Exponential function Definition and 161 Threads

- exp(f) = Laplacian(f) where f is a real valued function of two variables in an open domain.

Homework Statement The question is located here http://i51.tinypic.com/nex2q1.jpg Homework Equations The value I have been given for a) is 5 The value I have been given for b) is 5PI/6 The Attempt at a Solution Note that e^(a + bi) = e^a e^(bi) = e^a (cos b + i sin b). (e^a is...

Homework Statement \lim_{x \to \infty} \left(e^x-x \right )^{1/x} Homework Equations The Attempt at a Solution Should I be equating this to f(x), taking the log of both sides, using L'Hopital's rule on the resulting indeterminate quotient? If I do that, then I end up with a perpetual...

Homework Statement I am trying the find the derivative of the function: y= (2x-1)^(tan(3x)) Homework Equations Chain rule, in this case: f'(g(h(x))) * g'(h(x)) * h'(x) exponent rule: where (d/dx) a^x = (a^x) ln a The Attempt at a Solution I feel as if to apply the...

I really appreciate if anyone could indicate me how to handle this inverse Laplace transformation (ILT): L-1[Exp(-c0*Sqrt(a(s)))/Sqrt(a(s))] where a(s)=(s2+c1s)/(c2s+c3) c0,c1,c2,c3 are all constants. I searched some literatures regarding the ILT of Exp funtions but no such form. I used...

Homework Statement If there is a quantity T comprised of two other quantities such that T=L+B, and quantity T and B are both increasing in every period at a fixed percent such that %growth T > %growth B, it will be true that %growth L > %growth T. It will also be true that as n approaches...

I am having difficulty trying to rearrange this equation to solve for one unknown variable. i=(io)exp(z*α*F*∆V)/(R*T) How would I rearrange this equation if i wanted to solve for io, also how would I rearrange this equation to solve for say α.

Homework Statement According to the Inverse Function Theorem, for every z_0 \in C there exists r > 0 such that the exponential function f(z) = e^z maps D(z0; r) invertibly to an open set U = f(D(z_0; r)). (a) Find the largest value of r for which this statement holds, and (b) determine the...

Homework Statement I need to integrate the following function: y = \int e^{-\frac{n}{\omega}cos \omega t} dt Homework Equations How should I solve this? Will this lead me to Bessel function? 2. My attempt a=\frac{n}{\omega} u=-a cos (\omega t) du = a\omega sin...

Homework Statement Hey guys. How do I solve this integral? http://img816.imageshack.us/img816/208/68315659.png I've not been doing this for a long time :) Thanks a lot. Homework Equations The Attempt at a Solution

I've tried and searched for a long time, and I haven't been able to prove or find a proof that the following sequence converges (without using another definition of the exponential function): \forall x \in \mathbb{R}. Prove that: \lim_{n \rightarrow \infty} (1+ x/n)^n exists. I can...

Hi, I need to solve the equation e^z = -3 The problems arises when i set z to a+bi e^a(cos(b) + isin(b),~b = 0 Then I am left with e^a = -3 However you're not allowed to take the log of a negative number. Also i know that cos(\pi) + isin(\pi) = -1 Obviously 3e^{(\pi*i)} is a solution, but...

Hello every one, I was doing my research and then I simply struck at a point. The point is that i do not know how to solve the following Integral. I am not at all bad at doing math but some times I got blanked. so, here is the Integral, Integral (infinity,u) exponent^(-u) du...

Hello, How can I find the limit of x^{k}\mathrm{e}^{-ax^n} for x\to +\infty. a,k,n are positive reals. I even have troubles solving it for k=1, a=1, n=2. Any hint?

Hi everyone, here is this integral I can't find solution: \int_0^\infty \frac{x e^{-x}}{A+Bx} dx A and B are constants. I'm going crazy, I don't think there is even an analytical solution.

Homework Statement A particular species of mollusk is distributed according to a Poisson process with an unknown density of lambda per cubic meter of water. A sensor is constructed that can detect these species that are within 4 m, and n readings are collected over a year to try and measure...

Homework Statement Find the second derivative of: e^{ax} and e^{-ax} Homework Equations The Attempt at a Solution The book that I am using seems to have been very vague on how to take the derivatives of exponential functions. I am aware that: \frac {d(e^{x})}...

