Exponential Definition and 1000 Threads

I know that the z for one particle would be e^-βμB(-S) + e^-βμB(-S+1) +...+ e^-βμB(S) but I have no idea how to equate this to the solution with two sinh in a quotient.

since euler's identity states e^pi*i+-1 is it okay to write down e^pi*= e * -1/e = -1 and would for integers the same rule apply where a^pi*i= a * -1/a = -1 or is it only for the constant e? this might be a stupid question, but i cant find an answer and im curious

My attempt and solution : $$2^x=3^y+509\Longrightarrow 2^x-512=3^y+509-512\Longrightarrow 2^x-2^9=3^y-3$$ $$\Longrightarrow 2^9(2^{x-9}-1)=3(3^{y-1}-1)$$ $$\Longrightarrow (x,~y)=\boxed{(9,~1)}$$ İs there any solution?

Hello frens, How should one approach this sort of integral? Any tips would be appreciated. Let's say we have $$ \int_{(1)}^{(2)}\exp\left[ a+b\exp\left[ f(x) \right] \right]dx$$ ...where the limits of integration are not important. Any tips? Thanks!

Given a vector of numbers, say [exp(-a t) ] for t - [1, 2, 3, 4, 5] and choose maybe a = -2.4, how can I approximate -2.4 from using Laplace transform methods? I know you can use regression for this, but I'd like to know the Laplace transform (or Z-transform since it is discrete) approach.

I am looking for a closed form solution to an integral of the form: $$ \int_0^\infty \frac{e^{-Du^2t}u \sin{ux}}{u^2+h^2} du $$ D, t, and h are positive and x is unrestricted. I have tried everything, integration by parts, substitution, even complex integration with residue analysis. I've...

In the book "Calculus by Michael Spivak" it says that a^x = e^xln(a) is a definition. And I am not convinced to accept this as true without a proof.

part (a) Asymptote at ##x=0.5## part (b) ##\dfrac{e^x}{4x^2-1}= -2## ##e^x=2-8x^2## ##2e^x= 4-16x^2## ##16x^2=4-2e^x## ##x^2= \dfrac{4-2e^x}{16}## ##x=\dfrac{\sqrt{4-2e^x}}{4}## part (c) ... ##x_{2}=0.2851## ##x_{3}=0.2894## ##x_{4}=0.2881## ##x_{5}=0.2885## ##x_{6}=0.2884##...

Find the limit $$\lim_{x\to \infty} x\left[\frac{1}{e} - \left(\frac{x}{x+1}\right)^x\right]$$

Hey all, I saw a formula in this paper: (https://arxiv.org/pdf/physics/0011069.pdf), specifically equation (22): and wanted to know if anyone knew how to derive it. It doesn't seem like a simple application of BCH to me. Thanks.

I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am...

Preface: I have not done serious math in years. Today I tried to do something fancy for a game mechanic I'm designing. I've got an item with a variable power level. It uses x amount of ammo to produce f(x) amount of kaboom. Initially it was linear, e.g. fL(x) = x, but I didn't like the scaling...

Hi all, I have another second order ODE that I need help with simplifying/solving: ##p''(x) - D\frac{e^{\gamma x}}{A-Ae^{\gamma x}}p'(x) - Fp(x) = 0## where ##\gamma,A,F## can all be assumed to be nonzero real numbers and ##D## is a purely nonzero imaginary number. Any help would be appreciated!

Solve $4^x+3^x=12$ without using CAS?

Hi, my question regard the possibility to consider a generalization on the product integral (of type II). The product integral is defined in analogy to the definite integral where instead the limit of a sum there is a limit of a product and, instead the multiplication by ##dx## there is the...

I did only the the first three prop. And on a means we have, on pose or posons means let there be , propriétés means properties, alors meand then. I apologize i am a french native speaker and i am busy so i cannot rewrite this in entirely english

This is a pedagogical /time management / bandwidth / tradeoff question, no argument that learning the complex exponential derivation is valuable, but is it a good strategy for preparing for first year Calculus? my 16YO son is taking AP precalc and AP calc next year and doing well, but struggled...

\tiny{s464\\6.6} solve the exponential equation. $3^x-14\cdot 3^{-x}=5$ Rewrite $\quad 3^x+\dfrac{14}{3^x}=5$ $\times$ $\quad 3^x \quad 3^{2x}+14=5\cdot 3^x$ quadratic $\quad 3^{2x}-5\cdot3^x-14 =0$ Factor $\quad (3^x-7)(3^x+2)=0$ Discard $\quad 3^x =7$ hence $\quad...

It is my second "energy state diagram problem" and I would want to know if I am thinking correctly. First I have done some function analysis to get a glimpse of the plot: - no roots but ##\lim\limits_{x\to-\infty}U(x)=\lim\limits_{x\to+\infty}U(x)=0## - y interception: ##U(0)=-U_0## - even...

GRAPH WITH VALUES: Sorry I have a small dilema, I don't know if this is a exponential or polynomial function. I'd think its exponential but it doesn't have same change of factors.

My trial : I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##. So any help about the right answer is appreciated!

Does this curve look quadratic or exponential

For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that $$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$ which generalizes what I initially...

Hi everybody We can't differentiate ##x^x## neither like a power function nor an exponential function. But ##x^x=e^{x\mbox{ln}x}##. So ##\dfrac{d}{dx}x^x=\dfrac{d}{dx}e^{x\mbox{ln}x}=x^x(\mbox{ln}x+1)## And here comes the doubt: prove the domain of ##x^x## is ##(0, +\infty)## Why is only...

A bacterial population x is known to have a growth rate proportional to x itself. If between 12 noon and 2:00 pm, the popilation triples. 1. What is the growth rate pf the given problem? 2. Is the number of bacterial population important to the problem? 3. At what time should x become 100...

