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

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1answer
23 views

How to prove the gaussian functions are linear independent?

Assume that I have N Gaussian functions with different means $\mu_i$ and variances $\beta_i$, How to prove $e^{-\beta_i(x-u_i)^2}$ are linear independent? 1$\le$i$\le$N
3
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1answer
24 views

Calculating a bound on the norm of a matrix exponential

The problem is this: Let A be a square $n \times n$ matrix, and define $$e^A=\sum_{k=0}^\infty \frac{1}{k!}A^k$$ Find a bound for $\lvert e^A \rvert$ in terms of $\lvert A \rvert$ and $n$. I was ...
8
votes
8answers
241 views

Why is the differentiation of $e^x$ is $e^x$?

$$\frac{d}{dx} e^x=e^x$$ Please explain simply as I haven't studied the first principle of differentiation yet, but I know the basics of differentiation.
0
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0answers
33 views

Transforming a logarithmic expression?

Do you know any nice way to rewrite $\log(1-e^{A})$?
1
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1answer
45 views

Solving an equation of the type $axe^{qx} + be^{rx} + cx + d = 0$

I need to solve an equation of the type, $axe^{qx} + be^{rx} + cx + d = 0$ I tried but couldn't solve it. Does anyone have an idea how to solve this(for x)? Thanks
11
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4answers
1k views

Exponential growth of cow populations in Minecraft

Minecraft is a computer game where you can do many things including farm cows. When fed wheat, cows in Minecraft breed with each other in pairs and produce one baby per pair. After about 20 minutes ...
1
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1answer
50 views

Find $(1+i)^i$ in simpler terms, without imaginary exponents. [duplicate]

I was asked to find $(1+i)^i$, I don't know what to do when there is an imaginary component in the exponent. since $1+i=\sqrt{2}e^{-\frac{1}{4}i \pi}$ then $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ ...
0
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0answers
18 views

Integral of a Poisson-Exponential Joint Distribution

I'm considering a joint distribution of Exponential variable $X$ and Poisson variable $Y$. the thing is, I can't figure out a way to derive the $pdf$ for such a distribution. I know that ...
1
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2answers
163 views

How one can implement the equation with “$i$” in it?

I have an equation: $$f(t)=c(e^{i2\pi\frac{n}{T}t}+e^{-i2\pi\frac{n}{T}t})$$ ...for $t\in(-\pi,\pi)$, and with $T=2\pi$. I have to draw a plot of the function $f(t)$ for $n\in\left \{0,1,2,5 \right ...
0
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2answers
58 views

simple exponential

This is taken from an example given in Gilbert Strang's Linear Algebra. The topic is not relevant, but I don't understand the following: $\left(1+\frac{0.06}{N}\right)^{5N} = e^{0.30}$ How is this ...
1
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0answers
20 views

What does it mean for a probability to “increase exponentially”?

In Wikipedia's description of the Metropolis algorithm, I see the phrase: The probability of rejection increases exponentially as a function of the number of dimensions. This obviously can't ...
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2answers
73 views

TAMS TOURNEMENT: exponential question (very hard) [closed]

What is the sum of the roots of $(2−x)^{2012} −x^{2012} = 0$ Any tips or solutions to this question would be greatly appreciated!
1
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0answers
34 views

Differentiation of $\exp(A)$

Let's say we have $${\sigma(\exp(a\cdot X^{-1} \cdot a^\mathrm{T}))}/{\sigma X}$$ when I know that the term inside the exponent is essentially a scalar. Should I differentiate according to ...
0
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0answers
15 views

Continuously Compound Interest [closed]

If $5,000 per year flows uniformly over an 8 year period and earns 3% interest, compounded continuously, then the present value, to the nearest dollar, is...?
0
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3answers
32 views

prove that this is a cauchy sequence

Prove that $\{x_n\}$ = $e^{-n}$ is a Cauchy sequence. I tried to prove this by proving that, For all $ϵ>0$, there is a positive $N$ s.t. for all $n>N$, $|e^{-n}|< ϵ$ For all $ϵ>0$, ...
1
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1answer
25 views

Given the density function: $\frac{1}{2}\exp\left(-\frac{x}{2}\right), \space x > 0$ find $P\left(\sum_{i=1}^{81}X_i > 170\right)$

Suppose that $X_1,X_2...X_{81}$ are independent random variable with the same probability density function $$\frac{1}{2}\exp\left(-\frac{x}{2}\right), \space x > 0$$ Find ...
7
votes
1answer
80 views

Does $\exp(2ir\pi)$ equal $1$? What's wrong?

Since $e^{ix}=\cos x+i\sin x$, thus $e^{2\pi i}=\cos2\pi+i\sin2\pi=1$ Now I take arbitrary real number $r$ then $e^{i2\pi r}=(e^{i2\pi})^r=1^r=1?$ But this cannot be true since $\cos2\pi ...
2
votes
2answers
86 views

Is the function $y = a^x + b$ exponential?

