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

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0
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1answer
35 views

Solve slope intercept equation for two points and the maximum starting value?

I have two points (x2,y2) and (x3,y3) that represent points in an exponential decay curve of discounted cash flows (x2 is less than x3): My question is: What is the decay curve equation for the ...
1
vote
1answer
21 views

Taking the logarithm of $e^{-x}<b$

If I have an inequality as $e^{-x}<b$ where $b,x$ are positive , can I take the logarithm on both sides and say, $-x<=ln(b)$
1
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2answers
49 views

Solve exponenital integral equation

$$\frac{1}{\sqrt{2\pi}\sigma_1 }\int_x^\infty\exp(-\frac {t_1^2-1}{2\sigma_1^2})dt_1 + \frac{1}{\sqrt{2\pi}\sigma_2 }\int_x^\infty\exp(-\frac {t_2^2-1}{2\sigma_2^2})dt_2 = a $$ $$\sigma_1 , \sigma_2 ...
7
votes
4answers
91 views

A function with a property $f(x+y)=f(x)f(y)$

A function with the property $f(x+y)=f(x)f(y)$ is well known exponential function, $f(x)=a^x$. My question is, how do you prove if there is no other function with this kind of property? Edit: I ...
3
votes
3answers
61 views

Random Walk And Stochastic-Processes

Assume that $P(X_i = 1) =1/2, P(X_i =-1)= 1/4,\text{ and }P(X_i = 0)=1/4$. Consider the random walk starting at 1 given by $$S_n = 1 + X_1 + X_2 + \cdots + X_n$$ where $X_1,X_2, ...$ are i.i.d. ...
3
votes
5answers
555 views

Where did it come from? (derivative of exponential)

We all know this rule: $\text{If } y = a^{f(x)} \text{ then } y' = a^{f(x)} \: f'(x)\ln a$ In my book there is the example: Find $\frac{d}{dx}\left((x^{2} + 1)^{\sin x}\right)$ According to ...
3
votes
3answers
39 views

Value of sine of complex numbers

I stumbled upon a problem with evaluating the sine function for complex arguments. I know that in general I can use $$ \sin(ix)=\frac{1}{2i}(\exp(-x)-\exp(x))=i\sinh(x). $$ But I could also write the ...
0
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1answer
33 views

Solving Laplace Transform of $-e^{-at}u(-t)$

I have found a problem in applying Laplace Transform to $-e^{-at}u(-t)$ I am doing these steps: $$ = - \int_{-\infty}^{+\infty} e^{-at}u(-t) e^{-st}dt$$ $$ = - \int_{-\infty}^{0} e^{-at} ...
1
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0answers
13 views

exponentialfunction poofs with generating functions

I have two things to proof. $(e^{ax})^{-1}=e^{-ax}$ and $(e^{ax})^{m}=e^{(ma)x}$ I know the power series of $e^x$ and $e^{ax}$ and that $e^{ax}e^{bx}=e^{(a+b)x}$ I tried it forward and backward: ...
0
votes
1answer
17 views

Bounding the growth of f(u)

Given a function $f(u) \le 2{\sqrt u}\,f({\sqrt u}) + 1$. We need to bound the growth of $f(u)$. If we expand this by recursively substituting, we get some series like $1 + \left\{ ...
0
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2answers
37 views

Determine limit of indeterminate form.

If the question is $$\lim_{x\to\infty}(e^x+1)^{\frac1x}$$ Do you just say that because $\lim_{x\to\infty}\frac1x$ is $0$, the original function has limit approaching 1, without caring the $e^x$?
6
votes
2answers
126 views

Negative solution to $x^2=2^x$

Just out of curiosity I was trying to solve the equation $x^2=2^x$, initially I thought there would be just the two solutions $x=2$ and $x=4$, but wolfram shows that the two equations intersect at not ...
0
votes
1answer
22 views

Determine residues of $\frac{e^{-\sqrt{z(z+r))}}}{1+\alpha\sqrt{z(z+r)} + (1-\alpha \sqrt{z(z+r)})e^{(-\sqrt{z(z+r)})}}$

I have tried to determine residues of the below function via Mathematica and Matlab, but they lead me nowhere. For small enough $\alpha$, I figured out what are the poles, but nothing about residues. ...
0
votes
0answers
15 views

vector exponential

So I know that matrix exponentials exist and I stumbled across this webpage which writes about vector exponentials. http://delta.cs.cinvestav.mx/~mcintosh/comun/summer99/mcintosh/node46.html ...
0
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1answer
38 views

How do you find inverse for certain exponential function?

