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Questions tagged [exponential-function]

For question involving exponential functions and questions on exponential growth or decay.

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Exponential function expansion

Can someone please explain why this is true? $$ \exp\left(\ln(x) + \frac{\ln(x)}{x} + \frac1x + O(\frac{1}{x^2})\right) = x \left(1 + \frac{\ln(x)}{x} + \frac{1}{x} + O(\frac{\ln(x)^2}{x^2})\right) $$...
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Integral form of a limit

Consider: $$\lim_{n \rightarrow\infty} \sum_{k=0}^n {n\choose k} \frac{1}{n^k}$$ One might consider using the binomial expansion to obtain: $$\lim_{n \rightarrow\infty}\left(1+\frac{1}{n}\right)^n $$ ...
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Applications of the logarithm property $(\log_pq)(\log_rs)=(\log_rq)(\log_ps)$

I recently came across this intriguing property of logarithms: $(\log_pq)(\log_rs)=(\log_rq)(\log_ps)$, which is linked to a result known as the "chain rule" for logarithms. This seems to be less well-...
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How to isolate x in equation with ln(erfc(x))?

I was wondering whether it is possible to isolate $x$ in this equation: $$ f = 1- \exp\left(\frac{xt}{C^2}\right)\operatorname{erfc}\left(\frac{\sqrt(xt)}{C}\right) $$ I have not come further than $$...
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Expectation of some function with exponential form under Gaussian distribution

Suppose we have a real-valued function $g \colon \mathbb{R} \to \mathbb{R}$ (here, assume that we do not know the closed-form of $g$), and define its composition function $f$ by $f(x) := \exp (g(x))$. ...
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Scientific notation doesn't make sense?

I've already looked at this and this question. However, using R, I get an output like 4.97e+00. By the other questions this should mean $4.97 \times 10^{+00} = 4....
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169 views

How can I solve this infinite exponent tower?

Using calculus (or algebra), how would I solve an infinite exponent tower such as this? $$c_0x^{c_1x^{c_2x^{c_3x^{.^{.^.}}}}}=a$$ Where $c_0=1$ and $c_{n+1}=\frac{c_n}{2}$ for $n=0,1,\ldots$ and $a&...
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Calculating Directional Derivatives

Q) Let $f(x,y,z)=xy^2z^3+ exp(xyz)$. Find the directional derivative of $f$ at the point $(1,1,1)$ in the direction to the point $(-1,2,3)$. A) I have been given the following formula, $D_u(f(a,b,c))=...
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Must $f' = \Omega(f)$, if $f = \Omega(e^n)$?

For some example functions that grow more slowly than $e^n$, like $2n^3$, or any polynomial, their derivative grows more slowly than them (for $2n^3$ it goes down to $6n^2$). For $e^n$, its ...
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what is the Integral of Exponential function??

I need to calculate the integral of $$\int_{0}^{u}e^{-\alpha t-\frac{\beta}{t}}dt$$ From one book I came to knnow $$\int_{0}^{\infty} e^{-\alpha t-\frac{\beta}{4t}}dt = \sqrt{\frac{\beta}{\alpha}}...
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Logarithmic Paper

enter image description here I have a question related to logarithmic paper. Does the midpoint shows 0.5*(10^(-1))? Or less than that because of increase of logarithmic also? I am curios about the ...
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System with exponential and log

Solve the following system in real numbers: \begin{cases} \log_2(x+y)+4=2^x+2^y \\ \frac{x+y}{4}+\frac{xy}{x+y}=1 \end{cases} I used the fact that $\frac{xy}{x+y} \leq \frac{x+y}{4}$ in order to get $...
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Fourier Transform Question Help

To find the fourier transform of $$f(x) = e^{-3|x|}cosx$$ By using Euler's formula $$cosx=\frac{e^{ix}+e^{-ix}}{2}$$By definition, $$\mathcal{F[f(x)] = \hat{f}(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\...
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Weird result regarding “infinitely explosive” differential equations

Firstly, take the family of differential equations $\dot x = > \frac{dx}{dt}=x^\alpha$, for any $\alpha \in \mathbb R$ The solution to these equations is $$(\text{for } \alpha=1):x(t)=x_0e^t$$ $$(\...
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How to find a variable, within an exponential, in a sum series

I am trying to solve for D in the following equation: $$ \frac{M_t}{M_\infty} = 1 - \frac{32}{\pi} * \sum_{n=1}^{25}\frac{exp(-q^2_nDt/R^2)}{q_n^2} * \sum_{p=0}^{100} \frac{exp(-(2p+1)^2)\pi^2Dt/H^2)}...
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Solving a limit to prove Big Theta

