Exponential Definition and 1000 Threads

Homework Statement The electric field is described by E = \frac{C e^{-br}}{r^2} , find the charge density and then integrate over all-space and show it's zero. [/B]Homework Equations E = \frac{C e^{-br}}{r^2} \\ \\ \nabla \cdot E = \frac{\rho}{\epsilon_0} \\ \\ \rho(r) = Q \delta^3...

Hey! How does the operator of angular momentum operates in exponential form? $$ e^{-i\theta J}\vert l, m \rangle = ?? $$ where $$J\vert \Psi \rangle = J\vert l, m \rangle$$ and $$J^2\vert \Psi\rangle = \hbar^2 l(l+1)\vert \Psi\rangle $$ Also, how do you operate $$J_-$$ and $$J_+$$...

Homework Statement Hi, I'm new here. I'm really rusty, I resume my career this year, and I'm reading 'the spivak book', (for Calculus 1). Making some exercises, I get curious about how to solve this: x+e^x=4 I would love if someone could give me any trick Homework EquationsThe Attempt at a...

Homework Statement The life times, Y in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.6 years. A tester makes random observations of the life times of this particular brand of lightbulbs and records them one by one as either a success if...

How to solve: a1e-k1x+a2e-k2x+...+ane-knx =0 for x? For example in simple case of n=1,2. a1e-k1x+a2e-k2x=0 the solution will be x=In (a1/a2) / [ k1-k2]. But for terms >2 what will be the solution?

Homework Statement Problem(s): Suppose that X has an exponential distribution with mean equal to 10. Determine the following: (a) P(X > 10) (b) P(X > 20) (c) P(X < 30) (d) Find the value of x such that P(X < x) = 0.95. Correct answers: (a) 0.3679 (b) 0.1353 (c) 0.9502 (d) 29.96 Homework...

Homework Statement Here is a math problem I would to make sure I have understood correctly. a) Billy goes to be bank. He deposite 500 dollars in his account. The bank offers a interest of 2 percent a year. Sally goes to the bank 4 years later.She deposite 250 dollars at the same bank...

The no. of real solution of 4x^2 = e^x. \bf{My\; Try::} Let f(x) = 4x^2-e^x\;, Then f'(x)=8x-e^x and f''(x)=8-e^x and f'''(x)=-e^x<0\; \forall x \in \mathbb{R} So f'''(x) = 0 has no real roots, So Using \bf{IVT} equation f''(x) = 0 has at most one real roots. and equation f'(x) = 0 has at...

Homework Statement In 2000 the population of a country was estimated to be 8.23 million. In 2010 the population was 9.77 million. Assume that the number of people P(t) in millions at time t (in years since 2000) is modeled by the exponential growth function. P(t) = Aekt Find P(t) giving the...

Homework Statement Given that the function f can be expanded in a power series of a and a^\dagger, show that: [a,f(a,a^{\dagger})]=\frac{\partial f }{\partial a^\dagger} and that [a,e^{-\alpha a^\dagger a}] = (e^{-\alpha}-1)e^{-\alpha a^{\dagger} a}aThe Attempt at a Solution I've tied using...

Homework Statement Find the eigenvectors and eigenvalues of exp(iπσx/2) where σx is the x pauli matrix: 10 01 Homework Equations I know that σxn = σx for odd n I also know that σxn is for even n: 01 10 I also know that the exponential of a matrix is defined as Σ(1/n!)xn where the sum runs...

So I've seen this type of integral solved. Specifically, if we have ∫e-i(Ax2 + Bx)dx then apparently you can perform this integral in the same way you would a gaussian integral, completing the square etc. I noticed on wikipedia it says doing this is valid when "A" has a positive imaginary part...

Could you help me with this questions as well? This is what I have done: 2^x - 2^-x/ 2^x +2^-x 2^x - 1/2^x/ 2^x +1/2^x u -1/u/u+1/u =1/3 3u² - 3/ 3u² +3= U u² - 1- u = 0 u² - u - 1 =0 (u -1) (u+1) =0 U =1 or -1

Hi, does anyone know how to integrate e^-x (sin(nπx))? I have tried part integration but that goes on until infinity... and I am not sure how to use the substitution method...Please help! I have tried taking e^-x as U but then I end up getting the entire canceled off then...

Homework Statement In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have...

