Questions tagged [exponential-function]
For question involving exponential functions and questions on exponential growth or decay.
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Evaluating tricky definite complex integral and getting numerical result as a single complex number
For given constants $a\in \mathbb{R}$ and $b \in \mathbb{R}$ and I would like to numerically evaluate the following complex integral
$$\int_{-1}^{1} (1-|x|) \exp(ia(b+x)^2) \, d\mathrm{x}. $$
I am not ...
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How to prove exponential functional identity knowing that it is a solution to a first order ODE and knowing its Taylor expansion
Establish the identity
$$E(ax)E(bx) = E[(a+b)x]$$
knowing that
$y = E(px)$ satisfies $y' - py = 0$ and
$E(px) = \sum_{n=0}^\infty\frac{(px)^n}{n!}$
An additional hint the textbook gives ; "...
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integral of power of gaussian CDF function multiplied by exponential
I am trying to solve the following integration (closed-form solution) or numerically:
$$
\int_{0}^{x}\left[1-erfc \left(\frac {x-\alpha}{sqrt(2) \sigma}\right)\right]^n \lambda e^{-\lambda \alpha} d\...
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Another proof of e is irrational?
I find some proof here, but can I prove it in the following way?
Assume $e$ is rational, then $e=\frac{p}q$, both $p$ and $q$ are positive integers. By Lagrange's Remainder Theorem,
$$e^x=1+x+\frac{1}{...
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Finding smallest number returned by an integral [closed]
I wanna preface this: I never formally learned integrals and differentials
So, I'm currently helping a friend study for an exam, however... there's a specific exercise here, which is kinda out of my ...
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Can I rewrite an exponential with product of powers as product of two terms?
I have what I thought was a relatively simple problem but cannot quite pinpoint the answer. I want to do a regression of something like $a \times \mathrm{e}^{-b/x}$. However, I can only regress the ...
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Which holomorphic functions have constant argument on rays from the origin? On circles centered at the origin?
Multiples $$f(z) = c z, \qquad c \in \Bbb C \setminus \{0\},$$ of the identity function on $\Bbb C \setminus \{0\}$ trivially all satisfy the following special condition:
Condition A: All points on a ...
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What's the expected time to get on bus, given bus coming time is $\exp(\lambda)$ and there is 0.6 chance bus is not full?
What's the expected time to get on the bus, given bus coming time is $\exp(\lambda)$ and there is 0.6 chance the bus is not full?
Let $T \sim \exp(\lambda)$ be the time of the bus coming. Then $E[T] =...
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How do we show $\int_{[0,\:1)^2}\exp\left(-\frac{\left\|x-\frac{x_i+x_j}2\right\|^2}{\sigma^2}\right)\:{\rm d}x=\pi\sigma^2$ here?
Let $D=[0,\sqrt N)^d$ for some $N\in\mathbb N$, $x_i,x_j\in D$ and $\sigma>0$. In equation $(30)$ of this paper I've read that, if $d=2$, then \begin{equation}\begin{split}\int_D\exp\left(-\frac{\...
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Integration by Parts Table Technique With Exponential and Polynomial Higher Order
I come to you with a rather simple question in need of a reference or two from more knowledgable sources. Here was the simple integration by parts problem:
However, a rather peculiar table (book-...
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How to find optimal parameters in Gaussian basis set?
I decide to ask my question (https://physics.stackexchange.com/questions/755354/how-to-find-optimal-parameters-in-gaussian-basis-set) in MATHEMATICS part of StackExchange too, because I think the ...
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Is the Lambert W function the Newton flow of the exponential function?
Is this right?
The Lambert W function, denoted by $W(z)$, is defined as the inverse function of $f(z) = ze^z$. In other words, if $w = W(z)$, then we have $z = w e^w$.
The continuous Newton's method ...
