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

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1answer
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Definition of exponential function -

A lot of textbooks offer a definition of the exponential function such as this: $$\exp(x):=\sum_{k=0}^{\infty}\frac{x^k}{k!}.$$ a) Show that the given definition for $\exp$ is correct, ...
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28 views
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2answers
26 views

Excel's EXP function compared to a series expansion

I am comparing the results of a series expansion of $e^x$ to Excel's $\mathop{EXP}(x)$ function. Should I expect them to be the same? Excel's gives $\mathop{EXP}(10) = 22026.4657948067$. However, ...
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1answer
95 views

For which complex $a, b, c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
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1answer
23 views

How is derived it? E(exp(-bdt))=E(1-bdt) [on hold]

I have following transformation: E(exp(-bdt))=E(1-bdt). But I don'tknow how is it derived? Could anyone clarify? Thank you!
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2answers
245 views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...
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0answers
26 views

Distribution of the minimum of two exponential random variables

$X$ and $Y$ are two exponential random variables with rate 1 and 2. lets define random variable $Z$ such that: $z_i = min(x_i,y_i)$, where $i =1,2,3,...N$. Let $V$ be another random variable and ...
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3answers
49 views

Finding X from Exponential Equations

$$2^x \cdot 4^{1-x}= 8^{-x}$$ I wrote all the base numbers as a power of 2 but I'm not sure what to do after.
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4answers
41 views

Unknown both as a exponent and as a term in an equation

Let's say I have an equation $e^{x-1}(x+1)=2$. According to Solving an equation when the unknown is both a term and exponent it's impossible to solve this using elemetary functions. If so, then how do ...
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1answer
12 views

Expectation of geometric summation of exponentail random variables

Let $\{\tilde x_i, i = 1,2,\ldots\}$ be a sequence of iid exponentially distributed random variables with parameter $\lambda$ and let $\tilde n$ be a geometrically distributed random variable with ...
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3answers
32 views

function bounded by an exponential has a bounded derivative?

here's the question. I want to be sure of that. Let $v:[0,\infty) \rightarrow \mathbb{R}_+$ a positive function satisfying $$\forall t \ge 0,\qquad v(t)\le kv(0) e^{-c t}$$ for some positive constants ...
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1answer
20 views

Inverse of the complex exponential function, considered as a multivariable function

Consider the complex exponential function $g: \mathbb{C} \to \mathbb{C}, z \mapsto e^z$. When identifying $\mathbb{C}$ with $\mathbb{R}^2$ in the natural way, then $g$ can be considered as a ...
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2answers
19 views

Comparison between two exponentail random variables

A and B are exponentially distributed with parameter $\alpha$ and $\beta$. A and B race repeatedly. $N_b$ denotes the number of times B wins before A wins his first race. Find $P (N_b = n )$ for $n ...
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2answers
79 views

Need help with $e^x=1/x$

I've tried everything. I expressed $x$ and I got $x=\ln{1\over x}$, and don't know what to do. Original question is to find $e^x-{1\over x}=0$. There is a solution I've typed it in Wolfram
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2answers
36 views

Solve for $x$ in: $e^{2\ln(x)-\ln(x^2+x-3)} = 1$

So the question is to solve for x in: $$e^{[2\ln(x)-\ln(x^2+x-3)]} = 1$$ I took the natural log of both sides, and simplified. Here is what I've gotten it down to: $$2\ln(x) = \ln(x^2+x-3)$$ And I'm ...
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0answers
13 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
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2answers
43 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ The correct answer is $\frac ...
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0answers
24 views

Why does integrating a complex exponential give the delta function?

How come, when we integrate a complex exponential from $ -\infty $ to $ \infty $, we get a scaled delta function? $$ \begin{align} \int_{-\infty}^{\infty} e^{i k x} \; dk & = 2 \pi \delta \left ( ...
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2answers
36 views

Find the sum of the roots of the exponential equation

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum. I'll simplify it first as: $$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$ Let ...
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1answer
23 views

Why is this true - easy question concerning asymptotics of exponential

Suppose $\lambda > 0$ is constant as $t \searrow 0$. In my lecture notes it is written that $\left(1+\sum_{k=1}^{\infty} \frac{(-\lambda t)^k}{k!}\right) \lambda t = \lambda t + o(t)$ and ...
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4answers
129 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
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1answer
42 views

How to find the expected cost of an exponential probability?