Hi, I need help to add an exponential trendline on my graph. I have three graphs and I have succeeded with the first one but somehow the exponential trendline will not show on the two other graphs even though I have selected the option multiple times. I also tried to re-do the graph in a new...

Homework Statement superman has been disabled by a nearby amount of kryptonite, which decays exponentially. If Superman cannot regain his power until 90% og the kryptonite disintegrates, then how long will it be before he regain his powers? Use r=-0.138629. Round to the nearest day. a. 4days...

No. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$

Hello PF, I just found a curious integral. I wondered if it comes from a bigger group of integral definitions: \int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a) Where Si(x) is the sine integral function \mathrm{Si}(x)=\int_0^x \frac{\mathrm{sin}x}{x}\mathrm{d}x I proved the...

Homework Statement [/B] I need to differentiate the exponential function i = 12.5 (1-e^-t/CR) and I need to plot a table so that I can do a graph of i against t but I'm not sure how. (CR is the equivelant of Capacitance 20 Micro Fards and Resistance 300 Kilo Ohms)Homework Equations [/B] How do...

Hello, and thanks for welcoming me in the forum of Physics Forums. I just found a curious integral that I solved by Taylor series. I wondered if it comes from a bigger group of integral definitions: ##\int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a)## Where Si(x) is the sine...

I'm modelling a variable output Y which has a value of 1 at x=0. I've noticed that in the system I'm modelling, as x increases, y increases at an exponentially decreasing rate, up until a limit of around 1.53. I view this as changes in x causing the Y value to increase by a max of 53%. The...

< Mentor Note -- Thread moved from the Calculus forum to HH/Calculus > Hey everyone, I've started my new course and have been given revision and tutorial questions to do at home, this question has been bugging me for days now! I need to make "t" the subject to allow me to progress into the...

Demonstrate that ##|e^{z^2}| \le e^{|z|^2}## We have at our disposal the theorem which states ##Re(z) \le |z|##. Here is my work: ##e^{|z|^2} \ge e^{(Re(z))^2} \iff## By the theorem stated above. ##e^{|z|^2} \ge e^x## We note that ##y^2 \ge 0##, and that multiplying by ##-1## will give us...

Hi, I'm building a fluid model and using the method of characteristics to solve it. I'll not go into the details as they aren't necessary. Basically I have two points $(-\epsilon,70)$ and $(\epsilon, 0)$ and need to create an exponential curve between them. Could someone please tell me of a way...

So, I am wanting to vary a parameter in an equation with respect to time. Vary mass flow [ m(t) ] for an exponential decay to half its original value in around 60 seconds. I know the regular decay equation where: m(t)=m0*exp(-At) but I want the value to approach a steady state at 60 seconds...

Hey guys, if I have an integral of the form \int d^{3}x \hspace{2mm} e^{i(k\cdot x)}, how do I evaluate this? Thanks a bunch...

Homework Statement The graph goes through the points (-2, 13) and (0, 5) and has the horizontal asymptote y = 4. f(−2) = ____ therefore: ____(B^____ ) = ____ b = The Attempt at a Solution f(−2) = 13 therefore: 1 (B^-2 ) = 13 b = ? not sure Thank you

hello guys .. first of all , iam not sure that i should type this thread here . so excuse me for that in this problem i can understand the part until it's said that w=0 then x(t)=1, which is periodic for any value of T but i can't understand the part after that in the case of w is not equal...

Homework Statement Hi all. For some reason I have been having a lot of difficulty with this problem in Peter Petersen's text. The problem is Prove: ##d(exp_p(tv), exp_p(tw)) = |t||v-w| + O(t^2 )## Homework Equations The exponential map is the usual geodesic exponential map. And ##d(p,q)## is...

Hi all, I was wondering what the relationship between the Riemannian Geometry exponential map and the regular manifold exponential map and for the reason behind the name.

Hello everyone, I have spend whole day but still not figure out an inverse Laplace transform. Hope someone can help me. The question is in the attachment. I'm trying to extract u^2/4D^2 out the bracket to match the standard inverse table, but it seems difficult to deal with the square root...

When I was first introduced to a derivation of the taylor series representation of the exponential function here (pg 25): http://paginas.fisica.uson.mx/horacio.munguia/Personal/Documentos/Libros/Euler%20The_Master%20of%20Us.pdf I noted the author, Dunham mentioning that the argument was non...