How do you find inverse for $y=\frac{e^x-e^{-x}}{2}$?
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2answers
12 views

Problems with term transformation

my math book gives the following question: A company sells phones and models the daily sales with the following function: $$f(t) = k*(t-15)*e^{-0,01t}+k*15$$ I have to find the value for t, so that ...
0
votes
2answers
67 views

Equivalence of definitions of $e^x$

Let $e$ defined as $\lim\limits_{n\to\infty}(1+\frac{1}{n})^n$. Let now $f,g:\mathbb{R}\to\mathbb{R}$ defined as: $$f(x)=\sup \{e^r\mid r\in \mathbb{Q}\text{ and } r<x\}$$ And ...
3
votes
1answer
77 views

Defining $e^{x}$ as an infinite number of integrals

I was thinking that using the Taylor expansion of $e^{x}$ we use a summation of derivatives is it possible to write it using integrals? Something along the lines of $\int\int\int...\int dxdxdx...dx$ ...
10
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7answers
701 views

What are your favorite relations between e and pi? [closed]

This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for ...
5
votes
5answers
174 views

Find the limit $\lim_{x\to\infty} f(x)= \lim_{x \to \infty} \left(\frac{x}{x+1}\right)^x$

I need to find the following: $$\lim_{x\to\infty} f(x)= \lim_{x \to \infty}\left (\frac{x}{x+1} \right )^x$$ I know that this limit = $\frac{1}{e}$ from plugging it into a calculator, but I have to ...
0
votes
2answers
65 views

How to solve equations like $8n^{2} = 64n\log_2(n)$

How do I solve this equation: $$ 8n^{2} = 64n\log_2(n) $$ I've plotted logarithmic curves for both functions and found that they intersect at $n = 44$, but I've no idea how to solve the equation.
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2answers
51 views

Differentiate an exponential integral

Would you guide me differentiating this integral for $m$: $$\frac{\text{d}}{\text{d}m} \int_{x=-\infty}^m\int_{y=n}^{+\infty}\exp\left(-\left[\left(\frac{x-a}{b}\right)^2 - ...
0
votes
0answers
11 views

Reverse fourier transformation of exponential

What is the result to this formula: $$F^{-1}\left(\exp(\hat{f})\right)$$ with $F^{-1}$ the inverse fourier transformation? I tried to start with $$F^{-1} = ...
0
votes
1answer
12 views

Normalizing a biexponential function so its time integral is always unity

I'm reading a paper that uses the following function: \begin{align} f(t) &= \frac{1}{\tau_1-\tau_2}(e^{-t/\tau_1}-e^{-t/\tau_2}). \end{align} The authors state that The normalization adopted ...
1
vote
1answer
23 views

Joint moment generating function problem

X and Y have the following joint moment generating function: $M_{X,Y}(a,b) = \Large \frac{4}{5}[\frac{1}{(1-a)(1-b)}+\frac{1}{(2-a)(2-b)}]$ Find E(XY) I have gone through this problem several ...
0
votes
2answers
31 views

What is the proper adjective/adverb for a power function?

I have a function where space grows as a power of time: $x= at^2$. In my report, I've been using the adjective 'exponential' or adverb 'exponentially' to describe the expansion with time. However, ...
0
votes
1answer
36 views

Integrating over discontinuities

I have the following integral: \begin{align} \int^T_0e^\tau I(\tau+t^0)d\tau. \end{align} In this integral, $I(t)$ is a function with period $T$. At each time $T, 2T, \ldots$, $I$ is increased by a ...
0
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4answers
31 views

Derivative of log and exponential function

How would you find the derivative of: $$y= e^\sqrt{x} + \ln\sqrt{x}$$ Would I simply use the product rule?
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0answers
2 views

Derivation in exponential function

I am working on the GNLSE with the split-step method, but I do not understand every step. The (for this question) important part of the GNLSE is $$\frac{\partial A}{\partial z}=(\hat{D}+\hat{N})A$$ ...
2
votes
3answers
68 views

Compute $\lim\limits_{x \to 0} \frac{e^x-1}{x}$ without using derivatives [duplicate]

How to compute $\lim\limits_{x \to 0} \frac{e^x-1}{x}$ without using derivatives? Every method I can think of gives me some indeterminate form.
2
votes
2answers
30 views

Confused With Simplifying Exponents

If you had $x^{2/4}$, would that simplify to $x^{1/2}$? If you were to simplify $x^{2/4}$ to $x^{1/2}$, x cannot be negative for a real solution... But if you left it as $x^{2/4}$, x would be able ...
1
vote
2answers
24 views

Finding the decay rate of some element

I have a question that in short reads: If some element decays exponentially with a half life of 25,000 years, how long will it take for 99.9% of it to decay away? Using the exponential decay ...
0
votes
1answer
63 views

Solving line intercept equation for exponential decay using two points?

I have two points (x1,y1 and x2,y2) that represent points in an exponential decay curve (discounted cash flows): Exponential Decay using varying Discount Rates The limits of my mathematics is using ...
1
vote
1answer
31 views

Show that $\exp(x)-1=\mathcal{O}(x)$ for $x\to 0$

Find a function $g(x)$ that is as simple as possible s.t. $\exp(x)-1=\mathcal{O}(g(x))$ for $x\to 0$. Claim. Such a possible function is $g(x)=x$. Proof. Using the definition of the class ...
1
vote
2answers
30 views

What could the ratio of two sides of a triangle possibly have to do with exponential functions?

