < ($x^n\cdot$ x) + (x^(n+1)+x) I simplified to this: < 2x^(n+1) ...
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3answers
81 views

Compute $e^A$ where $A$ is a given $2\times 2$ matrix

Compute $e^A$ where $A=\begin{pmatrix} 1 &0\\ 5 & 1\end{pmatrix}$ definition Let $A$ be an $n\times n$ matrix. Then for $t\in \mathbb R$, $$e^{At}=\sum_{k=0}^\infty ...
0
votes
1answer
47 views

Least squares fit to a an exponential equation with one unknown

I have this equation $$y = s - cx^{1.85}$$ where s is a known integer and c is unknown. I want to use the least squares method to find the best value of c that fits a set of points. I've used ...
5
votes
5answers
77 views

Limit with number $e$ and complex number

This is my first question here. I hope that I spend here a lot of fantastic time. How to proof that fact? $$\lim_{n\to\infty} \left(1+\frac{z}{n}\right)^{n}=e^{z}$$ where $z \in \mathbb{C}$ and ...
2
votes
5answers
77 views

How to solve integration of a product of an exponential and a trigonometric function?

Preparing for the exam I bumped into this integral and I just can't get hold on it. It's an integration of a product of an exponential and a trigonometric function. It's going in an endless loop for ...
4
votes
1answer
51 views

Integration of Exponential and Logarithms, $\int_{z-1}^z \log(\frac{1}{z-y}) \exp (-| y| ^{3}) \, dy$

The integral I am dealing with is: $$\frac{3}{2 \Gamma \left(\frac{1}{3}\right)}\int_{z-1}^z \log \left(\frac{1}{z-y}\right) \exp \left(-\left| y\right| ^{3}\right) \, dy$$ where $z\in \mathbb{R}$ ...
0
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1answer
33 views

What points help identify a cubic and an exponential graph?

What are the different points on the graph we need to know which would help determine the equation of the curve. For example in a quadratic graph, we can determine the equation if we know either any ...
1
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3answers
62 views

$\sum_{1}^{\infty}\int_{n}^{n+1} e^{-\sqrt{x}} dx,$ converge or diverge?

Since $$D^{-1} e^{-\sqrt{x}} \big|_{x := u^{2}} = D^{-1} e^{-u} Du^{2} = 2D^{-1} e^{-u} u = -2(u+1)e^{-u} + C = -2(\sqrt{x} + 1)e^{-\sqrt{x}} + C,$$ we have $$\int_{n}^{n+1} e^{-\sqrt{x}} dx = ...
3
votes
1answer
78 views

Does every power of $2$ of the form $2^{6+10x}$, $x\in\mathbb{N}$, have $6$ as one of its digits?

I was thinking about this today. $2^6$ is easy, $64$ and then $2^{10}$ is roughly $1000$ so $2^{16}$ is really $(2^6)(2^{10})$ or $(2^6)\cdot 1024$ and this nicely gives a result that starts with $6$ ...
2
votes
2answers
132 views

Is $-e^{i\pi} = 1$?

Since $e^{i\pi} = \cos \pi + i\sin \pi = -1,$ a suspicious argument is to proceed to conclude that $$-e^{i\pi} = 1.$$ However, this leads to $$-e^{i\pi} = e^{0}.$$ Is the above reasoning wrong?
3
votes
3answers
71 views

Question about the exponential function.

For $x\in\mathbb R$ we define $$\exp(x) := \sum_{n=0}^\infty \frac{x^n}{n!}. $$ This is the standard definition of the exponential function, e.g. given by Rudin in the introduction to Real and Complex ...
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0answers
126 views

sum of exponential series with power increasing by geometric series

Is there any way to reduce the following summation... $2^{ar^0}+2^{ar^1}....+ 2^{ar^n} $ to a simple equation? I feel like I can pull out a $2^a$ somehow and then treat it as a normal series but I ...
3
votes
3answers
60 views

Exponential equation.

Find all $ y \in \mathbb{Z} $ so that: $$ (1 + a)^y = 1 + a^y \;,\; a \in \mathbb{R}$$ I have tried to use the following formula: $$ a^n - b^n = (a - b)(a^{n - 1} + a^{n - 2}b + a^{n - 3}b^2 + ... + ...
1
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1answer
33 views

How to prove that $f(x) = x^ε - \log x$ is $\infty$ when $x\to\infty$?

I'm trying to prove that the function $x^ε$ is "bigger" than $\log x$ when $x\to\infty$, for every $ε>0$. Or to put it in a more formal way: For every $ε>0$, there exists a constant $N$ for ...
5
votes
4answers
95 views

$\sum_{n=1}^{\infty} \frac{n^2}{ n!}$ equals [duplicate]

$ \sum_{n=1}^{\infty} \frac{n^2}{ n!} $ equals I'm not able to convert in any standard series? Any hints?
0
votes
1answer
22 views

For small x, Taylor Series to determine constants n and C in the approximation

cos(x) - e^(-(x^2)/2) I have tried writing the taylor expansion for cosine and e but do not know how to combine them in one. In addition to this, i do not know how to approach the problem whilst ...
0
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0answers
46 views

derivative of matrix exponential

How to express $\nabla\exp(i\theta(\mathbf{r}))$ in terms of $\nabla\theta(\mathbf{r})$ where $\theta$ is a Hermitian matrix of $n\times n$? Here $\nabla$ means calculating the gradient wrt. ...
2
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0answers
31 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / ...
3
votes
1answer
54 views

Convenient notation, or something more?

A little while ago I happened across a curious formula that blew my mind (no idea what it's called): $e^{\frac{d}{dx}}f(x)=f(x+1)$ I played around with it a bit and managed to prove it using the ...
0
votes
3answers
59 views

Is exp(-x) convex?

Is $f(x)=e^{-x}$ a convex function? I know that $e^x$ is convex. If I take the second order derivative of $f(x)$: $$f''(x)=e^{-x}$$ Then we can see for all the $x$, $f''(x)>0$. I'm not sure ...
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0answers
63 views

distribution of sum of double exponential random variables

I want to find out whether there is a concise expression (i.e. not a convolution) for the distribution of a random variable A which is the sum of $n$ i.i.d. rv's $B_i$, which are themselves double ...
0
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0answers
26 views

Finding the correction factor of a model

If I have a model, and data against that model. The model says that it should be linear, and the data begins linear and drops away from the line as x decreases. Is there some function I can multiply ...
0
votes
3answers
51 views

solve the equation with superscripts

I need help solving the below equation. First of all, I am not even sure if it can be solved, but I hope it can. $$ 2^{3+x} - 2^{-x} = 2^{3} - 2^{0} $$ Thank you
0
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1answer
27 views

Product of complex exponential

I'm having trouble resolving this issue on complex numbers involvendo principle of induction. As I show that: $$e^{i\theta_1} e^{i\theta_2}\cdots ...
1
vote
1answer
93 views

Series of exponential function

I had a thought today and I've tried to see if it is a thing. I'm certain it is a thing, I just don't know how to search for it. We have the Taylor series which is a summation of monomials: ...