Homework Statement I don't understand why they took M(e)=1 , and how the proceed on with the proof. Thanks in advance(= Homework Equations The Attempt at a Solution

Homework Statement Hi all Is infinity a minimum of the exponential function f(x)=ex or not? I personally do not think so since it is a limit, but I wanted to ask you guys to be 100% sure.

Homework Statement -Find the intervals on which f is increasing or decreasing -Find the local maximum and minimum values of f -Find the intervals of concavity and the inflection points f(x)=xex f'(x)= ex+xex then I must solve for x when the function equals zero to find my critical numbers...

Hi all, I'm trying to solve the definite integral between 0 and inf of: exp(a*x^2 + b*x + c) --------------------- dx 1 + exp(m*x + n) with a,b,c,m,n real numbers and a < 0 (negative number so it converges). I've read in the forum's rules that I have to post the work that I have...

I'm trying to find the laplace transform (if possible) of exp(-2r/a)r^2. --- I did integration by parts to check that the integral of exp(-2r/a)r^2 from 0 to infinity is a^3 / 4. but i cannot get the laplace transform to work out. the answer mathematica online gives would have to have the term...

I have a problem with an exponential function. I am wondering if an exact solution is possible, or if I have to write the solution as a logarithm of an unknown. A formula says that E(z)=E(0)^{-kz}, where E is light intensity and z is depth in water. My objective is to find the constant k. I...

Homework Statement 10(1 + e-x)-1=3 Homework Equations The Attempt at a Solution I'm supposed to solve for x, but I don't know how to go about this. I tried dividing the 3 by the 10, but after that I don't know what to do. I believe I should use ln on both side, but that's after...

Homework Statement This topic is under linear system differential equation.Solve the system by using exponential method. Just want to ask the expansion of exponential function Homework Equations e^x=1+x+(x^2)/2!+(x^3)/3!+... The Attempt at a Solution then how about the e^(-x)=...

Homework Statement This topic is under linear system differential equation.Solve the system by using exponential method. Just want to ask the expansion of exponential function Homework Equations e^x=1+x+(x^2)/2!+(x^3)/3!+... The Attempt at a Solution then how about the e^(-1)=...

Hey everyone, I'm new to using the Maxima software and I'm having some trouble. When I enter the following formula to be evaluated: diff(1-exp^(-t/R*C),t); I get the following output: \frac{log\left( exp\right) \,C}{{exp}^{\frac{t\,C}{R}}\,R} This doesn't look right, even if I...

J.J. Sakurai Modern Quantum Mechanics p. 74 It says, [A,H] = 0; H|a'> = Ea' |a'> where H is the Hamiltonian A is any observable |a'> is eigenket of A then, exp ( -iHt/h)|a'> = exp (-iEa't/h)|a'> where h is the reduced Planck's constant. I want to know WHY ? and besides, I would...

Homework Statement Find the limit as x tends to zero of: (e^-x - cos x)/2x Homework Equations lim_x->0 e^-x = 1 lim_x->0 cos x = 1 lim_x->0 sin x / x= 1 The Attempt at a Solution Hi everyone, Here's what I've done so far: (e^-x - cosx)/2x = [(e^-x)^2 - (cosx)^2] / 2x(e^-x...

what does the exponential function e^x have to be multiplied by to get e^-x?

Homework Statement Int(-infinity to +infinity) exp[i(t^3/3 + at^2 + bt)]dt = 2pi*exp[ia(2a^2/3 - b)]*Ai(b-a^2) O.Vallee gives this formula in his book, "Airy Functions and Applications to Physics" but there are no proof of this formula. I tried to prove this formula, but I failed. Would you...

Homework Statement Species A doubles every 2 hours and initially there are 6 grams. Species B doubles every 5 hours and initially there are 14 grams. Homework Equations The Attempt at a Solution I've tried graphing this, but I don't think I have the right equations down. I don't...

I am looking a formula to compute the derivitative of e^{a x^2} with respect to x n times, where a is a constant. \frac{d^n}{dx^n}e^{a x^2} I am going to find the result of above derivitative when x -> 0.

