Questions tagged [exponential-function]
For question involving exponential functions and questions on exponential growth or decay.
7,947
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Bound for the upper level set
If $\int_X exp({a|f_j(x)|})d\mu(x)\le C$ for all $j$ (for some $a>0$) with $f_1,\dots,f_n$ on the probability space $(X,\mu)$ and $L^2$ norm of $f_j$ is bounded by $\eta^j$ for all $j$, for some $...
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Error term, for infinite product formula, for the exponential
For this infinite product, what is the error or remainder term? Thanks!
$$\prod_{k=0}^na_k=\left(1\pm\frac{x}{2^n}\right)^{\pm2^n}$$
If x >= 0 then,
$$a_0=1+x,\qquad a_{n+1}=\left(1+\frac{x^2}{2^{n+...
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How to construct real exponentiation?
I have been trying to rigorously define real exponentiation. Online there doesn't seem to be ANY definition of real exponentiation that covers every case of base and exponent.
In school and on non-...
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Any proof of $(E+A+\frac{1}{2!}A^2+\cdots)(E+B+\frac{1}{2!}B^2+\cdots)=E+(A+B)+\frac{1}{2!}(A+B)^2+\cdots$ which doesn't use norm of matrices?
I am reading "Linear Algebra" by Ichiro Satake.
Let $A_0,A_1,\dots$ be a sequence of complex $n\times n$ matrices.
We define $A_0+A_1+\cdots$ converges if $a_{ij}^{(0)}+a_{ij}^{(1)}+\cdots$ ...
- 1,324
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Best Statistical Method for Determining When an Exponential Function goes Non-linear?
I have a somewhat complicated question and as someone with no data analysis background, I'm searching for advice/recommendations for the best data analysis "tool" or method to do what I need....
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2
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Are $\lim_{(n,m) \to (\infty,x\infty)} (1 + 1/n)^m$ and ${\substack{n \to \infty \\m = nx}} (1 + \frac1n)^m$ correct alternative definitions of $e^x$? [closed]
I know $e^x =\lim_{n \to \infty} \left(1 + \frac xn\right)^n$. But I'd like to use a different notation (for a video). I'm not sure if this is correct, but here it is: $e^x = \lim_{(n,m) \to (\infty,x\...
- 81
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Non-linear, continuous, strictly increasing, unbounded function with a constant average rate of change?
I have a strictly increasing, continuous, non-linear, unbounded (above and below) function $w:\mathbb{R}\rightarrow{} \mathbb{R}$ such that for fixed $x,y\in \mathbb{R}$, it holds that for some ...
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Proving existence and uniqueness of the number $e$ as the exponential function base with derivative 1 at $x=0$ with $\lim_{h\to 0} \frac{e^h-1}{h}=1$.
I was just revisiting the definition of the number $e$ in Section 4.3 of Zbigniew Nitecki's Calculus Deconstructed. I'll start by saying I really like his ground-up development of the exponential and ...
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To solve the Integral $\int \frac{2^x-3^x}{2^x+3^x}dx$? [closed]
Can someone help me solve the integral
$$\int\dfrac{2^x-3^x}{2^x+3^x}dx$$
Could someone provide a step-by-step solution or suggest an effective method?
Thank you!
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How to prove limit of argmin of function involving exponents $2(1-x)^p + x^p$?
Given
$$
f(x,p) = 2(1-x)^p + x^p,
$$
define its maximizer over the domain $x\in(0,1)$
$$
x^\star(p) = \mathop{\text{argmax}}_{0<x<1} f(x,p),
$$
and then consider:
$$
x^\text{lim} = \lim_{p \...
- 551
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Finding the point of maximum curvature on an exponential
I am trying to estimate the point of maximum curvature along an exponential decay function.
A little background: I am working with raw mass spectrometry data measuring helium-4 using a helium-3 spike ...