What exactly is an exponential function? Some of the sources at which I looked said that it's a function where the rate of change at $x$ $(f'(x))$ is proportional to the value at that point $(f(x))$, ...
0
votes
2answers
83 views

Solving $\sinh(ax) = bx$

I need an equation that expresses $x$ in function of $a$ and $b$. $$\sinh(ax) = bx$$ I'm a newbie in mathematics, and i don't know where to get help. I need to get this problem solved in order to ...
1
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3answers
33 views

Finding a closed-form formula for a sequence that is defined recursively

$$a_0 = 0, a_1 = 1 \quad \text{ and } \quad a_n = a_{n-1} + 2a_{n-2}\quad \text{ for }n\geq 2$$ a) Find $a_2,a_3,a_4,a_5$ b) Find a closed form-formula for $a_n$ I found the value to be ...
3
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1answer
56 views

How find steps to Question: $x = 10^{x-1}$. Answer: $x = 1$

I created an equation a bit ago where I knew the answer, but not how to solve it. Equation: $$x = 10^{x-1}$$ Answer: $x = 1$ I can not see to find any documentation related to this problem. I know ...
1
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0answers
16 views

Decisions on the order of integration with double integrals (when Deriving PDF via CDF) (Bank Problem)

Consider the following problem: Gandalf, Saruman and Radagast go to a bank together. There are two open counters which Gandalf and Saruman immediately go to get their service. Radagast goes to the ...
8
votes
3answers
391 views

Solution to differential equations $y(0)=1$ and $y^{(n)}=y+1$

When I was solving some differential equations, I asked myself the following: Is there a function has the following: $$y'=y+1$$ $$y''=y+1$$ $$y'''=y+1$$ $$......$$ $$......$$ If the initial value is ...
0
votes
1answer
27 views

Integral of Euler's formula

Why is $=\int\limits_{-\infty}^{\infty}\cos(-tx)dF(x)+i\int\limits_{-\infty}^{\infty}\sin(-tx)dF(x)=\int\limits_{-\infty}^{\infty}\cos(tx)dF(x)-i\int\limits_{-\infty}^{\infty}\sin(tx)dF(x)$? I know ...
0
votes
1answer
19 views

How to model function with unknown exponents?

I know the Cobb-Douglass function which describes the production quantity: $$Q(K, L) = A \cdot K^\alpha \cdot L^\beta$$ Also I do know multiple assignments of K ...
0
votes
0answers
11 views

compounding signups to a website

I am interested in figuring out how many people will sign up to a website. The figures are: 2000 people sign up per hour, that figure multiplies by 1.5 every day - so: Day 1 - 2000 per hour Day 2 - ...
1
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0answers
14 views

Can an integral in the exponent of an exponetial function be written as a product?

I am asked to simplify/calculate the following integral: $$\frac{\int d^3q \exp \left( -\beta C \int d^3q [u(q) u(-q)|q|^2] \right)|u(q)|^2}{\int d^3q \exp \left( -\beta C \int d^3q [u(q) u(-q)|q|^2] ...
0
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1answer
20 views

Fourier transform of a function with sine [duplicate]

I don't know how to compute the Fourier tranform of this function: $f(x) = \frac{\sin \pi a x}{\pi x}$ I know that $\frac{\sin \pi a x}{\pi x} = \frac{e^{i \pi a x} - e^{- i \pi a x}}{2i \pi x}$ ...
0
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2answers
49 views

How to get last digit of $7^{7^7}$

I want to find the last digit of $7^{7^7}$. I found out already that $7^7$ (mod 10) last digit is 3. But how do I use that to get the last digit of the whole thing? Thanks
1
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2answers
67 views

Transforming the exponential function of a sum into the sum of functions

Is there a way to transform the function $$\exp(A+B+C),$$ where $\exp(\cdot)$ is the exponential function, into a sum $$f(A)+f(B)+f(C)?$$
4
votes
2answers
95 views

$[1-x_n]^{n}\to 1$ implies that $n x_n\to 0$ as $n\to\infty$?

As the title says $[1-x_n]^{n}\to 1$ implies that $n x_n\to 0$ as $n\to\infty$? We know that $\left(1+\frac{x}{n} \right)^n \to e^x$ as $n\to\infty$. This implies (not sure why) that ...
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votes
2answers
63 views

Differentiability proof of exponential function $\sum_{n=0}^ \infty \frac{x^n}{n!}$

$$f(x)=\sum_{n=0}^ \infty \frac{x^n}{n!}$$ I want to prove that $f$ differentiable on $x$ in $[0,1]$. I am not clear with using the definition of differentiability. I can prove it is ...
0
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1answer
37 views

Express recurrence in closed form

I am having trouble understanding the process of expressing the following recurrence in its' closed form. First of all, I do not really understand what "closed form" means. If someone could elaborate ...
1
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1answer
34 views

Finding nth derivative of an exponential function and its value at the origin.

