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

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0
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2answers
35 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
23 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
55 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
vote
1answer
40 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
127 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
votes
2answers
11 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.
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1answer
24 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) ...
1
vote
3answers
73 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
26 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
76 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
61 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
44 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
votes
1answer
30 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
vote
3answers
59 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
76 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
122 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
62 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 ...
1
vote
0answers
43 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
55 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
vote
1answer
32 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 ...
-1
votes
1answer
28 views

How to integrate this expression? [closed]

I am evaluating this expression $\int_0^\infty x^2 exp(-ax^2+bx) dx$ for a,b>0, it has a closed form in terms of gamma when its limits is $[-\infty, \infty]$. badly need it guys.
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
13 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
votes
0answers
36 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
votes
0answers
29 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 / ...
2
votes
1answer
47 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
54 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 ...
0
votes
0answers
35 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
votes
0answers
22 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
46 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
62 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: ...
1
vote
4answers
83 views

How to show $\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$

I need to show the following: $$\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$$ For all complex numbers z. I don't know how to start this. Should I use l'Hopitals rule somehow?
0
votes
1answer
38 views

How power and exponential are related?

I have a model to fit but I am not sure if it is correct: Is $\exp(ax+bz+c)^d$ algebraically the same as $\exp(dax+dbz+dc)$? Edit what about this one? Is $[exp(ax+bz+c)+j]^d$ algebraically ...
0
votes
2answers
45 views

How to show that $\lim_{x\rightarrow \infty }(1+\frac{x}{n})^{n}=e^{x}$ [duplicate]

I have a problem: $E(x)=e^x$, $L(x)=\ln (x)$, $E^{-1}(x)=L(x)$. Show that $\lim_{n\rightarrow \infty }(1+\frac{x}{n})^{n}=e^{x}$ Hint: use $f(t)=\ln (1+xt)$ and look at $f'(0), x\neq 0$. I ...
0
votes
0answers
31 views

Hypothesis testing with exponential distribution

I have to show the size of a test is $a=1/3$. $X$ is a random sample of size $1$ from $exp$ distribution. Null hypothesis $H_0 = 1/2$ rejected in favor of $H_1 = 1$. If $f(x;1/2)/f(x;1) \leq 3/4$. ...
0
votes
1answer
9 views

Conditional cdf of exponential variable

Given the rate parameter for an exp r.v, I am able to calculate conditional pdf and mean on the condition of A > c. For conditional pdf I calculate P(A>c) and divide that by the pdf of the r.v. I ...
0
votes
2answers
77 views

Find the exact length of the arc of this curve

$y = 2e^x + (1/8)e^{-x}$ ... on the interval $[0, \ln(2)]$ I know am supposed to user the Arc Length formula, but I'm not sure if I have the derivative of this function correct. I came up with: ...
0
votes
1answer
18 views

Best way to prove all 3 solutions for exponential equation?

I was given the equation; $(x-7)^a=1$ where $a=(x-4)$ The 3 solutions are: $x=4, 6, 8$ When $x=4$, $(-3)^0=1$, which can be reached by setting $(x-4)=0$ because $n^0=1$ When $x=8$, $1^4=1$, ...
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0answers
27 views

A property of exponential of operators 2

Let $X$ be a Banach space. The other day I asked if all bounded operators $A:X\to X$ satisfy the following property: (P): All bounded nonzero trajectories $t\mapsto e^{tA}x$ satisfy $$\inf_{t\in ...
0
votes
1answer
42 views

How to solve the equation 5x=0.01^x [duplicate]

Hi I recently posted a this question earlier and got some excellent answers but to take it a little further I liked k170's answer however it contained a Lambert W Function in the answer and I was ...
2
votes
3answers
184 views

How to solve 5x=0.01^x

I just want to know how to solve: $$\ 5x=0.01^x$$ I have tried to use logarithms. It would be a huge help if someone could help because no matter what I do $\ x$ always gets stuck in a logarithm. ...
9
votes
1answer
104 views

Are these equations known?

Hello I found two equations that lead to constant e. I wonder if they are known. I think especially first one is most likely known but I couldn't find, it is hard to search google with all these ...
1
vote
1answer
21 views

What is this equal to? : $|A+B|^2$ where $A = P e^{ia}$ and $B = Q e^{ib}$

$A$ and $B$ are two complex numbers: $A = P e^{ia}$ $B = Q e^{ib}$ I would like to know what is this equal to? : $|A+B|^2$ Please also give a small proof if possible.
2
votes
2answers
32 views

Finding the interval after substitution

Given this problem $$8\cdot 3^{\sqrt{x}+\sqrt[4]{x}}+9^{\sqrt[4]{x}+1}\geq 9^{\sqrt{x}}$$ After simplifying I get $8\cdot 3^{\sqrt[4]{x}-\sqrt{x}}+9\cdot 3^{2\sqrt[4]{x}-2\sqrt{x}}\geq 1$ now ...
0
votes
1answer
21 views

Simplify expression with lambert w-Function

I have an expression and i am almost sure what it equals: $ e^{-W_{-1}\left(-\frac{log\left(x\right)}{x}\right)} $ I only need a simplified version of this expression for $x\geq e$. I assume: ...
0
votes
0answers
13 views

Moving Logarithmic function equation plotted on log log paper up or down on the y axis

I'm hitting a stump here. I have a logarithmic function plotted on log log paper so it's a straight line. So let's say I have this entire line plotted out on the log log paper....how would I simply ...
0
votes
1answer
50 views

If $E(z)= \sum _{n=0 }^{\infty }\frac {z ^n } {n! } $, how is $E(0) $ defined?

If $E(z)= \sum _{n=0 }^{\infty }\frac {z ^n } {n! } $, how is $E(0) $ defined? The exponential function for complex $z $ is defined in Rudin's principles as the power series $ \sum _{n=0 }^{\infty ...
0
votes
2answers
75 views

Solve $2^x=13 \bmod 3^4$

Solve $2^x=13\bmod 3^4$ I know $\log13=30\bmod 3^4$ and $\log16=15 \bmod 3^4 $ I've tried subbing $\log13/\log16$ for $2$ but I am not sure what to do next.