I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time. The time-domain form is, s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}} Here, \mu , \Omega and \lambda are constants. A quick...

I know that e^-ix = cos(-x)-isin(x), but if we have e^-iwx does that equal cos(-wx) - isin(wx)? Thanks

Just a quick question: Does anybody know if there is a closed-form solution to this rather simple-looking definite integral? ##F(\lambda) = \int_0^{\infty} \dfrac{e^{-x}}{1 + \lambda x} dx## If ##\lambda > 0##, it definitely converges. It has a limit of 1 as ##\lambda \rightarrow 0##. But it...

Ok, first I tried to show that ##A = \left \{a^{r}|r\in\mathbb{Q},r<x \right \}## does not have a maximum value. Assume ##\left\{ a^{r}\right\}## has a maximum, ##a^{r_m}##. By this hypothesis, ##r_{m}<x## and ##r_{m}>r\forall r\neq r_{m}\in\mathbb{Q}##. Consider now that ## q\in\mathbb{Q}|q>0##...

Good day and here is the solution, I have questions about I don't understand why when in the taylor expansion of exponential when x goes to infinity x^7 is little o of x ? I could undesrtand if -1<x<1 but not if x tends to infinity? many thanks in advance!

I'd be grateful for any formulation that describes this statistical process

Suppose we construct a set, adding at each step a polynomial number of elements. My impression that after we do countably infinite number of steps, the set will have countably infinite cardinality. But what happens if we add exponential number of elements each step? For instance, on step 0 we...

I try to proof it but i got stuck right here, i want your opinions Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?

I know it's probably an easy one, but I'm getting confused on how to treat that exponential numerator in order to escape from the indeterminate form ##\frac{0}{0}##

Hi, I was just thinking about different ways to use the Fourier transform in other areas of mathematics. I am not sure whether this is the correct forum, but it is related to probability so I thought I ought to put it here. Question: Is the following method an appropriate way to calculate the...

1) Since I want at least ##6## flights to come within ##2## hours, then the time interval between each should be, at worse, ##2/6=1/3## hours, and the probability is ##P(X\leq1/3)=1-e^{-1/3}##. 2) The probability that at best 5 airplanes arrive at the airport is...

Using the boundary conditions where psi is 0, I found that k = n*pi/a, since sin(x) is zero when k*a = 0. I set up my normalization integral as follows: A^2 * integral from 0 to a of (((exp(ikx) - exp(-ikx))*(exp(-ikx) - exp(ikx)) dx) = 1 After simplifying, and accounting for the fact that...

Starting from the definition of a matrix exponential as a power series, how would we show that ##(e^A)^n=e^{nA}##? I know how to show that if A and B commute then ##e^Ae^B = e^{A+B}## and from this we can show that the first identity is true for integer values of n, but how can we show it’s...

a) This looks somewhat like a pendulum problem (length ##\ell/2##). I reasoned there will be a clockwise rotation, and that the acceleration is due to the force of...

Summary:: Hello, my question asks if the complex exponential equation 4e^(-j) is equal to 4 ∠-180°. I tried to use polar/rectangular conversions: a+bj=c∠θ with c=(√a^2 +b^2) and θ=tan^(-1)[b/a] 4e^(-j)=4 ∠-180° c=4, 4=(√a^2 +b^2) solving for a : a=(√16-b^2) θ=tan^(-1)[b/a]= -1 b/(√16-b^2)=...

https://www.asi.edu.au/wp-content/uploads/2015/03/PhysicsASOE2013soln.pdf Q12 e) Working backwards, P = Ae^kt form, i.e. EAts = Emin e^(ln2/τ x t). Not sure how they get this formula in the first place with these values.

Consider two harmonic oscillators, described by annihilation operators a and b, both initially in the vacuum state. Let us imagine that there is a coupling mechanism governed by the Hamiltonian H=P_A P_B, where P_i is the momentum operator for the oscillator i. For example P_A =...

Hi, I want to know how the highlighted steps are arrived at in the first page. What are \(R_X (y), R'_X (y),F'_X (0) ? \)How \(R_X (0) = 1 ?\) Solution to differential equation should be \(R_X (y)=K*e^{\int{R'_X (0) dx}}\) But it is different. How is that? What is $-R'_X...

From the sketch, I know that this function is an even function. So, I simplify the Fourier transform in the limit of the integration (but still in exponential form). Then, I try to find the exponential FOurier transform. Here what I get: $$g(a)=\frac{2}{2\pi} \int_{0}^{\infty} e^{-x} e^{-iax}...

I managed to expand a general expression from the alternatives that would leave me to the answer, that is: I will receive the alternatives like above, so i find the equation: C = -sina, P = cosa So reducing B: R: Reducing D: R: Is this right?

The summary pretty much explains my question. I know that ##\left[ A, e^B \right]=0## if ##[A,B]=0## (and can prove it), but I can't figure out how to prove if it is or is not an "if and only if" statement. Thanks in advance!

Hello, i am testing batteries for a project of mine. I first measure the voltage on the battery (OCV, Open Circuit Voltage), and then i place a load resistor over the battery. This results in a voltage drop with a exponential decrease in voltage like in the image. But i am working with a...

Hi, I think this is a nitpicking question, but oh well let me hear your inputs. Actually I tried to solve this question straightforwardly, by Taylor expanding the exponential and showing that: \textbf{A}^n = \begin{pmatrix} a^n & nba^{n-1} \\ 0 & a^n \end{pmatrix} i.e. e^{\textbf{A}t} =...

Hi there, I have tried to do these questions but I don't understand. Any help would be appreciated!