Exponential Definition and 1000 Threads

the integral is: and according to mathematica, it should evaluate to be: . So it looks like some sort of Gaussian integral, but I'm not sure how to get there. I tried turning the cos function into an exponential as well: however, I don't think this helps the issue much.

Prove that $3^n\ge(n+3)^3$ for any natural number $n\ge6$.

Hello everyone, can anybody solve this limit? This is really tough one for me, thank you in advance.

##\int \frac{e^x (2-x^2)}{(1-x) \sqrt{1-x^2}} dx## I tried using substitution x = sin θ but still can not solve it. I guess I have to get rid the term ex but do not know how Thanks

Hi all. A problem has arisen whereby I need to maximize a function which looks like $$ f(A) = \mathbf{w}^T \left[\int_0^t e^{\tau A} M e^{\tau A^T} d\tau \right]^{-1} \mathbf{w} $$ with respect to the nxn matrix A (here, M is a covariance matrix, so nxn symmetrix and positive-definite, w is an...

I was just playing with the integral ##\int e^{ixa}dx## when I found something interesting. If you integrate from ##x = m2\pi/a## to ##x = n2\pi/a## where ##m## and ##n## are any two integers, the integral equals zero. On one hand, as we can in principle choose whatever values we like for ##m##...

From integration by parts, and using y(10) = 0, I get the equation ##2e^{3t-30} = \frac{|y-2|}{|y+1|}.## Let ##f(t) = 2e^{3t-30}##. Since it's for t>10, f(10) = 2, and we have ##2=\frac{|y-2|}{|y+1|}##. Depending on the sign I choose to use, I get either that y=-4 or y =0. Since ##t: 10...

Homework Statement: Solve the equation: (2x+5)^(x+1)=729 Homework Equations: (2x+5)^(x+1)=729 Here's my work: (2x+5)^(x+1)=729 x+1=ln(729)/ln(2x+5) x+1=ln(729/(2x+5)) e^(x+1)=e^ln(729/(2x+5)) e^(x+1)=729/(2x+5) From here, I don't know what to do. All I know is that the answer is x=2. But I...

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}e^{-n}\sum^{n}_{k=0}\frac{n^k}{k!}$

I'm just trying to follow the below And I understand all, I think, except what's happened to the term when A hits 1: [A,1] ? If I'm correct basically we're just hitting on the first operator so reducing the power by one each time of the operator in the right hand bracket thanks

I know that both 3^x and e^x can't be 0 for real x. Then x^2-4=0 is the only choice and we get x=2,-2. Am I right? Or should I add something?

I am conducting research into the matrix exponential, and I would like to discuss the mathematical development of the exponential, and eventually some applications in terms of quantum physics. Currently, my problem is tracking down the original usage of this technique. I am trying to find a...

If I take the exponential function e^t and take the derivative, I think I get the same e^t. Even if I keep doing it over and over, second, third derivative, etc. My admittedly naive question, though, is this symmetric? Meaning...if I take the the integral of e^t, do I just get the reverse or...

I have the following two problems that I need to solve: 1. Suppose that the service time for a student enlisting during enrollment is modeled as an exponential RV with a mean time of 1 minute. If the school expects 500 students during enrollment period, what is the probability that the...

So I am trying to understand and solve the problem mentioned in the title.I found a solution online: https://physics.bgu.ac.il/COURSES/QuantumMechCohen/ExercisesPool/EXERCISES/ex_9011_sol_Y09.pdf The problem is, I can't understand this step : I relly can't find out how the two expontential...

##cos(\omega)## is $$\frac{e^{j \omega } + e^{-j \omega }}{2}$$ ##sin(\omega)## is $$\frac{e^{j \omega } - e^{-j \omega }}{2 j}$$ I also know that ##cos(\omega - \pi / 2) = sin(\omega)##. I've been trying to show this using exponentials, but I can't seem to manipulate one form into the...

