Exponential Definition and 1000 Threads

The exponential of a complex number is a complex number. Does this extent to the quaternions and the octonions? Does the exponential of a quaternion give a quaternion? Does the exponential of an octonion give an octonion? Thanks.

I’ve read about the exponential map that for Lie groups the exponential map is actually the exponential function. But the exponential map is based on the geodesic ODE, so you need Christoffel symbols and thus the metric. But usually nobody gives you a metric with a Lie group. So how can I get...

How to Integrate it::: ∫e^(ax²+bx+c)dx Or in general e raised to quadratic or any polynomial. I am trying hard to recall but I couldn't recall this integration. I tried using By-parts but the integration goes on and on.

Homework Statement In my book, there is a formula that gives the amount (in grams) of Radium in a jar after t years (100 grams were initially stored): R = 100⋅e-0.00043⋅t The book asks me to sketch the graph of the equation. I decided to find a point where the time elapsed equals the...

Hello,I was wondering. Is the exponential function, the only function where ##y'=y##. I know we can write an infinite amount of functions just by multiplying ##e^{x}## by a constant. This is not my point. Lets say in general, is there another function other than ##y(x)=ae^{x}## (##a## is a...

This is not a physics question. Each time a ball bounces it will bounce to, let's say 75% of its previous height. (I am not interested in the time, energy or velocity, of the ball.) So if we drop it from 100 cm it will bounce back up to 75 cm, and on the next bounce it goes up to 56.25 cm and...

Homework Statement S = 1+ x/1! +x2/2! +x3/3! +...+xn/n! To find S in simple terms. Homework Equations None The Attempt at a Solution I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.

Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus...

There is nothing wrong with the well known $$e^{i\theta}=\cos\theta+i\sin\theta$$ for real ## \theta## but what about $$\int_{-\infty}^\infty~e^{i\theta(p)}\mathrm{d}p=\int_{-\infty}^\infty~\cos\theta(p)\mathrm{d}p+i\int_{-\infty}^\infty~\sin\theta(p)\mathrm{d}p$$ I have been trying to use...

Hello all, I am trying to solve the following problem: In the given graph, we see the function: \[f(x)=ka^{-x} , x\geq 0\] 1) Find k and a 2) Find x1 3) Show that an increase of 2 units in x brings a 50% reduction in the value of the function f. I have tried solving it, but taking two...

I can't understand the exponential function fit for this set of data works well: ExpDec2 exponential function fit 0 3,04 10 2,77 20 2,52 30 2,27 40 2,09 50 1,92 60 1,75 70 1,62 80 1,51 90 1,43 100 1,36 110 1,29 120 1,24 130 1,19 140 1,14 150 1,09 160 1,05 170 1,02 180 0,99 190 0,97 200 0,95 210...

I have a model where the probability is spherically symmetric and follows an exponential law. Now I need the probability density function of this model. The problem is the singularity at the origin. How can I handle this? P(r) = ∫p(r) dr = exp(-μr) p(r) = dP(r)/(4πr²dr) One way I tried to...

Say I have a function that represents the population growth of a certain country that can be written as ##f\left(x\right)=1.25\left(1.012\right)^t##, where t is in years. I can graph this function and it will look a certain way exponentially. I've looked at a ton of examples, and they're all...

1. The problem statement, all variables, and given/known data Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) =...

I'm taking an online class and I was doing some very simple exponential equations when this was thrown at me, and I have no clue how to solve it. 27^x=1/√3

Homework Statement I found an answer on the internet for this problem, but I'm not sure on one of the steps. The solution says, "Take ln of both sides to get rid of Ae. If we do that, then the right side will be ln(Ae^t/T). I don't see how using ln will get rid of Ae? Homework Equations Refer...

Let f(z) = $e^{e^{z}}$ . Find Re(f) and Im(f). I don't know how to deal with the exponential within an exponential. Does anybody know how to deal with this?

Hey! :o We consider the exponential distribution. I want to show that $$\mathbb{P}\left (\left |X-\frac{1}{\lambda}\right |\leq \lambda \right )\geq \frac{\lambda^4-1}{\lambda^4}$$ I have shown so far that \begin{align*}\mathbb{P}\left (\left |X-\frac{1}{\lambda}\right |\leq \lambda \right...