Name says it all. The two seem so unrelated? What's more, if I'm not mistaken the exponential version contains an imaginary part. I'm kind of ignorant about imaginary numbers, but does this mean that ...
0
votes
1answer
26 views

Limit laws how to write an integral as a max of a sum

Good Morning, I am not able to prove following equation: lim 1/n log()=... https://www.wias-berlin.de/people/koenig/www/GA.pdf Korollar 1.3.2 I thought about doing a Laplace Transformation, but I ...
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vote
2answers
24 views

Let $X$~$Expo(1)$ and $S$ be a random sign, find the PDF of $SX$ and compare to the Laplace PDF

The Laplace distribution has PDF $f(x) = \frac{1}{2}e^{-|x|}$ for all real x. Let $X$~$Expo(1)$ and $S$ be a random sign (1 or -1, with equal probabilities), with $S$ and $X$ independent. Find the ...
2
votes
1answer
23 views

Exponential decay, show that the probability that a particle decays in an interval is proportional to the length of the interval

Let $T$ be the time until a radioactive particle decays, and suppose that $T$~$Expo(\lambda)$. (a) Find the half-life of the particle. Median (half-life) occurs when CDF = $\frac{1}{2}, F_T(m) = ...
1
vote
1answer
24 views

The problem related to Exponential distribution.

Question: There are two batteries. A Battery's life is following the Exp(1/20) distribution. The other one's life is following the Exp(1/40) distribution. One day, a person randomly chose one ...
2
votes
6answers
118 views

limit of $\left( 1-\frac{1}{n}\right)^{n}$

limit of $$\left( 1-\frac{1}{n}\right)^{n}$$ is said to be $\frac{1}{e}$ but how do we actually prove it? I'm trying to use squeeze theorem $$\frac{1}{e}=\lim\limits_{n\to ...
1
vote
0answers
31 views

Is there a simple solution for $ \frac{d}{dy} \log\big( \int_{0}^{y} \exp(-a \frac{x^{3/2}}{y}+ b x)\, dx \big) $

What is the solution of $$ \frac{d}{dy} \log\left( \int_0^y \exp\left(-a \frac{x^{3/2}}{y}+ b x\right)\, dx \right) $$ Where a and b are constants. Even if the solution is an approximate solution. ...
0
votes
1answer
28 views

Differential of definite integral

What is the solution of $$ \frac{d}{dy} \int_{0}^{y} \exp(-y(x+1))\, dx $$ If there were no $y$ inside the exponential function the answer would be the exponential function.
0
votes
0answers
37 views

Transform cos to e function

What are the steps in order to transform the cosine function to the exponential function: $$ \left[\cos \left(\frac{k \pi} N\right)\right]^n \approx e ^ {\frac{-n}2 \left(\frac{k \pi} N \right)^2} ...
2
votes
1answer
33 views

Distribution of $Z$ from Moment Generating Function

Suppose that $X_1, X_2, ..., X_n$ are independent and identically distributed Exp(λ) random variables and let $Z = X_1 + X_2 + · · · + X_n$. Determine $M_Z(θ)$, the moment generating function of $Z$ ...
1
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2answers
37 views

Holomorphic Functional Calculus for Exponential Identity

For a Banach algebra $A$ with unit $e$, then for $a\in A$, I want to prove $$\exp((\alpha+\beta)a)=\exp(\alpha a)\exp(\beta a)\qquad\text{for all }\alpha,\beta\in\mathbb{C}$$ So far I have said let ...
2
votes
1answer
29 views

Moment Generating function hard example!

X is a random variable with density $$f(x)=2e^{-2x+2} , x\geq1$$ and 0 otherwise. Determine $Mx(θ)$, the moment generating function for X, and the values of θ for which $Mx(θ)$ is defined. Use ...
2
votes
2answers
34 views

exponential functions with constant

I'm in a pre-calc class, and we're looking at logarithms and exponential functions. One of the exercises I'm struggling with is: $$5e^{2x} = 6 + 29e^x$$ I would ususally multiply each side by a log ...
1
vote
1answer
37 views

Convergence/Divergence of Series for $e^{-1}$

How do I show that the series $\sum \frac{n^n}{n!} x^n$ diverges at $x =e^{-1}$? I understand that $a(n) = (\frac{n}{n+1})^n$ is a strictly decreasing function and therefore $(\frac{n}{n+1})^n > ...
0
votes
0answers
35 views

Expected values with e to the power of e^x as a factor

Suppose that the random variable X has an exponential distribute with mean four. Let the random variable Y=ln(X+1). Find E[Y] if $y=ln(x+1)$ then it follows that $x=e^y-1$ and that $dx/dy = e^y$ ...
2
votes
1answer
85 views

Show that $e^x=1+x+\frac{x^2}{2!}+…+\frac{x^n}{n!}+R_{n+1}$

Show that $\qquad$ $\qquad$ $e^x=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+R_{n+1}$ with $\qquad \qquad$ $0 \lt R_{n+1} \lt e^x \frac{x^{n+1}}{(n+1)!}$ if $0 \lt x$ and $\qquad \qquad$ $|R_{n+1}| \lt ...