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

learn more… | top users | synonyms (1)

0
votes
1answer
72 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
61 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
19 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
125 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
68 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
34 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
73 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
576 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
139 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
68 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
61 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
65 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
43 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
58 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
61 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 ...
2
votes
2answers
114 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
91 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
89 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
144 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
92 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 ...
1
vote
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
235 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
votes
0answers
35 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
54 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
57 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
32 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
44 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
30 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
23 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
181 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 ...
4
votes
2answers
257 views

If $x$ is rational, can $\log(1-x)/\log x$ be algebraic?

If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to an algebraic number, say $g$? $$\frac{\log(1-x)}{\log x} = g$$
3
votes
1answer
64 views

Why is the solution to $y' = y^n$ always in polynomial form EXCEPT when $n = 1$?

Could someone explain (intuition-wise) why the differential equation $$y' = y^n$$ for $n \in \mathbb{N}$ seems to always some kind of polynomial solution (or a ratio of polynomials, etc.) except ...
1
vote
3answers
42 views

Exponential distribution wait time probability

I would like to check my answer, I have been asked to work out the probability of value greater then 10 given an exponential distribution with a mean of 10. My intuition would be that this is equal ...
5
votes
7answers
158 views

Why does $e^{-(x^2/2)} \approx \cos[\frac{x}{\sqrt{n}}]^n$ hold for large $n$?

Why does this hold: $$ e^{-x^2/2} = \lim_{n \to \infty} \cos^n \left( \frac{x}{\sqrt{n}} \right) $$ I am not sure how to solve this using the limit theorem.
3
votes
3answers
61 views

Finding closed form expression for $ \sum_{n=1}^{\infty} n\left( e^{-n(n-1)} - e^{-n(n+1)}\right)$

I want to find a closed form expression for: $$ \sum_{n=1}^{\infty} n\left( e^{-n(n-1)} - e^{-n(n+1)}\right)$$ This isn't a homework problem, more of something that emerged as a side problem from ...
2
votes
3answers
50 views

Exponent $x$ tending to infinity

When an expression has $x$ tending to infinity and it's an exponent, like: $$\lim\limits_{x \to \infty} \frac 1{2^x},$$ is there some way to get a solid value for this limit (hopefully $0$, but at ...
1
vote
0answers
33 views

Exponential and Hyperbolic Functions Before Power Series

Is there a textbook that covers the exponential function and the possibly the hyperbolic functions from a precalculus geometrical viewpoint? I'm essentially asking for something like a trigonometry ...
0
votes
2answers
47 views

Closed form expression for Integral of exponential function

How to compute the following integral \begin{equation} \int_{n_0}^{\infty} \exp(- c x^\lambda) dx, \end{equation} when $0 < \lambda \leq 1.$
1
vote
2answers
36 views

Which is greater ? Sum of odd power terms or even power terms in the exponential Taylor series?

I came across this question, in a book. Define $f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{{(2n+1)}!} $ and $ g(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{{(2n)}!} $, where x is a real number. Then, ...
7
votes
2answers
89 views

Show that $p_n(a)\neq 0$ if $|a|=n$

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then ...
1
vote
1answer
287 views

Limit involving tetration

Let the notation be $a^{\wedge\wedge}b = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration. My mentor conjectured the following: Let $n$ be a positive integer, then let $A(n)$ be ...
1
vote
2answers
46 views

Link between exponential distribution and poisson probability mass function

Customers arrive randomly and independently at a service window, and the time between arrivals has an exponential distribution with a mean of 12 minutes. Let X equal the number of arrivals per hour. ...
0
votes
0answers
53 views

Normal Exponential Convolution proof

I am seeing a scientific paper where they explain background correction modelled as two random independent variables one with exponential distribution and the other with normal distribution. ...