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

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0
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2answers
51 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
66 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
52 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
vote
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 ...
4
votes
4answers
129 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
82 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. $$ \lim_{x \to \infty} \frac{1-e^2}{1+e^2} $$
8
votes
1answer
114 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 ...
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
54 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
65 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
48 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
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6answers
124 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
68 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
18 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
115 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 ...
3
votes
2answers
67 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
votes
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
72 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
558 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
133 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
66 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
59 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
61 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
42 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
55 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: ...
1
vote
3answers
59 views

Find a limit without using L'Hopitals rule 9

Can someone please show me how to do this without using L'Hopitals rule: $$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x$$ I know the limit is $e^a$, but I would like to know the steps taken to ...
0
votes
0answers
8 views

Conversion of cylindrical harmonic field into space-harmonic field for plane waves

It is well known that a plane wave can be represented by an infinite sum of cylindrical wave function of the form $\varphi^i(\rho,\phi)=e^{\left(-j\beta \rho ...
3
votes
2answers
113 views

Calculate $\lim \sin(2\pi n!e)$. [SOLVED]

I need to calculate $\lim \sin(2\pi n!e)$. I put it into Wolfram and saw that it is likely to converge to 0. Of course this would mean that the fractional part of $n!e$ should be always very close ...
1
vote
4answers
71 views

Definition of exponential function, single-valued or multi-valued?

If we define $$e^z=1+z+\frac{z^2}{2!}+\cdots$$ then it is single-valued. However, if we write $$e^z=e^{z\ln e}$$ then it is multi-valued. Besides, $a^z$ is multi-valued in general. It is kind of ...
1
vote
5answers
69 views

Limit of $\dfrac{(1+4^x)}{(1+3^x)}$?

I don't remember how to find the limit in this case. I take $x$ towards $+\infty$. $\lim\limits_{x\to \infty} \dfrac{1+4^x}{1+3^x}$ I do not know where to start. I would instinctively say that ...
0
votes
3answers
85 views

What's wrong with this?

What's wrong with this : $$e^{i\pi} = -1$$ $$\therefore e^{2i\pi} = 1$$ $$\therefore log \left( e^{2i\pi} \right ) = log(1) = 0$$ $$\therefore 2i\pi = 0$$
0
votes
4answers
136 views

How to solve $e^{ix}=i$?

This is a question related to another posted question: The answer to the following question "Find all solutions to: $e^{ix}=i$" is as follows: "Euler's formula: $e^{ix}=\cos(x)+i\sin(x)$, so: $ ...
0
votes
2answers
88 views

Transcendental numbers & logarithms

Given two coprime positive integers greater than one, say $n,\ m$ , where $n > m$ . How do we find the ratio $\dfrac{\log m}{\log n}$ in terms of $n$ and $m$ symbolically ? Claim: The ratio is ...
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2answers
140 views

Exponential equation $(x^x)^{2015}=2015$

Solve for $x$: $(x^x)^{2015}=2015$ Tried several times, but have no idea about how to start even.
0
votes
1answer
42 views

vector as production of matrices, trouble with exp()

A vector $(a_{11}x, a_{22}y, a_{33}z)$ can be expended as: $$\begin{align} \begin{pmatrix} a_{11}x \\ a_{22}y\\ a_{33}z \end{pmatrix} &= \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ ...
4
votes
4answers
226 views

Why is $e$ so special? [duplicate]

The number $e$ (and the exponentiation function $e^x$) appears in so many places in mathematics and engineering. There seem to be a multitude of applications of it. I want to know why.
0
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0answers
34 views

Simplify integrals

I want to further simplify $\frac{A_1}{A_2}$ integral and find a more simplified expression between them f(r) is defined as a function of $r$ (no further information about it exists): I applied ...
0
votes
0answers
25 views

Improper integral over product of exponentials: $\int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx$

I'm looking for a way to evaluate following integral $$ \int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx $$ f(x) resembles however a complex simulation and can therefore ...
2
votes
1answer
51 views

Covariance between squared and exponential of Gaussian random variables

Assuming the random vector $[X \ \ Y]'$ follows a bivariate Gaussian distribution with mean $[\mu_X \ \ \mu_Y]'$, and covariance matrix $ \left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X ...
2
votes
4answers
55 views

Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists the same as $\lim_{n \to \infty} (1+1/n)^n$ exists

My expanded question: Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists as $z$ goes through real values the same as $\lim_{n \to \infty} (1+\frac{1}{n})^n$ exists as $n$ goes through ...
3
votes
2answers
30 views

For fixed $x \geq 0$, find $\lim\limits_{n\to\infty}1-\left(\frac{n-\lambda}{n}\right)^{nx}$

For fixed $x \geq 0$, find $\lim\limits_{n\to\infty}1-\left(\frac{n-\lambda}{n}\right)^{nx}.$ Clearly, the object of interest is ...
2
votes
3answers
60 views

Prove that $f(x)=exp(-x-e^{-x})$ for $x\in \mathbb{R}$ is a p.d.f and find the c.d.f.

Prove that $f(x)=exp(-x-e^{-x})$ for $x\in \mathbb{R}$ is a probability density function and find the cumulative density function. I think that by proving that $f(x)$ is a pdf, it should be fairly ...
1
vote
1answer
43 views

Indeterminant in Summation

I have the following summation: $$\sum_{k=0}^\infty(1-e^x)^k=\sum_{k=0}^\infty\sum_{j=0}^k\binom{k}{j}(-1)^je^{jx}$$ Then $$e^{jx}=\sum_{i=0}^\infty j^i\frac{x^i}{i!}$$ So, ...
1
vote
1answer
27 views

Trinomial Theorem Solution Verification

I have the following: $$\left(e^{\omega_0x}+e^{\omega_1x}+e^{\omega_2x}\right)^n$$ where $\omega_k=e^\frac{2ki\pi}{3}$. We can change the above to ...
2
votes
0answers
22 views

Exponential Growth/Compound Interest confusion?? [closed]

So I have this problem: $\$800$ is invested at a rate of $6.5\%$ and is compounded monthly for $5$ years. So I use the formula $A = 800(1 + 0.065/12)^{12 \cdot 5}$; however, I do not get the correct ...
5
votes
8answers
177 views

Prove: $(1-\frac{1}{d+1})^d>\frac{1}{e}$

I need to prove that $\left(1-\frac{1}{d+1}\right)^d>\frac{1}{e}$. I guess that I have to use that $\left(1+\frac{1}{n+1}\right)^n\rightarrow e$ for $n\rightarrow\infty$ or better $<e$ or ...