Homework Statement 1. \int^{\infty}_{-\infty}e^{-ax^2 - bx^{\frac{5}{2}}}dx 2. \int^{\infty}_{-\infty}x^ne^{-ax^2 - bx^{\frac{5}{2}}}dx (n is integer) Homework Equations Does anyone can give me the integral in the closed form or introduce any useful references? Thank you.

Homework Statement Find the exponential function that passes through the points (2, 1) and (5, 7). Homework Equations y= Ce^ (kt) The Attempt at a Solution I can only get as far as substituting in 2 for t and 1 for y, then I'm completely stuck.

Can a convenient value for a be found without resorting to substituting numerical values for h in this expression? EDIT: I am trying to indicate, "as h approaches zero". EDIT: neither of the formattings worked; hopefully someone understands what I am asking...

Homework Statement I'm new to integration, and was attempting the integral of ex with respect to x. Homework Equations \intex dx The Attempt at a Solution Should I keep it in the format y=ex so that I use this in the calculation of the area? Or should I convert to ln y? I got stuck...

Homework Statement What is the derivative of ex3? also what is the derivative of (ln1/x)2Homework Equations The Attempt at a Solution is it 3x2e3x2? 2(ln1/x)(x)?

I have seen the following identity used. Exp[iw/2]-Exp[iw/2]=Exp[iw]-1 I can't find this in any book and I can't prove it myself. The left side equals 2isin(w/2) The right side equals cos(w)+isin(w)-1 On the face of it, that seems to make the identity absurd How can one go about proving...

Homework Statement e^(2x+9) - 4e^x -5 = 0 Homework Equations The Attempt at a Solution I changed this into: e^(2x+9) =4e^x + 5 I took logarithm of both sides: 2x+9=ln(4e^x +5) but i don't know what to do with the right hand side. what will be the easiest way to solve this...

Its true that if you integrate an exponential function from some time t0 to infinity it will converge to a finite value. However, is the same true if it is multiplied by say t, t^2, t^3,t^n. i.e. t*exp(-t) for example. the exp is decaying to zero faaster than t is, so it goes to zero...

The inverse of the exponential function... Homework Statement Find the inverse of the function = e ^ (x^3) Homework Equations The inverse of the exponential function = the natural logarithm of that same function The Attempt at a Solution inverse of f(x) = ln(x^3) ? This...

hello! there's this question I am working on and I am mostly through it (well, I hope I am at least), there's just one or two things still annoying me/need confirmation. Here's the question: 50 coins were flipped. For every coin that landed with the head showing upwards, it was taken away...

[SOLVED] Definite Integral of Exponential Function Homework Statement I have an integral that I need to solve for a quantum physics problem \int^{\infty}_{-\infty}e^{-a|x| - ikx}dx How would I go about solving this thing?

Dear All, I am searching for an upper bound of exponential function (or sum of experiential functions): 1) \exp(x)\leq f(x) or: 2) \sum_{i=1}^n \exp(x_i) \leq f(x_1,\cdots,x_n, n) . Since exponential function is convex, it is not possible to use Jenssen's inequality to get an upper bound...

Homework Statement Given that Q=Pa^t and Q doubles between t and t+d, prove that d is the same for all t. Homework Equations Q=pa^t The Attempt at a Solution This is what I've tried so far: Q_0=Pa^t and Q_1=Pa^{t+d} Then: \frac{a^{t+d}}{a^t}} \equiv 2 This is where I begin drawing blanks...

Homework Statement Express b^{x} as a function of logarithms. Homework Equations There are a couple of equations in the attempted solution. I can't say if they are actually relevant The Attempt at a Solution I've investigated the property y = log_{b}(b^{y}), and also...

good evening all! Homework Statement Determine the exact values of j^j Homework Equations j = sauare root of -1 The Attempt at a Solution stuck :cry: :cry: :cry:

Homework Statement find y', y = \frac{e^x}{1+e^x}Homework Equations derivative of e^x = e^x, quotient rule.The Attempt at a Solution The old man is back, sorry, and I don't seem to be able to enter this using the tex stuff. This is from Kline's Calculus, page 348. I proceed as follows: y' =...