If i had a bosonic field ##\phi(x)## and I took the exponential in the following way to get the operator $$W=e^{\imath f \phi(x)}$$ where ##f## is a parameter what effect would this have when acting on the vacuum ##|0\rangle##? Is it analogous to the space translation operator? Will it transform...

How can i continue from here, answer is x=0,-2

I am having so much trouble figuring this out, I would really appreciate some help. The question is: The following function, L, gives the approximate percent literacy rate in India t years after 1900. L(t)=5.3 x 1.025^t Which of the following equivalent functions shows, as a constant or...

$$\int_{-\infty}^{+\infty} exp (-[ax + bx^2]^2) dx$$ $$a\&b\in R$$

Namaste I seek a clarification on the periodicity condition of discrete-time (DT) signals. As stated in Oppenheim’s Signals & Systems, for a DT signal, for example the complex exponential, to be periodic, i.e. ej*w(n+N) = ej*w*n, w/2*pi = m/N, where m/N must be a rational number. Above is...

I encountered this problem is Hopital section...how do I even apply it?

Rewrite in exponential form: Log(6) 1294 = 4 Log(w) v = t Ln(1/4) = x Evaluate Log(4) 64 = ? Log(16) 4 = ?

Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...

Hello, Consider the system of linear homogeneous differential equations of first order dy/dx = A(x) y where x denotes the independent variable, A(x) is a square matrix, and y is an unknown vector-function...

Why is ##P(X>5|X>1) = P(X>4)## in case of an exponential distribution? Can anyone kindly explain it to me?

I used a matrix to calculate the Fourier transform of a lorentzian and it did generate a decaying exponential but that was followed by the mirror image of the exponential going up. I am referring to the real part of the exponential. If I use an fft instead I also see this. Shouldn't the...

The universe will expand exponentially when dark energy completely dominates the energy density of the universe. To clarify, does that make the present expansion 'quasi-exponential'?

Let a,b,c and n are real numbers.a-b = C I want to get rid of a,b and find the following expression in terms of C and n. How can I do that? (an-bn)= ? (in terms of C and n) Thank you.

Hello everybody! I was studying the Glashow-Weinberg-Salam theory and I have found this relation: $$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} =...

Homework Statement Consider the exponential probability density function with location parameter ##\theta## : $$f(x|\theta) = e^{-(x-\theta)}I_{(\theta,\infty)}(x)$$ Let ##X_{(1)}, X_{(2)},...,X_{(n)}## denote the order statistics. Let ##Y_i = X_{(n)} - X_{(i)}##. Show that ##X_{(1)}## is...

Given ln(ab) = b⋅ln(a) Then ln(1x) = x⋅ln(1) Also ln(2x) = x⋅ln(2) Say ln(2x) = ln(1x) Then Also x⋅ln(2) = x⋅ln(1) But, dividing both sides by x ln(2) ≠ ln(1) Similarly, x⋅ln(2) = x⋅ln(1) Dividing both sides by x and ln(2) 1 ≠ 0 But we know x = 0 as per the original statement...

Hi, I was trying to see where the equation N = No e-λt came from and it is derived from dN/dt = -λN which is discussed very well in this thread in post #2 (https://www.physicsforums.com/threads/derivations-of-the-decay-constant-equation.213312/). I understand the steps except for the reason why...

Homework Statement -2^x = y Homework EquationsThe Attempt at a Solution When I plug this function in my graphing calculator, it appears to be 2^x reflected across the x axis. This doesn't make sense to me. For example, for x values of 1 and 2, the value of y is not on the same half of the...

Homework Statement forumlate if its growth/decay is exponential I have equation that i intergrated and found Pressure over volume = work done of pressure P = 3.2c^-1.4 f(v) = -8v^-0.4 i set limits of 10x10^-6 --> 100x10^-6 and 10x10^-6 --> 100x10^-6 but i increase both values by 20x10^-6 every...