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Prove $x^{a+b} + y^{a+b} \le 2$ for $x^a+ y^a = x^b+ y^b$ $(a \ne b)$
Let $a$, $b$ distinct positive numbers and $x$, $y>0$ such that
$$x^a+ y^a = x^b+ y^b.$$
Show that $x^{a+b} + y^{a+b} \le 2$.
Notes:
It is easy to show that $x^b + y^b\le 2 $. Indeed, assume ...
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How to simplify $e^{-\sum_{i=-k}^k(k-|i|)x_i}$?
Consider the expression given by
$$
\large e^{-\Large\sum_{i=-k}^k(k-|i|)x_i}
$$
Is there a way of simplifying this expression?
For example, provided $\{x_i\}$ is bounded and "smooth" enough ...
- 3,068
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Why does compound interest lead to a function that is the derivative of itself?
There are two ways to derive the number $e$. The way it was originally found, was thinking of compound interest. Let's say the interest rate is $z$ per year. If the interest is compounded once at the ...
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$\lim_{x\to -\infty} x^x$ exists? and if so, what's its value?
Disclaimer: I am completely and utterly new here. Please openly correct me if I'm doing anything wrong.
I received the following question on a recent calc test:
$$\lim_{x \to - \infty} x^x=?$$
$$a) 1 \...
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Is this double exponential smoothing algorithm valid?
I use this double smoothed prediction algorithm:
https://cs.brown.edu/people/jlaviola/pubs/kfvsexp_final_laviola.pdf
Equations:
$$
First Smoothing Statistic:\;Sy_t' = \alpha y_t + (1-\alpha)...
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Solve interesting exponential equation in terms of x
I am having difficulty solving this equation in terms of $x$
$y =\dfrac{x^x}{x^{x^{\frac{x}{k}}}}$
I have been able to re-arrange exponents into the following:
$x^x x^{-\sqrt[k]{x^x}} $
$x^{x - \sqrt[...
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How to simplify $\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}$
Is it possible to simplify
$$
S(\mathbf{x})=\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}
$$
A few observations:
$\sum_{k=0}^{n}\sum_{i=0}^k x_{i}=\sum_{k=0}^{n}(n+1-k)x_k$
$ e^{\...
- 3,068
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Lion and wildebeest population modelling question issue
I am a student completing A levels and recently received the following question in an official mock exam for the Edexcel A-level in Mathematics. Having discussed the question extensively with my peers ...
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2
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L'Hopital's Rule for nested exponential function
I am trying to look for the following limit:
$$\lim_{x\to\infty} (xe^{1/x} - x)^x$$
I re-wrote the limit in the following way
$$\exp\left( \lim_{x\to\infty} x\ln \left[ x(e^{1/x} -1) \right] \right)$$...
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Condition on $h$ ensuring that $e^{-(c(h-k))^2}<\varepsilon$ for all $k\in\mathbb Z$
Let $\varepsilon\in(0,1)$ and $c>0$. Note that $$\varphi(x):=e^{-(cx)^2}<\varepsilon\Leftrightarrow|x|>\frac{\sqrt{-\ln\varepsilon}}c\tag1$$ for all $x\in\mathbb R$. Now let $h\in(-1,1)$.
...
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Faulty generalization of $(-1)^{z}$, is there a hole in my argument
I am an Vietnamese grade 12 student. We were studying about exponential functions until I read the part that writes:
For the function $f(x) = a^{x}$, we always assume a is an positive real number ...
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how to verify the inverse compensator of an exponential Hawkes process?
I derived an expression for the inverse compensator of the Hawkes process with a sum of exponential kernels that I wrote about at https://vixra.org/abs/1211.0094 and want to verify that it is correct.
...
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Differential for $2^x$ is $2^x \ln 2\, \Delta x$?
Here's a question from my calculus textbook:
Show that when $\Delta x \to 0$, the increment in the function $y = 2^x$, corresponding to an increment $\Delta x$ in $x$, is, for any $x$, equivalent to ...