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An inequality and two limits for series
In Corollary 2 of the paper [1] below, Qi and his coauthors derived the inequality
\begin{equation}\label{log-exp-last-proof-ineq}\tag{EI}
\Biggl[\sum_{j=0}^{\infty}\frac{1}{\binom{j+n+2}{n}}\frac{u^{...
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Estimate the exponential order of a non-continuously-differentiable function with integral inequalities
Suppose $V(x(t))$ is not continuously differentiable. There is a conclusion: If $V(x(t))$ holds
$$V(x(t))-V(x(t_{0}))\leq-\lambda\int_{t_0}^{t}V(x(s)) ds,$$
then
$$V(x(t))\leq e^{-\lambda(t-t_{0})}V(x(...
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Using Convexity to Compare Functions
I've been given the question:
Show $\frac{e^a + e^b}{2}$ > $e^\frac{a+b}{2}$.
The solution involves checking the convexity of both sides of this inequality, but I don't understand the intuition ...
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Can $\int e^{-|x|^a}\,dx$ where 0<a<1 be given an analytical solution? [closed]
As the title suggests. Are there any solutions to solving this integral analytically or by using a power series? $\int e^{-|x|^a}dx$ where $0<a<1$?
Any help, suggestions, or leads given are ...
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How does exponentials being eigenfunctions of differentiation relate to the solution of linear differential equations?
I know that exponentials are eigenfunctions of the derivative operator since $$\frac{d}{dx}\left\{ e^{\lambda x} \right\}=\lambda e^{\lambda x}$$ I also know that if a linear constant coefficients ...
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2
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How do you solve for x in the equation $(1+1/x)^{ 6x}=a$ [duplicate]
I would assume that it uses the Lambert W function in some way, but I cannot figure out a way to transform it into a form where W(x) would be useful
So far, I have not been able to get very far past ...
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Bounding a certain function
I'm currently watching a lecture where we have a function that looks like
$$C_{\alpha\beta} = \sum_{k=1}^{q^2 -1} \lambda_{k}^{t} {\rm tr}(o_{\alpha} u_k) {\rm tr}(o_{\beta} v_k).$$
Now, ${\rm tr}(o_{\...
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How to use BCH formula to compute mutiplication of group element?
i cant get the right answer.
Given the commute rule of (for example, the caculation below will be in static group):
we have: generator H, P, K, J, and commutation rule $[H,P]=0$, $[H,K]=0$, $[P,P]=0$...
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Evaluating $\int_t^Te^{-a_r(T-x)}\cdot\sqrt{s^2(f(a_n))^2+n^2(f(b_n))^2+2snp\,f(a_n)f(b_n)\;}dx$, where $f(k)=\frac1k(1-e^{-k(T-x)})$
Hello:) i am currently working on the Jarrow Yildirim model with a Two factor Hull & White process. In order to derive a expected value i have to calculate this deterministic integral:
\begin{...
- 43
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1
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$\lim_{x\to\infty}\exp{(x+iy)}$ [closed]
We can write $\exp{(x+iy)}=e^x\cdot e^{iy}$. Therefor, the limit is
$$\lim_{x\to\infty}{\exp{(x+iy)}}=\lim_{x\to\infty}{e^x}\cdot e^{iy}=\infty\cdot e^{iy}$$
My question is: what is this and what its ...
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Exponential decay curve fitting
I have an initial set of data (Column A-C), and I noticed the % Gain/Lost follows some sort of exponential decay. I've plotted it (data from A2-A13, C2-C13) and got $y=16.391e^{-2.246x}$ and $R^2=0....
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$\frac{\text {d}y}{\text {d}x} = e^y$ general solution $y = -\ln(-x+C)$ or $y = -\ln|-x+C|$?
Is the general solution for $\frac{\text {d}y}{\text {d}x} = e^y$
$$y = -\ln(-x+C)$$
or
$$y = -\ln|-x+C|$$
or something else?