I'm given the equation $(n^2+1)^{10}$ I have to find it's efficieny class, and prove it: It looks to be $Big Theta (n^{20})$ I've tried this using limits: $\lim_{n\to\infty}(n^2+1)^{10}/n^{20} = \...
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Zero's and fixpoints of $ f (\exp(b z)) $

Let $ b > 1$. Let $f(z)$ be -meromorphic over the entire complex plane. Also $f(z)$ maps the real line to the extended real line. And $f(z)$ is non-linear. Consider the strip $A$ such that $ 0 =&...
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Upper bounds on the norm of a matrix exponential

Let $A\in\mathbb{R}^{n\times n}$ be a non-negative matrix and let $D\in\mathbb{R}^{n\times n}$ be a signature matrix, i.e. a diagonal matrix having either $+ 1$ or $-1$ elements on its diagonal. ...
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Solution for $\int \frac{1}{1-we^w}dw$

I am looking for a solution or a method of approximation for : $$\int \frac{1}{1-we^w}dw$$ that came up while working on an ODE problem. Got any suggestions? Note: $w$ is also a one variable ...
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q-analogies of Exponential and Logarithmic functions

From several different sources, I found two different q-analogies of the Exponential function. According to this and this sources exponential function has two q-analogies given by: $$\exp_q(x)=\sum_{...
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Is there an addition formula for $ f(z) = \exp(z) + \sin(z) $?

I know $\exp(z) , \sin(z) , \sin^2(z) , z \exp(z) , \sinh(z) , \exp(z) \sin(z) $ all have addition formulas. I'm still confused about addition formulas. It seems hard to tell if a function has an ...
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Combining Partial Sum and Exponent functions

The Challenge I am facing is combining two formulas based for a game on command and Conquer. The Silo creates continuous productivity when left alone for a while with storage. Only looking at the ...
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Finding the maximum of an exponential sum

my problem is how to find the maximum of an exponential sum. The interested function is $f(x)=\sum^M_{m=1}\exp(a_mx^2+b_mx+c_m)$. $a_m$, $b_m$ and $c_m$ for all $m$ are known coefficients and $M$ is ...
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Find the close form of a summation with binomial coefficients and fractions

I was stuck by a problem for quite a long time, which is relating to the summation with binomial coefficients and fractions as below: $\sum_{i=1}^M\sum_{j=1}^K(-1)^{i+j}\binom{M}{i}\binom{K}{j}\frac{...
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Exponential family of distributions

A family of pdfs or pmfs is called an exponential family if $$f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$$ so, I wonder if this distribution belongs to the ...
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Inequality involving exponential integral

I tried to prove the following inequality: For $\tau\ge 1$ and $A>0$, $$\frac{1}{\tau} \exp\left(A\right)E_1\left(A\right)\le \exp\left({\tau A}\right)E_1\left({\tau A}\right),$$ where $E_1(x)=\...
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Generalizing an iterative logarithm integral.

First, some notation: $f^{\star0}(x)=x$ $f^{\star k}(x)=f\left(f^{\star k-1}(x)\right)$ So that for any integer $k$, $\log^{\star k}(x)=\underbrace{\log(\log(\dots(\log}_k(x))\dots))$. I then came ...
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Inequality proof on polynomials and exponential

I want to prove the following : Let $f:\mathbb{R}^+ \to \mathbb{R^n}$ and $C,a,b \in \mathbb{R}^+$ with $a<b$ then $\exists C'$ s.t. $\forall t \geq0$ : $$||f(t)|| \leq C (1+t)^{n-1}e^{-bt} ...
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Exponential map of a Lie group around $0$

I found the following on my notes of differential geometry: Proposition: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}=T_1G \cong \mathbb{R}^n$. The exponential map $\text{exp} \colon \...
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Sum of Logarithms of expoenetial functions

Given a matrix $P \in \Re^{n \times d}$, and a column vector $\theta \in \Re^d$. Assume that $\sum\limits_{i=1}^n \ln{(1+e^{P_i\theta})} \leq 1$, where $P_i$ is the $i^{th}$ row in $P$. What can be ...
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When is $\frac{2 n f(n)}{n !}$ in the order of some fixed power of $n$?

I would like to know when $\frac{2 n f(n)}{n !}$ is $O (n^b)$ where $b$ is a constant. Here, $n$ is a positive integer. My attempt: $$ \frac{2 n f(n)}{n !} = \frac{2 n f(n)}{\sqrt{2 \pi n} (\frac{n}{...
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Solve $x = \sqrt[x]2$

How does one solve $x = \sqrt[x]2$ for $x$? This can be otherwise stated as $x = 2^{1/x}$ Raising both sides to the power of $x$: $x^x = (2^{1/x})^x$ $x^x = 2$ But I don't know where I can go from ...
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What's the worst sequence that still leads to a converging series?