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How do i write the exponential decay for this? [closed]
At this time, the number of people who use a particular health plan is 19,250. The number of people who use the plan is expected to decrease by 14% each year.
Write an equation of the type you just ...
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1
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Solve for $x$: $10^x \equiv 34 \pmod{49} $
I found the generalization that $10^{2n} \equiv 2^n \pmod{49}$, but I haven’t been able to prove it or guarantee that it holds for all nonnegative integers. Is there a reason this holds?
Another ...
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Closed form of an averaging series
Consider the expression
$$
f(\mathbf{x},n)=\sum_{k=0}^{n} \frac{e^{-k^2\overline{x_k}}-e^{-(k+1)^2\overline{x_k}}}{(2k+1)\overline{x_k}}
$$
where $\mathbf{x}=[x_{-n},\cdots,x_{-2},x_{-1},x_0,x_1,x_2,\...
- 3,068
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Derivation of quaternion logarithm
I'm trying to understand how the inverse of the quaternion exponential was derived. Given the definition of the quaternion exponential, $$e^Q=e^{a+bi+cj+dk}=e^{a+v} = x+yi+zj+wk = e^a\frac{v}{|v|}\sin(...
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How to solve the equation $(1-e^{-αa})(1-e^{-βa}) = 1/2$ for a?
Here is what I tried:
$(1-e^{-αa})(1-e^{-βa}) = 1/2$ implies $e^{-ln(2)} -e^{-βa} -e^{-αa} + e^{-a(β+α)} = 0$
But from here I don't know how to get the value of "a", any help ?
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Inequality involving exponentials and integrals
I am trying to understand the following inequalities, but I am lost in the step from (1) to (2). Can't find where the $|v|^{-(p-1)/p}$ comes from. I guess I am missing some elementary inequality ...
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Solving for i in mortgage payment formula in M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1].
This is my first post ont his website, so please excuse me if i missed some rules (please let me know and i'd be more than happy to edit my question accordingly).
I'm trying to figure out what level ...
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Minimizing a function $f$ of the form $f(x)=\prod_if(x_i)$, is it possible to ensure $x_1,\ldots,x_j$ minimizes $\prod_i^jf(x_i)$?
Let $\varphi:\mathbb R\to\mathbb R$ with $$\varphi(x)\xrightarrow{|x|\to\infty}0,$$ $d\in\mathbb N$ and $$f_d(h):=\prod_{i=1}^d\sum_{k\in\mathbb Z}\varphi(h_i-k)\;\;\;\text{for }x\in\mathbb R^d.$$ ...
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Extension of a well known formula for the Euler number into the complex domain
It is well known that $\lim_{n \to \infty } (1+\frac{r}{n})^{n}= e^{r}$ for every real number $r$, where $e$ denotes the Euler number. For every complex number $z$ a meaning is given to $e^{z}$ (which ...
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Finding the value of this exponential limit
The limit is given by
$$
\lim _{x \rightarrow+\infty}\left(e^{\sqrt{x+1}}-e^{\sqrt{x}}\right)^{\frac{1}{\sqrt{x}}}
$$
I don’t know where to start
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A Series Representation for the Natural Logarithm Not Comprised of Exponential Terms
Using the Riemann Integral and the fact that the indefinite integral of $\frac{1}{x}$ is $\ln|x| + C$ I get
$$\int_a^b \frac{1}{x}dx = \lim_{n \to \infty} \sum_{i=1} ^{n} \frac{1}{a+\Delta xi}\Delta x=...
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Exponential function using binomial approach and approximation by power series, both using python. Why different results?
Why using power serie the result was so different ? I imagine it happened due to the truncation, I used 20 terms, are more terms needed ?
EDITED: The code is below
...