Here are the steps I'm taking:
$$\begin{align} \frac{\text {d}y}{\text {d}...
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Integral the product of exponential function and Bessel functions [duplicate]
How to solve $\int_{0}^{R} r e^{a r^2}J_{0}(br)dr$. The solution of $R=\infty$ can be found in
How do I integrate this exponential + Bessel function term?
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Determining the witnesses (constants $C_0$ and $k_0$) when showing $c^n \in O(n!)$ ($c > 1$)
I'm having a hard time finding the constants/witnesses $C_0$ and $k_0$ that show $c^n \in O(n!)$. That is $|c^n| \leq C_0|n!|$ for $n > k_0$ ($c > 1$).
To be clear, I understand we can prove the ...
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Determining the witnesses (constants $C_0$ and $k_0$) when showing $n^d \in O(b^n)$ ($b > 1$ and $d$ is positive)
I'm having a hard time finding the constants/witnesses $C_0$ and $k_0$ that show $n^d \in O(b^n)$ ($b > 1$ and $d$ is positive). That is $|n^d| \leq C_0|b^n|$ for $n > k_0$
I understand you can ...
- 933
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A root finding problem
While solving a partial differential equation, I obtained the following function:
$$ F(z)=(k^2 - z^2)J_1^2(z)-z^2J_0^2(z) \tag 1$$
where $k \in (1, \sqrt2)$ and $z$ is a complex variable. I need to ...
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Definite integral of $e^{-ax^2+bx+cx^{-1}-dx^{-2}}$
I need to find a close-form expression for the following definite integral:
$$
\int_{0}^{\infty}\exp\left(-ax^{2} + bx + cx^{-1} -dx^{-2}\,\right)\,{\rm d}x
$$
where $a,b,c,\ \mbox{and}\ d$ are all ...
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Examples of Taleb's explanation of "exponential"?
Am requesting some examples (of actual mathematical functions) that fit what Taleb (@nero) describes here:
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How do I take this 2D integral of an exponential function that has a truncation?
I have a 2D distribution that goes like $f(x,y) = exp(-\frac{x^2 +y^2}{v_t^2})$
that I want to truncate at a given radius away from the origin. The truncation line has the form $y(x) = \frac{x^2 - a^...
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Solve the differential equation that define exp(x)
In the wikipedia page for the exponential function in the "formal definition" section I found this statement:
Solving the ordinary differential equation $y'(x)=y(x)$ with the
[initial ...
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e{ } and e( ) Notation
I was reading An Introduction to the Theory of Numbers by Hardy and Wright, and on pages 66 and 67, I encountered the notation, as following
What does the $e( )$ and $e\{\}$ notation mean? Any help ...
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Is the intersection of a function and its inverse on the line $y=x$ equivalent to the function being strictly increasing? [closed]
Is the following statement true?
The functions $f$ and its inverse $f^{-1}$ intersect each other on the line $y=x$ if and only if $f$ is a strictly increasing function.
I think this statement is not ...
user1340729
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Closed form of $f(n) = \prod_{m=2}^{n-1}( e^{\pi i n/m} - e^{-\pi i n/m})$
For context, I am not a mathematician, but I like to do explore some math concepts a couple of times a year. I have been playing with an idea for the past few years or so and a while back I asked this ...
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Question on exponential functions and drawing suitable lines on the graph
This question is from an exercise in an Edexcel Further Pure Mathematics book. Parts a, b, and c were fairly easy for me. For part a, I found the corresponding values of y using a calculator and in ...
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2
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Examples of expansions of the exponential of a sum of two matrices [closed]
The exponential function of a matrix is fundamental in mathematics, physics and beyond.
One can define it using the power series
$$
e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}
$$
For any matrix $M$ ...
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1
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What to consider when taking kth root on both sides of equality
Say I have the following expression:
$10^{l} = a^{k}$
If I take the kth root of both sides, does that mean we get:
$10^{\frac{l}{k}} = a$
We don't have to consider anything with plus or minus?