The length $X$ of of a call follows the exponential distribution with mean $2$ minutes. In dollars, the cost of of a call of $x$ minutes is $3x^2-6x+2$. Find the expected cost of a call? The addition ...
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1answer
49 views

Integration Of exponential Function

I have tried almost everything, but can't solve this integral. $$\int e^{-1/x^2} \, dx $$
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5answers
962 views

Proof of the derivative of ln(x)

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) ...
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1answer
59 views

Prove $e^x$ limit definition from limit definition of $e$.

Is there an elementary way of proving $$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n,$$ given $$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n,$$ without using L"Hopital's rule, Binomial Theorem, ...
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0answers
20 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
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1answer
33 views

solution of a simple exponential relation [closed]

I have a simple exponential relation that want to solve it and find domain for x and y. The relation is as follow: $$e^x-e^y<k$$ $$x>y>0$$
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2answers
110 views

Show that $\frac{(x^2 + y^2 )}{4} \leq e^{x+y-2}$

Show that \begin{equation} \frac{x^2 + y^2}{4} \leq e^{x+y-2} \end{equation} is true for $x,y \geq 0$. As far, I have prove that \begin{equation} x^2 + y^2 \leq e^{x}e^{y}\leq e^{x+y} ...
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0answers
43 views

Solve $e^{\sqrt{x^{2} - x - 1}} = |x|$

Is it possible to obtain the solution of $$e^{\sqrt{x^{2} - x - 1}} = |x|$$ in closed form? I know that $x$ must be somewhere between $\displaystyle\frac{\sqrt{5} + 1}{2}$ and $2$ after trying some ...
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2answers
45 views

Very easy question about infinitesimals [duplicate]

how can I prove that: $$ \lim_{x\to 0} \frac{e^{-1/x^2}}{x} = 0 ? $$ I suppose that the exponential "goes" to $0$ faster than linear, but I'm not sure.
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0answers
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exponential distribution using memoryless property [closed]

X1~Erlang(2, r1), X2~Erlang(2, r2), X1, X2 are independent. Pr(X1 < X2) using memortless property(forgerfulness)
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2answers
93 views

Product limit with exponentials

Find an explicit formula for the limit: $$\lim_{n \rightarrow \infty} n \prod_{k=2}^{n} (2 - e ^ {\frac 1 k})$$ I am not asking for convergence proof since I know the sequence is decreasing and ...
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3answers
52 views

How do I integrate this exponential + Bessel function term?

I would like to integrate this in my research: $\int_0^\infty s e^{i bs^2}J_0(a s)$, where a and b are both real and greater than zero. Integration by parts seems like the obvious first step, but ...
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1answer
100 views

How to integrate $\int^{\infty}_{-\infty} e^{-2\pi^2/x^2} dx$?

I am wondering how can i integrate this quantity above? Here it is again, $$\int^{\infty}_{-\infty} e^{-2\pi^2/x^2}dx.$$ Thanks a lot.
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Exponential operator on function; can it be simplified?

Suppose I have two operators $A(t),V(t)$. There is also a parameter $t \in [0, \infty]$. Moreover I have a continuous function $f(t)$ which satisfies $A(s)f(t)=0$ for all $s \in [0,\infty]$. How can I ...
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2answers
43 views

Integral of logarithm of exponential function

I am trying to solve this integral: $$\int \log\left(1 + \frac{1}{\pi}\exp\left(\frac{-x^2}{2a^2}\right)\right) dx$$ where $a$ is some fixed constant. The bounds of this integral are $-a$ and $a$, ...
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2answers
62 views