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Solving $|x-1|^{\log_2(4-x)} \le|x-1|^{\log_2 (1+x)}$
Let us solve
$|x-1|^{\log_2(4-x)}\le|x-1|^{\log_2 (1+x)}......(*)$
Let $4-x>0 ~\& ~1+x>0$.......(1)
Case 1: $|x-1|\le 1\implies 0\le x\le 2$........(2)
We get $\log_2(4-x) \color{red}{\ge} \...
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How to solve $2^{\sin^2(x)}=\cos(x)$
$$2^{\sin^2(x)}=\cos(x)$$
How can I find the solution?
This is what I did :
$$\ln(2^{\sin^2(x)})=\ln(\cos(x))$$
$$\sin^2(x)\ln(2)=\ln(\cos(x))$$
$$(1-\cos^2(x))\ln(2)=\ln(\cos(x))$$
$$\cos^2(x)\ln(2)+\...
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2
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136
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Hints on solving $x^{2x}-(x^2+x)x^x+x^3=0$
Solve this equation over $\mathbb{R}^+$: $x^{2x}-(x^2+x)x^x+x^3=0$
I’ve been trying to solve this exponential equation but can’t get the answer because normal substitution ($y=x^x$) isn’t working. ...
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Simplifying an arbitrary constant.
Could someone explain me this simplification?
I cannot understand exact reason why $c_1$ is before $\exp$ function without being in another one.
Screenshot presents end of solution of this ...
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2
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78
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Is there a proof $e^ae^b=e^{a+b}$ using its power series expansion?
We know $e^ae^b=e^{a+b}$. It iss a standard property of exponentiation; that it provides an isomorphism from $(\mathbb{R},+)$ to $\mathbb{R}^+,\times)$.
However, I don't see any natural reason for ...
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Trouble with integration of following: $ \int^\infty_0 \exp(a(x)+b(x))a'(x)dx$
I tried to solve it by partial integration but I am getting similar expression in loop.
I saw similar stuff before that people use gamma function to solve this type of question but I can not handle it ...
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4
answers
105
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Prove $\lim \limits_{h \to 0} \frac{a^h-1}{h} = \ln (a)$ without using L'hospital's.
I'm a Calc 2 student and was curious as to why $\frac{d}{dx}a^x = a^x\ln a$. Using the limit definition you can arrive at $\frac{d}{dx}a^x = \lim \limits_{h \to 0} \frac{a^x(a^h-1)}{h}$ so the part ...
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0
answers
27
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How to deduce if a decay graph is polynomial or exponential?
I have a graph of the following shape -
shape of the graph
I am trying to measure whether this is an exponential decay or a polynomial decay. Since both kinds of functions have the same shape, I am ...
-1
votes
1
answer
18
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Finding the perfect damp for a sinusoid graph
Below is a sinusoid function that is defined by this equation:
$ 9.36 \times\sin(0.95x-2.06)+13.64 $.
Would it be possible to create a function that produces a dampening close to the following:
enter ...
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1
answer
91
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Function to approximate infinite series $\sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n+1)!}$
I'll provide background information at the bottom of the post for those who are curious, but the problem at hand is finding a function to approximate the value of
$$\sum_{n=0}^\infty (-1)^n \frac{x^{...
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0
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Reference books for $e^x$
I'm in search of a book in which I can learn about exponential functions.
Is there a good book in which I can learn about the history of $e$ , definition of $e^x$ ?
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Closed form for integrals of an exponential with rational function argument?
I am trying to find the closed form for the following integral:
$$I=\int_{-\infty}^{\infty}e^{-a^2(x-b)^2+\frac{c}{1+x^2}}\ {\rm d}x$$
where $a,b,c\in\mathbb{R}$, and $a\ge0$. I know that the integral ...
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votes
2
answers
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Behavior of $y = x^x$ when $x$ approaches zero
I am trying to understand the behavior of $y = x^x$ when $x$ approaches zero. When $x \le 1$, $y$ initally decreases but becomes larger at some point (around $x = 0.35$) and starts approaching one as $...
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