Exponential Definition and 1000 Threads

I'm trying to prove eA eB = eA + B using the power series expansion eXt = \sum_{n=0}^{\infty}Xntn/n! and so eA eB = \sum_{n=0}^{\infty}An/n! \sum_{n=0}^{\infty}Bn/n! I think the binomial theorem is the way to go: (x + y)n = \displaystyle \binom{n}{k} xn - k yk = \displaystyle...

Homework Statement Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem. Find b, such that maximum of the magnitude of the frequency response function...

Hello all, I am trying to figure out how to solve for the complex conjugate of the following: (-0.5)^abs(x) Thanks for your help. -Brian

I'm taking a signals and systems class and the textbook (Signals and systems by Oppenheim) says the CT complex exponential of the form x(t) = C eat with C and a complex is a periodic signal. I fail to see how. Let C = |C| ejα (exponential form of a complex number) and a = r + jω (rectangular...

Homework Statement Solve \sqrt{-e^{(i2\pi)/3}} Homework Equations The Attempt at a Solution I seem to be missing something simple, as I take: \sqrt{-1} = i then, e^{(1/2)*(i2\pi)/3} which comes out as: ie^{i\pi/3} however, the solution is: -ie^{i\pi/3}, and I can't seem to see where...

Homework Statement Determine the image of the line segment joining e^(i*2*pi/3) to -e^(-i*2*pi/3) under the mapping f = e^(1/2*Log(z)). Homework Equations The Attempt at a Solution The line joining the two points: {z | -0.5 < x 0.5, y = sqrt(3)/2} f = the principle branch of...

Homework Statement A bacteria culture starts with 840 bacteria and grows at a rate proportional to its size. After 3 hours there will be 2520 bacteria. How long will it take for the population to reach 1030 ? 2. The attempt at a solution dP/dt = k*P dP/P = k*dt *Integrated both...

Homework Statement prove that: 1-exp(-iwt)= 2i*sin(wt/2) Homework Equations exp(iwt)= cos (wt) + i*sin(wt) The Attempt at a Solution I attempted to express the exponential into sum of cos and sin and considering t=2*t/2 in order to obtain an argument like (t/2) (using...

Homework Statement I need to prove that (1+x_{1})·...·(1+x_{n})≥(1-Ʃ^{n}_{i=1}x_{i}^2)e^{Ʃ^{n}_{i=1}x_{i}} with all 0≤x_{i}≤1 I've already proven that (1+x_{1})·...·(1+x_{n})≤e^{Ʃ^{n}_{i=1}x_{i}} with all 0≤x_{i}≤1 and (1-x_{1})·...·(1-x_{i})≥1-Ʃ^{n}_{i=1}x_{i} with all 0≤x_{i}≤1 , but...

Homework Statement Big problem with exponential function proof assignment, need some help. let x≥0 and for every k\in N there is n_{k}\in N and x_{k1}≥...≥x_{k_{nk}} and x_{k1}+...+x_{k_{nk}}=x. Proof: if lim_{k→}∞ x_{k1}=0 then lim_{k→}∞ (1+x_{k1})·...·(1+x_{k_{nk}})=exp(x)=e^{x}

Why is e^{\int \frac{dt}{t}} = e^{ln|t|} = t as apposed to |t|? I don't understand what happened to the absolute value operator. Thanks for any help. I understand that e^{x}>0. Is this the justification? But I don't understand why you can't have a negative t in e^{ln|t|} because you would...

Homework Statement Solve 2^{k-2}=3^{k+1} Homework Equations The Attempt at a Solution 2^{k-2}=3^{k+1} \frac{2^{k}}{2^{2}}=(3^k)(3^1) \frac{2^{k}}{3^{k}} = 4 \cdot 3 \frac{2^{k}}{3^{k}} = 12 What do I do next to solve for K?

Homework Statement Show that one can write U=exp(iC), where U is a unitary matrix, and C is a hermitian operator. If U=A+iB, show that A and B commute. Express these matrices in terms of C. Assum exp(M) = 1+M+M^2/2!...Homework Equations U=exp(iC) C=C* U*U=I U=A+iB exp(M) = sum over n...

Homework Statement Let X1, X2 be exponential RVs with parameter λ, μ respectively. (The question does NOT say that they are independent but I think this must be a typo?! If not I have even less idea how to do the question). Let Y= min {X1, X2} and Z = max {X1, X2} Let W = Z-Y...

Homework Statement show that the Laplace transform of e^(At) = (sI - A)^(-1) \mathcal{L}\left\{ e^{At} \right\}(s) = \left(sI - A \right)^{-1} The Attempt at a Solution I find \left( e^{At} \right)_{ij} = \sum_{k=0}^{\infty} \frac{(A^k)_{ij}t^k}{k!} and since \mathcal{L}\left\{...