Homework Statement Let X and Y be independent exponential random variables with parameters a and b. Calculate E(X|X+Y). Homework EquationsThe Attempt at a Solution I'm pretty sure I have it, just want to make sure. Joint density for X and Y is abe^(-ax)e^(-by) for x,y>0. Let Z=X and W=X+Y so...

How can we show that the following equation has infinitely many solutions e^z-z^2=0. Thanks

I have a curve I obtained in COMSOL representing a electric field for m structure. I have to figure how to determine when the curve starts to become exponential decaying at certain regions. Any idea how to determine when the curve starts becoming exponential decaying from a plot?

I've been reading Mermin's book on Quantum Computer Science, and in the section in which he discusses the construction of a QFT using 1-Qbit and 2-Qbit gates, he makes reference to some expressions involving linear operators that I'm not familiar with (at least if I've seen them before I've...

So i have a graph representing iodine 131 y axis of 0-100 iodine remaining in the body (%of dose) x asis 0-60 Time in days with the points plotted (24,12.5) The equation of the graph is D = D0 (b) t/k So i have figured out "k" which is 8 for iodine half life. Which it asked me to do in the...

Hello all, I've been stuck on this question for a while and it's annoying the stew out of me! I know it's a basic definition type of question, but I can't seem to understand it. Can any of you help? Question: Let X be a random variable and A be an event such that, conditional on A, X is...

Have any of you been tracking the rate of new Ebola infections? It seems that there is pretty clear exponential growth, and the virus is anything but contained. Is anyone else bothered by this? By looking at this just mathematically, some huge swath of people will become infected. At a...

I asked to differentiate the given function using exponential function with sin(√3t + 1) I turned it into Im[e^(√3t+1)i] then I multiplied it by e^t which gave Im[e^t*e^(√3t +1)i] then I applied usual algebra to differentiate but I get a (t+√3ti +i) as the power of e when I try...

Homework Statement I'm working on this problem: Let \hat{U} an unitary operator defined by: \hat{U}=\frac{I+i\hat{G}}{I-i\hat{G}} with \hat{G} hermitian. Show that \hat{U} can be written as: \hat{U}=Exp[i\hat{K}] where \hat{K} is hermitian. Homework Equations...

Hi, I am struggling for some time to solve the following integral: $$ \int_{-n}^{N-n} \left( \frac{e^{-j\pi(\alpha-1)\tau}}{\tau} - \frac{e^{-j\pi(\alpha+1)\tau}}{\tau} \right) d\tau $$ N is a positive integer, n is an integer, \alpha can be a negative or positive rational number. I want to...

I'm reading about the exponential family of distributions. In my book, I have the expression f(x|\theta) = h(x)c(\theta)exp(\sum_{i=1}^{k} w_i(\theta) t_i(x)) where h(x) \geq 0 , t_1(x), t_2(x),...,t_k(x) are real valued functions of the observation x, c(\theta) \geq 0 , and...

So this question is from Khan Academy. I understood the first part and chose the correct function, but the second question(from 40 degrees to 30 degrees change) explanation confused me. _____________________________________________________________________________ QUESTION: Ajay made a...

Homework Statement Solve for x: 8100(\frac{7}{6})^{x} = 11,000 2. The attempt at a solution x = log_{9450}(11,000) = 1.02 I know this is not the right answer, but I am not sure what I am doing wrong.

Homework Statement Express the following signal in terms of ramp, exponential, and parabolic functions, where t should be consistently shifted Homework Equations signal simplification techniques, not known to me The Attempt at a Solution It seems you can express this in terms of...

Hi All, I have searched for hours but cannot figure this out. I need to find the equation for an exponential curve but I only have the x and y values. Is there a way to do this without using excel? I tried to follow this on YouTube: https://www.youtube.com/watch?v=ta4MZS7w2VA but the equation...

Hey guys, Few more questions for the problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: The first one starts off easy but I found that it gets progress more challenging later on. So the rate of spread should be p'(t) which is: p'(t) =...

Hello! :-) I' have a problem... I fitted experimental data with a second order exponential theoretical curve: y = A1*exp(-x/t1)+ A2*exp(-x/t2) + y0 and I have these data for 4 repetitions of the same experiment: first time: Reduced Chi-Sqr = 6,81066E-4 Adj. R-Square = 0,41005...