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

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0
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0answers
22 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 ( ...
0
votes
2answers
29 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 ...
0
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1answer
20 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 ...
8
votes
4answers
123 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: $2^x=8$ ...
0
votes
1answer
40 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 ...
1
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1answer
49 views

Integration Of exponential Function

I have tried almost everything, but can't solve this integral. $$\int e^{-1/x^2} \, dx $$
7
votes
5answers
926 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) ...
1
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1answer
55 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, ...
2
votes
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 $x+b$: ...
-2
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1answer
33 views

solution of a simple exponential relation [on hold]

I have a simple exponential relation that want to solve it and find domain for x and y. The relation is as follow: $$e^x-e^y<k$$ $$x>y>0$$
3
votes
2answers
109 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} ...
0
votes
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 ...
1
vote
2answers
44 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.
-2
votes
0answers
9 views

exponential distribution using memoryless property [on hold]

X1~Erlang(2, r1), X2~Erlang(2, r2), X1, X2 are independent. Pr(X1 < X2) using memortless property(forgerfulness)
1
vote
2answers
89 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 ...
1
vote
3answers
51 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 ...
1
vote
1answer
98 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.
0
votes
0answers
9 views

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 ...
0
votes
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 $a$, ...
1
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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 ...
2
votes
1answer
44 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 ...
1
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6answers
121 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 ...
4
votes
4answers
112 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 ...
-1
votes
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. $$ \lim_{x \to \infty} \frac{1-e^2}{1+e^2} $$
-4
votes
1answer
28 views

exponential independent random variables [closed]

I am currently doing some exam revision for an upcoming exam and am stuck on the following question. I assume I must do something with the exponential function where $\lambda= \dfrac 13$ but am ...
7
votes
1answer
100 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 ...
-1
votes
0answers
44 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 $\beta$ ...
2
votes
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 ...
2
votes
1answer
49 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: $x_1=0$ $x_2= 4.96511$
0
votes
2answers
55 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 ...
0
votes
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 ...
1
vote
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
1
vote
6answers
121 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.
0
votes
1answer
64 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 ...
0
votes
1answer
60 views

How to solve the following exponential equation?

How to solve the following exponential equation? $h_1 = x - yq_1^z $ $h_2 = x - yq_2^z$ $h_3 = x - yq_3^z$ here $x$, $y$, $z$ are unknown and $h_1$, $h_2$, $h_3$, $q_1$, $q_2$, $q_3$ are ...
1
vote
1answer
58 views

Does $\Delta \geq \max\{2em, \lg P + \lg \frac{1}{\epsilon} \}$ guarantees that $(\frac{em}{\Delta})^\Delta \leq \frac{\epsilon}{P}$?

I saw this in a technical paper which made a leap I can't follow, it tries to solve an inequality $(cx)^{-x} \leq y$, which it then says it is satisfied when $x \geq -\ln {y}$. I can't make the ...
1
vote
1answer
16 views

How to convert this limit involving arctan into an exponential?

As part of a larger proof I'm working on (convergence in distribution of a random variable to a certain cdf), I need to show that: $\lim_{n\to\infty} ...
5
votes
6answers
108 views

Proof of Euler's formula that doesn't use differentiation?

So I saw a 'proof' of the sine and cosine angle addition formulae, i.e. $\sin(x+y)=\sin x\cos y+\cos x \sin y$, using Euler's formula, $e^{ix}=\cos x+i\sin x$. By multiplying by $e^{iy}$, you can get ...
0
votes
1answer
76 views

Exponential & logarithmic functions [closed]

How can you solve the following logarithmic exponential expression for $y$ in terms of $x$, Where both $x$ and $y$ are positive distinct real numbers and $0<x<y <1$ ...
3
votes
2answers
62 views

Why are these representations of e the same? [duplicate]

I heard that $e$ can be defined as the limit as n approaches infinity of $(1 + 1/n)^n$, but I also heard that $e$ is also defined as the sum of the reciprocals of the factorials from $0$ to $\infty$. ...
0
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2answers
33 views

Proving exponential inequalities

I'm currently revising for an upcoming exam and am stuck on the following question. I have completed a similar question that involved cos and the mean value theorem I used the triangle inequality too, ...
2
votes
1answer
70 views

on the sum of an infinite series

Got stuck with this series: $$ \sum_{k=0}^\infty \frac{1}{(\theta+2+k)(\theta+1)^{k+1}} $$ which should be equal to $$ \int_0^1 \frac{t^{\theta+1}}{\theta+1-t}\textrm{d}t $$ But why? Which is the ...
7
votes
2answers
536 views

Is there ANY possible way to solve this equation?

So I came up with this equation and it just seems like I can't solve it AT ALL for '$a$' $$a*b^a = c$$ EDIT: By the way, I'm only taking $b^a$, not both $b$ and $a$, just in case anyone was ...
1
vote
6answers
108 views

Calculate $\lim\limits_{x\to 0}\left(\frac{\sin(x)}x\right)^{1/x^2}$

How to calculate $$\lim_{x\to 0}\left(\frac{\sin(x)}x\right)^{1/x^2}?$$ I know the result is $1/(6e)$.
3
votes
6answers
127 views

Is there proof show that $\log x$ is undefined and make no sense at $ x=0$?

I was asked by someone: why $\log x$ is undefined at $x=0 $? Is there proof show that $\log x$ is undefined at $x=0$? Note(01):: log is the inverse function of the exponential function. note(02): ...
0
votes
1answer
64 views

Solving an exponential equation for yield curve rates

I'm preparing for an exam and one of the topics is pricing bonds where bootstrapping a yield curve is used. Among other things, it involves solving exponential equations which is a recurring problem. ...
1
vote
1answer
52 views

Recursive definition of a Gevrey-class function

Given the following Gevrey-class function $\Phi:\mathbb{R} \rightarrow \mathbb{R}$ $$\Phi_{s,T}(t) = \begin{cases} \begin{align} 0 \quad & t \le 0 \\ 1 \quad & t \ge T \\ ...
4
votes
3answers
56 views

How can a complex exponential represent a real world quantity?

Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex ...
0
votes
2answers
37 views

Arithmetic sequence of exponentials

There is an arithmetic sequence $2^a, 3^b, 4^c$ such that a,b and c are positive integers. The question posed is to find ALL possible ordered triplets $(a,b,c)$. The constant difference d between ...
0
votes
1answer
52 views

Proving that exponential growth at rate r equals exponential growth at rate 1 to the power of r

This is a fairly basic result, but I could not find anything about it here. How do you prove that the following relationship exists, and where does the basis for it originate from: ...