I apologize in advance if any formatting is weird; this is my first time posting. If I am breaking any rules with the formatting or if I am not providing enough detail or if I am in the wrong sub-forum, please let me know. 1. Homework Statement Using Euler's formula : ejx = cos(x) + jsin(x)...

using this equation ##1-\sqrt{1-x^2/c^2}## where c = 1 and x = 0.0 - 1.0 the speed of c for example ##1-\sqrt{1-.886^2/1^2}## = y = 0.5363147619 gives me the y values. How do I find the inverse? How do find for x inputting the values of y? Thank you.

Hi all: I really do not know what to ask here, so please be patient as I get a little too "spiritual" (for want of a better word). (This could be a stupid question...) I get this: eiθ=cosθ+isinθ And it is beautiful. I am struck by the fact that the trig functions manifest harmonic...

For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?

Homework Statement given the formula m=n*e^(-nt) show that the maximum of this curve is at m=1/(t*e^(1)). 2. The attempt at a solution I can show this graphically but I am curious if it is possible to do it by hand?

Homework Statement Consider two operators A and B, such that [A,[A, B]] = 0 and [B,[A, B]] = 0 . Show that Exp(A+B) = Exp(A)Exp(B)Exp(-1/2 [A,B]) Hint: define Exp(As)Exp(Bs) as T(s), where s is a real parameter, differentiate T(s) with respect to s, and express the result in terms of T(s)...

x^4 = (5x+6)^2 then what is the multiplication product of all values x can take? i tried taking square roots of each and wrote in absolute value and found 6, 1, -1 (may be wrong) already but there must be more or different because it is not even in answer choices and the answer should be -36

Hello all, I need assistance in solving this exponential equation. \[54\cdot 2^{2x}=72^{x}\cdot \sqrt{0.5}\] The final answer should be 3/2. My strategy was to try and bring to a state where the exponents are equal. We know that 54 is 6 times 9. We also know 72 is 8 times 9. The solution...

Homework Statement Show ##\int_{0}^{1}e_n(x)\overline e_k(x) dx = 1## if ##n=k## and ##0## otherwise. Homework Equations ##e_n(x) = e^{2\pi inx}##. The Attempt at a Solution Consider 2 cases: case 1: ##n=k##. Then ##\int_{0}^{1} e_n(x) \bar e_k(x) dx = \int_{0}^{1} e_n(x)e_{-k}(x) dx =...

I was wondering if anyone could point me to a set of rules for finding the base of an exponential function? I can figure out that the base of $$f(x)=7^x$$ is 7 and the base of $$f(x)=3^{2x}$$ is 9 but even though I know $$f(x)=8^{\frac{4}{3}x}$$ has a base of 16, I don't know how that answer was...

Homework Statement a = [1 1;4 1] Homework Equations R = M^-1 * a * M X = M * e^(R*t) * M^-1 * x M is matrix of eigenvectors. The Attempt at a Solution lambda = 3, -1 initial conditions: x = [1 1]' at t = .1 eigenvectors: k1 = [1 2]' k2 = [1 -2]' M = [1 1;2 -2] M^-1 = [.5 .25...

Say I have ##e^{2\pi i n}##, where ##n## is an integer. Then it's clear that ##(e^{2\pi i})^n = 1^n = 1##. However, what if replace ##n## with a rational number ##r##? It seems that by the same reasoning we should have that ##e^{2\pi i r} = (e^{2\pi i})^r = 1^r = 1##. But what if ##r=1/2## for...

I'm trying to figure out how to solve 45-9x=1/(8x-2) for x. I've gotten as far as 22(5-9x) = 1/(23(x-2)) 2(5-9x) = 1/(3(x-2)) 10 - 18x = 1/(3x-6) But, I'm not sure how to deal with the fraction. EDIT: swapped 45-9x=1/(xx-2) with 45-9x=1/(8x-2)

Above is a question I need some help with. I can get to a certain point, but I can't get beyond. Can somebody with knowledge about this help?

Hello I'm having trouble solving for this exponential equation : 16^{x}-(5,4)^{x}-6=0 I used some logarithms properties but can't get anything close to the following solutions here I tried using log base 16 : log16(16^{x})-6=log16((5,4)^{x}) ; then x - xlog16(5,4)=6 ; factorizing x ...