I have a function defined as $f(x) = e^{-\frac{1}{x^2}}, $if $ x\ne0$; $0$ if $x =0$. where $f:[0,\infty) \to \mathbb{R}$ I am asked to prove the following: (a) that the nth derivative is of the ...
1
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1answer
13 views

Application of Borel-Cantelli for sequence of two parameters

Let $(A_{m,\ell})_{\ell \geq 0, m \geq 0}$ be a sequence of events in some probability space. How to show by using Borel Cantelli that, if $$\sum_{\ell \geq 0, m \geq 0} P(A_{m,\ell}) < \infty,$$ ...
3
votes
2answers
40 views

Relation on $\int_1^x\exp{t^2}dt$

Could you give me some leads to show the following relation : $$\forall x>0, \int_1^x\exp{t^2}dt = \frac{1}{2x}\exp{x^2} + \frac{1}{4x^3}\exp{x^2} - \frac{3}{4}\mathbb{e}+ \frac{3}{4} \int_1^x ...
0
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1answer
53 views

If $x^a + x^b = x^c + x^d$ how do $a ,b , c , d$ relationship are?

I used to solved these equation style and it's accidentally found an answer from matching $a, b, c,$ and $d$ relationship when $x^a + x^b = x^c + x^d $ (I assume that $ab = cd$) and found that's ...
2
votes
2answers
46 views

Find all real $a$ such that $6a^2+3=9^a$

Find all real $a$ such that $6a^2+3=9^a$ The problem seems to be very easy, but now i can't see an easy way to find if there are other roots than $1$. Tried using the derivative but that didn't help ...
1
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1answer
50 views

Power series $e^{-x^2}$

How would I create a power series of $f(x)=e^{-x^2}$ around $x_0=1$ without using a Taylor series? I need to know this for my upcoming exam so I would be really grateful to anyone who could show me ...
0
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1answer
33 views

exponential term evaluation doesn't make sense in this example

I am studying for my final and doing some practice questions, but I am confused by something: Here the solution says k at 0 we get N/2, but there is no way that answer is correct. If k is at 0 the ...
0
votes
1answer
19 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
0
votes
5answers
145 views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
1
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3answers
73 views

The limit of $e^{\frac{1}{x^2 + y^2}}$ as $(x, y) \to (0, 0)$

$$\lim_{(x, y) \to (0, 0)} e^{\frac{1}{x^2 + y^2}}$$ This really should be a simple limit question, I've done similar things many times before, but I'm very out of practice with limits and cannot for ...
0
votes
1answer
14 views

The time to fix a TV,is an exponential random variable with parameter $\lambda=\frac{1}{2}$.What is the probability that a fix take more than 2 hours?

I got the following question to solve: The time to fix a TV in hours, is an exponential random variable with parameter $\lambda = \frac{1}{2}$. What is the probability that a fix take more than 2 ...
2
votes
1answer
36 views

Rewriting $e^{-a|t|}$

here I have to prove the fourier transform of $e^{-a|t|}$ , the beginning of the proof is to rewrite $e^{-a|t|}$ as: $e^{-at} U(t) + e^{at} U(-t)$, I know how to continue the proof starting from this ...
0
votes
3answers
26 views

How did we come to this inequality $|\frac{e^x-1}{x}-1|\leq|\sum_{n=1}^{\infty}{\frac{|x|^n}{(n+1)!}}|$

How did we come to this inequality $|\frac{e^x-1}{x}-1|\leq|\sum_{n=1}^{\infty}{\frac{|x|^n}{(n+1)!}}|$. Using $e^x=\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$. I have this inequality in the proof of ...
5
votes
2answers
59 views

Prove or disprove that $p_n > e^{p_n - p_{n-1}}$ for large enough $n$.

Let $p_n$ denote the $n$-th prime. Prove or disprove that for large enough $n$ we have $$p_n > e^{p_n - p_{n-1}}.$$ The inequality trivially holds for all the twin primes larger than $7$. With ...
0
votes
1answer
28 views

How to obtain and graph a function that first grows exponentially and then decays exponentially?

I would like to know what the equation of a curve is if it grows exponentially (let's say it doubles each time). This would be: $f(x)=2^x$ But then I would like the line to exponentially decay (let ...
0
votes
1answer
17 views

Rate of convergence of the difference of two exponentials

I would like to find the convergence rate of the following function: $$f(x) = |e^{-ax}-e^{-bx}|,$$ with $a,b>0$ and $x\to+\infty$. By finding the convergence rate, I mean finding the largest ...
1
vote
1answer
37 views

Is the exponential map for $\text{Sp}(2n,{\mathbb R})$ surjective?

For $\mathfrak{g} := {\mathfrak s}{\mathfrak p}(2n,\mathbb{R})$ and $G = \text{Sp}(2n,{\mathbb R})$, is the exponential map \begin{equation} \text{exp} : \mathfrak{g} \to G \end{equation} surjective? ...