As a background, I'm initially interested in sequences $a_n$ giving rise to functions $\sum_{n=0}^\infty a_nx^n$ for $x\in(0,1)$ and their diverging behavior for $x$ to $1$. E.g. the geometric series $...
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Integral of third order polynomial exponential

I am looking for approximated or exact solution of \begin{align} I = \int_R \exp(cx^3-ax^2+bx)dx \end{align} where $a,b,c$ are complex numbers defined as: \begin{align} c &= \frac{1}{3}i\pi\phi'''...
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Formal Power Series Composition with Exponential

I have seen formal power series expressed as $$B(z) = \sum_{i=1}^{\infty} b_i{z}^i,$$ but then also as $$B(z) = \sum_{i=1}^{\infty} \frac{b_i}{i!}{z}^i.$$ Is there a significant difference between ...
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Newtons Law Of Cooling (And Heating)

Rule is: $D= A.e^{-kt}$, Where: $k,a$ are elements of real numbers, $D$ is the difference between the temperature of the item and the surrounding air, and t is the time in hours since the object ...
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Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
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Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that the ...
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What is different between $\frac{1}{1+\lambda x}$ and $\exp{(-\lambda x)}$

I want to choose a function $f(x)$ which has properties: $f(x)$ closes to $0$ when $x$ goes to $+\infty$ . I have two option for that $f(x)=\frac{1}{1+\lambda x}$, where $\lambda$ is tuning ...
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Confidence interval for exponential - is it the shortest possible?

The confidence interval for an exponential distribution is said to be: $$\frac{2n\bar{x}}{\chi^2_{1-\alpha /2,2n}}<\frac{1}{\lambda}<\frac{2n\bar{x}}{\chi^2_{\alpha /2,2n}}$$ In general we aim ...
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Solve for deceleration in exponential decay equation

$$y = y_0 + v_0\cdot d\frac{1 - d^{t}}{1 - d}$$ $y$ = final position $y0$ = initial position $v0$ = initial velocity $d$ = deceleration $t$ = elapsed time How do I solve for $d$? Specifically, ...
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Taylor of $\ln(f(exp(x))))$?

Let $ f(x) = \sum a_n x^n$ Such that The $a_n$ are real and $f(a),f ' (a) , f " (a) > 0 $ for any real $a > 0$. Let $ \ln(f(exp(x))) = \sum b_n x^n $. Let $c_n = a_n - b_n$. For a given $f$ ...
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exponential generating function for the number of ways to arrange marbles in a line

Say we have red, green, and blue marbles that we are arranging in a line of length n. We need to use an even number of blue marbles, at least two red marbles, and at most two green marbles. I am ...
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Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ https://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
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175 views

How can I apply Rouche's Theorem here?

How many solutions lie in the left half-plane? $$f(z) = z^3+2z^2-z-2+e^z=0$$ My work so far: Factoring the polynomial, moving the exponential term over to the RHS, and taking the modulus of both ...
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Integrate exponential with modulus

I need to prove the following integral relation: $$\lim_{T \to (1-i \epsilon) \infty} \mathrm{\int_{-T}^T dt_1 dt_2 e^{-ia |t_1|} \cdot e^{-ib |t_2|} \cdot e^{-ic |t_1-t_2|}} = \frac{-2}{(a+b)(b+c)} + ...
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exponentialfunction poofs with generating functions

I have two things to proof. $(e^{ax})^{-1}=e^{-ax}$ and $(e^{ax})^{m}=e^{(ma)x}$ I know the power series of $e^x$ and $e^{ax}$ and that $e^{ax}e^{bx}=e^{(a+b)x}$ I tried it forward and backward: $(...
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74 views

Compute $\int_0^1 \frac{\exp(u^3 - 4/3 u) du }{\sqrt {1 - u^2}} $

Compute the integral $$\int_0^1 \frac{\exp(u^3 - 4/3 u) du }{\sqrt {1 - u^2}} $$ I assume contour integration is the easier way ?
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Memoryless property implies $E(\max\{X_1,X_2,\ldots,X_n\})- E(\min\{X_1,X_2,\ldots,X_n\})=E(\max\{X_1,X_2,\ldots,X_{n-1}\})$

$X_1,X_2,\ldots,X_n \sim \mathrm{Exp}(\lambda), \quad \text{i.i.d.}$ How to show the following equation by using memoryless property: $$E(\max\{X_1,X_2,\ldots,X_n\}) - E(\min\{X_1 , X_2, \ldots, X_n\...
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Rewriting the trigonometric representation of a stochastic process into exponential form

Fenton and Griffiths (2008) in Risk Assessment in Geotechnical Engineering give the following representation of a stochastic process: $$Z(x_j) = \sum_{k=0}^{N} A_k \cos(\omega_k x_j) + B_k \sin(\...