Homework Statement Find an equation of the tangent line to the curve xe^y+ye^x=1 at point (0,1). Homework Equations I do not recall seeing the Implicit Function Theroem before, I even went back in my book (Stewart Calculus 6th) to check. I found this post but it does not help me...

1. i) For what values of a with the expression log2a: a) yield positive numbers? b) yield negative numbers? c) yield numbers greater than 1? d) yield numbers between 0 and 1? e) be zero? f) be undefined? Solution: 1. a) a > 1 b) 0 < a < 1 c) ? d) ? e) a = 1...

Homework Statement A new car that costs $30,000 has a book value of $18,000 after 2 years. a) Find a linear model that represents the value 'V' of the care after 'x' years b) Find an exponential model of V= ab^(x) that represents the value 'V' of the car after 'x' years...

Homework Statement Consider the exponential function f(x)= ab^(x) a) Show that (f(x+1))/f(x) = b b)Use the results from part (a) to explain why there is no exponential function of the form f(x)= ab^(x) whose graph passes through the points (0,4) (1,4) (2,8) (3,24) and (4,72)...

e-z2 where z is a complex number a+ib

How do you go about sketching y as a function of t for t≥0 y= e(0.5t + i(√7/2)t) - e(0.5t-i(√7/2)t) I know it goes through the origin, and the gradient is positive here. But I'm unsure on how to deal with the imaginary numbers when I have a graph of y vs t.

Looking at obtaining the unbiased estimator of σ^2, i know how to do it for σ, to obtain 1/x(bar), was wondering how to obtain for σ^2, i guess you just don't square bot sides. It should tale the form of kƩ(x_i)^2. many thanks any help would be much appreciated.

Hi all, am stuck with the integral with fraction in exponential The equation I=∫exp(bz)/(a+iz)*exp[(6*a^2-ik*w^2*z)/(z^2+a^2)]*dz I already tried to partial fraction the 2nd exponential term, then i tried to perform integration by parts but it doesn't work well.i tried substitution too...

Homework Statement If k is a natural number, prove that 2^{n} > n^{k} for all n \geq k^{2} + 1.Homework Equations We need to use a proof by induction.The Attempt at a Solution Let's do the case when k = 4. We check the base case directly: 2^{17} = 131072 > 83521 = 17^4 Suppose 2^{n} > n^{4} for...

Hi all, i need help with integration of exponential of inverse tangent, could not find it in table of integrals the whole equation is I=∫A/[w(a+z^2)^1/2]*exp(zb)*exp[(-i*arctan(z/(a)^1/2)] -am trying to integrate by parts but stuck at the arctan part Thank you.

Homework Statement Hi, I'm asking the usual "what formula gives me these results" question, but with a twist; I'm attempting to make a "virtual" analogue multimeter that replicates a physical piece of equipment, for training purposes. The idea is that a random MegaOhm value will be generated...

Homework Statement Prove that for any matrix A, the following relation is true: det(e^{A})=e^{tr(A)} The Attempt at a Solution PROOF: Let A be in Jordan Canonical form, then A=PDP^{-1} where D is the diagonal matrix whose entries are the eigenvalues of A. Then...

1. Let f(x)=sin(e^x) a. Find 2 values of x satisfying f(x)=0 b. What is the range of f(x) c. Find the value(s) of x that maximize f on [3.8,4] (use calculus) 2. y=e^x if and only if x=ln y 3.a. x=-infinity because the limit of e^x as x approaches -infinity is 0. and also...

Exponential function with 3 parameters?? Homework Statement A researcher suggests that the population, P, at time t can be modeled by P(t) = K / (1 + Le^(-Mt)), where K, L and M are parameters. Use technology to estimate and interpret K, L,and M. Construct the researcher's model using...

Hi, Suppose I have this matrix A: Why is the matrix exponential like so: when A is not simple (eigenvalues not distinct)? Thanks.

Homework Statement integrate the function F(t)=1-exp(-∫1/t+2 dt) limit of integration from 0 to t Homework Equations The Attempt at a Solution F(t)=1-exp(-ln(t+2)+ln2) =1-exp(ln(2-t))*exp(ln2) =1+2t-4 =2t-3 what do you think about my solutions?

I have been studying for the GRE and taking note of various approximations to use on the exam, but I am having a difficult time finding a way to evaluate the following without the aid of a calculator e^{-x}. The GRE practice book has a problem to which the answer is e^{-10} = 4.5 \times...

Homework Statement Find dy/dx. y=x^{lnx}Homework Equations The Attempt at a Solution ln y= ln x^{ln x} ln y= (ln x)(ln x) Taking the derivative now: \frac{1}{y}y'= (\frac{1}{x})(\frac{1}{x}) \frac{1}{y}y'= (\frac{1}{x})^2 Multiply by y: y'= (\frac{x^{lnx}}{x^2}) But it's not the right...

Homework Statement Given the exponential function and its inverse, where a > 0 Exponential Function: f(x)=a^x Inverse function: f^{-1}(x)=log_ax a) For what values of a do the graphs of f(x)=a^x and f^(-1)(x) intersect?The Attempt at a Solution I have no idea how to start this question. Off...