Hello all, I have a data which look like reversed exponentially modified Gaussian (EMG) function and interested to fit the data with with reversed EMG function. After searching on internet I found the EMG function, which is given below...

Hey! :o I want to show that $\displaystyle{\lim_{x\rightarrow \infty}\frac{e^x}{x^{\alpha}}=\infty}$ and $\displaystyle{\lim_{x\rightarrow \infty}x^{\alpha}e^{-x}=0}$ using the exponential series (for a fixed $\alpha\in \mathbb{R}$). I have done the following: $$\lim_{x\rightarrow...

Homework Statement Homework EquationsThe Attempt at a Solution [/B] Det( ## e^A ## ) = ## e^{(trace A)} ## ## trace(A) = trace( SAS^{-1}) = 0 ## as trace is similiarity invariant. Det( ## e^A ## ) = 1 The answer is option (a). Is this correct? But in the question, it is not...

Hi, I'm new to this forum. This semester I took Calculus I and just took the final yesterday. There were a few questions that were unexpected that I didn't know how to handle. This integral has got me stumped.\int_{0}^{1} e^{x}/(1 + e^{2x}) \,dx The techniques I know at this point include u...

Homework Statement Homework EquationsThe Attempt at a Solution I understand there is no vertical asymptotes and can usually get the horizontal ,but can't understand with the exponential.

Homework Statement Is there a way to simplify the following expression? ##[cos(\frac {n \pi} 2) - j sin(\frac {n \pi} 2)] + [cos(\frac {3n \pi} 2) - j sin(\frac {3n \pi} 2)]## Homework Equations ##e^{jx} = cos(x) + j sin(x)## The Attempt at a Solution ##cos(\frac {n \pi} 2)## and...

Hey! :o I am looking at the following: Show that $\displaystyle{\text{exp}(1)=\sum_{k=0}^{\infty}\frac{1}{k!}=e}$ with $\displaystyle{e:=\lim_{n\rightarrow \infty}\left (1+\frac{1}{n}\right )^n}$. Hint: Use the binomial theorem and compare with the partial sum $s_n$ of the series...

Hey! :o I am looking at the following: 1) A machine produces $100$ gram chocolate. Due to random influences, not all bars are equally heavy. From a long series of observations it is known that the mass X of a chocolate is distributed normally with parameters $\mu = 100$g and $\sigma = 2.0$g...

Homework Statement I am trying to solve an equation. 128^b - 127^b = 147.058. Homework EquationsThe Attempt at a Solution I have tried numerical methods like Bisection method and Newton-Raphson, but I need analytical solution. Thank you.

Homework Statement Find the trace of a ##4\times 4## matrix ##\mathbb U=exp(\mathbb A)##, where $$\mathbb A = \begin {pmatrix} 0&0&0&{\frac {\pi}{4}}\\ 0&0&{\frac {\pi}{4}}&0\\ 0&{\frac {\pi}{4}}&0&0\\ {\frac {\pi}{4}}&0&0&0 \end {pmatrix}$$ Homework Equations $$e^{(\mathbb A)}=\mathbb P...

I'm kinda just hoping someone can look over my work and tell me if I'm solving the problem correctly. Since my final answer is very messy, I don't trust it. 1. Homework Statement We're asked to find the Fourier series for the following function: $$ f(\theta)=e^{−\alpha \lvert \theta \rvert}}...

Homework Statement Show that if ##λ##and ##V ## are a pair of eigenvalue and eigenvector for matrix A, $$e^Av=e^λv$$ Homework Equations ##e^A=\sum\limits_{n=0}^\infty\frac{1}{n!}A^n## The Attempt at a Solution I don't know where to start.

For reference: Engineering Circuit Analysis, Hayt & Kemmerly, 4th ed, 1986, page 345. Given a series RL circuit, the phasor current is I(s) = Vm/(R+sigma L), where S = sigma, w (omega) = 0. Thus we are dealing with only a exponential forcing function. Obviously I(s) goes to zero as sigma...

Homework Statement Show that ##e^x = x## does not have any solutions, and show that ##\sec x = e^{-x^2}## has only one solution. Homework EquationsThe Attempt at a Solution Here is my proof of the first proposition: Since ##e^x## is concave up on ##\Bbb{R}##, it must lie above all of its...

http://www.nat.vu.nl/~tvisser/nonexponential.pdf I always thought that an unstable system will decay exponentially but I recently learned that it obeys the quadratic law in the short time and power law in the long time. Can somebody tell me the equation that governs the quadratic law and power law?