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How to calculate $\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x$
As the title mentioned, I want to calculate
\begin{equation}
\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x,
\end{equation}
where $n$ is a positive integer, $c$ is a positive real number in the ...
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Can a parabola be exactly replicated using exponentials?
I came across this interesting artifact of a problem I was helping a friend solve where I used Euler's exponential forms of sine and cosine which later became of the form $e^{ix} + e^{-ix}$ on one ...
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How can we show that this compelx sum of products actually converges to $0$?
I'm evaluating an integral and ended up with the expression $$\sum_{n\ge2}\prod_{i=1}^n\alpha_i\int_0^\infty\sum_{i=1}^n\left(\prod_{\substack{j=1\\j\ne i}}^{n-1}\frac1{\alpha_j-\alpha_i}\right)e^{-\...
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Implicit function equation $f(x) + \log(f(x)) = x$
Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that
$$
f(x) + \log(f(x)) = x
$$
for all $x \in \mathbb{R}_{>0}$?
I have tried rewriting it as a differential equation ...
- 2,005
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2
answers
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Conditional expected value for exponential dsitribution
I'm meet with the following problem: A given gas station two fuel pumps, $A$ and $B$, whose refilling time follow an exponential distribution with rates $\lambda_A$ and $\lambda_B$, respectively. ...
- 13
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0
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81
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Conflicting answers when calculating limits [duplicate]
We are asked to evaluate the limit $$\lim_{x \rightarrow \infty}\frac{e^x}{{\left(1+\frac1x\right)}^{x^2}} $$
Applying L'Hospital's rule, we get the correct answer to be $\sqrt e$. However if we apply ...
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Exponential series with fractional exponents $\sum_{n=0}^{\infty} \frac{x^{\delta n}}{\Gamma(\delta n+1)}$
$$
\mbox{I´m trying to prove that}\quad
\sum_{n=0}^{\infty} \frac{x^{\delta n}}{\Gamma(\delta n+1)}\quad\mbox{converges on}\ \mathbb{R}_{\geq 0}
$$
For $\delta =1$ we would simpy have $\exp\left(x\...
- 329
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0
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Smooth periodic extension of a function on $[0,1)^d$ to $\mathbb R^d$
Here is my goal: Given $n\in\mathbb N_0\uplus\{\infty\}$, I want to construct a function $q:[0,1)^d\to(0,\infty)$ such that $q\circ\iota\in C^n(\mathbb R^d)$, where $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;...
- 13.8k
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Understanding the domain of a function raised to another function
So the question is to find the sum of all real solutions to the equation $\left(x^{2}-5x+5\right)^{x^{2}+4x-60} = 1$.
The solution intends us to take 3 cases:
The first case is to set the base equal ...
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Problems with understanding Dirac Delta Properties [closed]
I need to check equation ${\bf 6.15}$ in an $\tt arxiv$ paper for my work, but I don't get why the Dirac Distributions change and why there is that exponential ...
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2
answers
81
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What formula would I use to calculate total income over $x$ years with a fixed salary increase rate.
I'm wanting to calculate how much total money I would have earned over x amount of years, but while accounting for increase in pay at the end of each year (or maybe even continuous?)
Lets say for ...
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Convergence rate of $(1+x/n+\dots)^n$
It is a well known fact that
$$\lim_n (1+x/n+O(n^{-3/2}))^n=e^x$$
For example, this is a key step in the standard proof of the central limit theorem.
What can we say about the rate of convergence of ...
- 5,507
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votes
1
answer
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Existence of a smooth function with some boundary conditions
Given $\epsilon > 0$, can we find a smooth function $f:[0,\epsilon]\rightarrow [0,1]$ satisfying the following conditions?
$f(0)=1$ and $f(\epsilon) =0$.
$f^{(n)}(0+) = f^{(n)}(\epsilon-)=0$, i.e.,...
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