Homework Statement Suppose that X=time to failure for a component has an exponential distribution with lambda =.25. Suppose that 9 of the components are selected and their failure times noted. Compute the probability that 3 of the components fail between times 1 and 2, and 4 of the components...

Hello I have been modeling the rotation curves of galaxies. Can help in calculating the rotation velocity for a thin exponential disk. I use the method of Kent S. http://articles.adsabs.harvard.edu//full/1986AJ...91.1301K/0001310.000.html My implementation in C #...

hi, if you had a litre container full of water with negligible air and nothing else and heated it at a linear rate from 50 degrees centigrade upto 200 degrees centigrade does the steam pressure rise linearly or exponentially? the measurements are plucked from the air, its the rate of steam...

A piece of rock contains 10^20 atoms of a particular substance. Each atom has an expoentially distributed lifetime with a half-life of one century. How many centurites must pass before there is about a 50% chance that at least one atom remains. What assumptions are you making? answer...

We are currently studying logarithmic and one of the subsection is on solving exponential equations. I cannot, for the life of me, solve this equation or similar equations. My hopes are that some of you guys can solve it for me and demonstrate the entire process. The question is...

I am looking to find an arbitrary digit of let's say 23^234. This one has 219 digits. so let's find the 187th digit... This isn't homework or anything, I just think it would be interesting and I couldn't find it on google. I would rather use simple number theory that can be taught to a...

Homework Statement find the Fourier transform, using the definition of the Fourier transform \widehat{f}(\nu)=∫^{∞}_{-∞}f(t)e^{-2 \pi i \nu t}dt, of the function f(t)=2 \pit^{2}e^{- \pi t^{2}}Homework Equations I have the answer: (1-2{\pi \nu^{2}})e^{- \pi \nu^{2}} The Attempt at a Solution...

Homework Statement Use the contour integral \int_{C}\frac{e^{pz}}{1+e^z}dz to evaluate the real integral \int^{\infty}_{- \infty}\frac{e^{px}}{1+e^x}dx 0<p<1 The contour is attached. It is a closed rectangle in the positive half of the complex plane. It height is 2i∏. Homework...

Hi all, Could you please answer my following question related to an exponential random variable? Thank you. Let X represent the waiting times at a telephone in an office. Assume that X is an exponential random variable with parameter λ: P(X < t) = 1 - e^{-λ*t} At each time when the...

k guys.. i'd like to try to figure out how to get to the derrivatives on my own so really just looking for a right or wrong. 1. f(x) = e^(3x^2+x) find f'(2) f'(x) = e^(3x^2+x) * (6x+1) f'(2)= e^(3(2)^2+2) * [6(2)+1] f'(2)=13e^14 ... correct?2) find slope...

Typical radio active decay question Half time = 1 year λ = ln 2 here Q 1: if we have 1024 atoms at t=0, what is the time at which the expected number remaining is one. Easy, I get 10 years Q 2: The chance that in fact none of the 1024 atoms remains after the time calculated in c...

Prove that cos z = 0 if and only if z = π/2 + kπ, where k is an integer. The solution is : cos z = (eiz+e-iz)/2 = 0 <=>eiz+e-iz)=0 <=>e2iz=-1 <=>2iz=πi +2kπi <=>z=π/2 + kπ But i don't understand how does these 2 steps work <=>e2iz=-1 <=>2iz=πi +2kπi how can a exp function...

Homework Statement Given is \sum_{n=-N}^{N}e^{-j \omega n} = e^{-j\omega N} \frac{1-e^{-j \omega (2N+1)}}{1 - e^{-j\omega}}. I do not see how you can rewrite it like that. Homework Equations Sum of a finite geometric series: \sum_{n=0}^{N}r^n=\frac{1-r^{N+1}}{1-r} The Attempt at a...

Homework Statement Consider the system of equations \begin{eqnarray*} e_1 &=& Ak^p + Bh^q \\ e_2 &=& A(k/2)^p + Bh^q \\ e_3 &=& Ak^p + B(h/2)^q \\ e_4 &=& A(k/2)^p + B(h/2)^q \end{eqnarray*} Suppose that the e_i are known, as well as k and h. Find A, B, p, and q. Homework...

If anybody knows could they please tell me how it is known that there exists some number 'e' with the property that the derivative of ex is itself. I have a feeling this might require some definition of ab. Surely this isn't a complicated problem, but for some reason I can't work it out or...

Integral descriptive: \int _{0}^{\pi }\! \left( a+{\it k1}\,\sin \left( \theta+{\it beta1} \right) \right) {e}^{{\it k2}\,\sin \left( \theta+{\it beta2} \right) }{d\theta} I think if beta1 and beta2 come to zero,i may be a typical integral.However,in this integral,beta1 and beta2 are...