Hi, I'm kind of stuck with this theorem stating that: if A is an unipotent matrix, then exp(log A) = A and also if X is nilpotent then log(exp X) = X Does anyone know any good approaches to prove this? I know that for unipotent A, logA will be nilpotent and that for nilpotent X, exp(X)...

Homework Statement Accidents at a busy intersection follow a Poisson distribution with three accidents expected in a week. What is the probability that at least 10 days pass between accidents? Homework Equations F(X) = 1- e-λx μ = 1/λ The Attempt at a Solution Let x = amount of time between...

Homework Statement Can this function be integrated analytically? ##f=\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32 \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right),## where ##a##, ##b## and ##L## are some real positive...

From the book Calculus made easy: "This process of growing proportionately, at every instant, to the magnitude at that instant, some people call a logarithmic rate of growing." From Wikipedia: "Exponential growth is feasible when the growth rate of the value of a mathematical function is...

The square of X is equal to 4 times the square of Y. If X is one more than twice Y, what is the value of X?

I can solve equations like 4^(x) = 16 or 5^(x + 1) = 25. However, there are exponential equations that a bit more involved. The following equation has two exponentials on the left side. Solve for x. 5^(x - 2) + 8^(x) = 200

Homework Statement Evaluate each of the following expressions without using a calculator. 1) log216√8Solve for the unknown value in each of the following equations without using a calculator. 2) 3(x+4)−5(3x)=684 3) 7(42x)=28(4x) Homework Equations Exponent law for multiplication The...

Hi all, Can anyone teach me this problem ? Thanks The life of a tiger is exponentially distributed with a mean of 15 years.If a tiger is 10 years old, what is the expected remaining life of the tiger? A 5 years B 10 years C 15 years D Longer than 15 years

I have a large quantity N, which starts off equal to a determinable value N0. Over a short time ∆t, the value of N changes by -∆t*(B*N - C) where B and C are determinable constants. Am I correct in thinking I can turn this into: dN/dt = -(B*N - C) How do I get this into a formula for N at...

Homework Statement Solve ## \frac{d^2y}{dt^2} + \omega^2y = 2te^{-t}## and find the amplitude of the resulting oscillation when ##t \rightarrow \infty ## given ##y=dy/dt=0## at ##t=0##. Homework EquationsThe Attempt at a Solution I have found the homogenious solution to be: ##y_h = A\cos\omega...

Given that an unstable particle has a constant probability of decaying per unit time, why is it said that its chance of surviving falls exponentially?

Homework Statement "Prove: ##∀n∈ℕ,7|[3^{4n+1}-5^{2n-1}]##" Homework EquationsThe Attempt at a Solution (1) "We take the trivial case: ##n=1##, and notice that ##3^5-5=238## and ##7|238## because ##7⋅(34)=238##." (2) "Now let ##n=k## for some ##1<k∈ℕ##. Then we assume that...

It's been a long time since I've worried about this - but could someone help me make Pr(x) the subject (I can't remember if it's possible, if it's not, I'd love a brief explanation): T = S [(1-Pr(x))^N] + Pr(x) Thanks in advance!

Is there a closed form for the constant given by: $$\sum_{n=2}^\infty \frac{Ei(-(n-1)\log(2))}{n}$$ (Where Ei is the exponential integral)? Could we generalize it for: $$I(k)=\sum_{n=2}^\infty \frac{Ei(-(n-1)\log(k))}{n}$$ ? My try: As it is given that k will be a positive integer, I have...

Homework Statement Let A be a Hermitian matrix and consider the matrix U = exp[-iA] defined by thr Taylor expansion of the exponential. a) Show that the eigenvectors of A are eigenvectors of U. If the eigenvalues of A are a subscript(i) for i=1,...N, show that the eigenvalues of U are...

Find x, if 2^x =8x.

Homework Statement they say 1. ##e^{ln x}= x ## and 2.##e^-{ln(x+1)}= \frac 1 {x+1}## how can we prove this ##e^{ln x}= x ## and also ##e^-{ln(x+1)}= \frac 1 {x+1}##? Homework EquationsThe Attempt at a Solution let ## ln x = a## then ##e^a= x, ## a ln e= x,## →a= x, where ## ln x= x