Fit exponential with constant

I have data whic would fit to an exponential function with a constant. So y=aexp(bt) + c Now I can solve an exponential without a constant using least square by taking log of y and making the ...
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1answer
46 views

Expressing e as an infinite series: finding values for similar series

I am supposed to be using the fact that $e = \sum_{n=0} ^\infty \frac{1}{n!}$ to find the value of $\sum_{n=0} ^\infty \frac{1}{2n!}$. Is there some method for substitution when dealing with infinite ...
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6answers
123 views

Best way find $\lim_{x\to 0}( \frac {\sin x}{x})^{\frac 1x}$

$\lim_{x\to 0}( \frac {\sin(x)}{x})^{\frac 1x}$ $$$$ I can use Tailor to get to $\lim_{x\to 0}(1+\epsilon(x))^\frac 1x$ $$$$ $(\epsilon(x)\underset{x\to\infty}\to 0) $ $$$$ but does that mean that ...
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4answers
121 views

Why $y=e^x$ is not an algebraic curve?

Why $y=e^x$ is not an algebraic curve over $\mathbb R$? I can say that is not a algebraic curve over $\mathbb C$ because $e^x$ is a periodic function, but what about $\mathbb R$? EDIT: I don't want ...
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1answer
64 views

Limit of $(1-e^2)/(1+e^2)$ as $x$ approches negative infinity [closed]

So, I am having trouble solving this limit. I have racked my brain many times to solve it. Any help is appreciated. $$ \lim_{x \to \infty} \frac{1-e^2}{1+e^2} $$
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1answer
103 views

Cosh and Sinh analogs

We know that $$\cosh{x}+\sinh{x}=e^x$$ and that his can be expressed as $$\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=\frac{(e^x+e^x)+(e^{-x}-e^{-x})}{2}=e^x$$ and this works out nicely because the ...
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0answers
45 views

Compute a complicated integral

I am trying to compute an integral function in the context of studying the behavior of complex systems. I arrive to this integral: $\int e^{\beta \ln(1-e^{-\alpha u})} du$ where $\alpha$ and $\beta$ ...
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1answer
69 views

Prove limit converges in definition of $e.$

I've looked up several related questions, but they do not answer what I am looking for. Please give link if this is a duplicate. What I eventually want to know is why ...
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1answer
50 views

Solve $\exp(x)(5-x)=5$ by hand

Is there a way to solve this equation by hand? $\exp(x)(5-x)=5$ Solutions: $x_1=0$ $x_2= 4.96511$
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2answers
57 views

Derivative of function $f(x) = \sqrt{2x}+ \sqrt{2/x}$

The derivative of function $$f(x) = \sqrt{2x}+ \sqrt{2/x}$$ Here's what I did, $$f(x) = \sqrt{2x}+ \sqrt{2/x} \\ = (2x)^{1\over2} + ({2\over x})^{1 \over 2}\\\\$$ $$f'(x)={1\over 2}(2x)^{-{1\over ...
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1answer
9 views

With $e_i\sim\exp(1)$ why does $\prod_{j=1}^n\exp(ite_j) = (\frac{1}{1-it})^n$?

We have $S_n=\sum_{i=1}^ne_i$ with $e_i\sim\exp(1)$ why does $\prod_{j=1}^n\exp(ite_j) = \left(\frac{1}{1-it}\right)^n$? I just want to understand the following line from my notes and hope it is ...
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1answer
47 views

Integral using gamma and beta functions

I can't solve this, no matter how I try $$\int_{-\infty}^{+\infty} \frac{e^{2x}}{4e^{3x}+9}\,dx$$ Thanks in advance
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6answers
122 views

How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$

How to prove $$1+x \leq e^x~\forall x \in \mathbb{R}$$ I'm stuck, I tried taking logs but didn't know how to proceed.
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1answer
65 views

What is the most intuitive explanation for euler's identity? [duplicate]

Is there any intuitive explanation for: $$e^{i\pi} + 1 = 0$$ About whether this question is a duplicate, what is asked for is not a proof but an explanation that helps with the not-so-intuitive ...