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

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2answers
44 views

Intersection of a power function with a line: how to compute?

How to compute $x$ from $$q x^p = 1 - x$$ where $x$ and $q$ are positive, while $p$ is a real number? When $p > 0$: it's two monotonic functions, one increasing and one decreasing, and having ...
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0answers
46 views

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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1answer
68 views

number of permutations of [n] for which all cycles have even length

I'm looking to find the number of permutations of [n] for which all cycles have even length, call that number $f_n$. I've seen here: Number of permutations of a specific cycle decomposition that the ...
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0answers
39 views

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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2answers
48 views

On Proving that $e^x$ is continuous at $0$ utilizing a limit result.

I was assigned the task to prove that $e^x$ is continuous in $x=0$ utilizing the fact that $$\lim_{x \rightarrow 0} \frac{e^x - 1} {x} = 1 $$ I think I am supposed to show that from the fact that for ...
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4answers
1k views

Showing that the exponential expression $e^x (x-1) + 1$ is positive

I'm looking at $$ f(x) = e^x (x-1) + 1$$ I'm having the feeling (based on the application where I am using it), that $f(x)$ should be strictly positive for $x > 0$. Indeed, Wolfram Alpha plots ...
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4answers
61 views

How to fit data to an asymptotic exponential?

I have 3 points that I must adjust to the following formula: $$ C = a' \cdot (1-e^{\alpha \cdot t})$$ The magnitudes I know are $C$ and $t$, and I have to obtain $a'$ and $\alpha$. I know that ...
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1answer
54 views

Solving $\operatorname{cis} x \operatorname{cis} 2x \operatorname{cis} 3x \dots \operatorname{cis} nx=1$

Given the equation: $$\operatorname{cis} x \operatorname{cis} 2x \operatorname{cis} 3x \dots \operatorname{cis} nx=1$$ How can I solve it? I know that $\operatorname{cis} x=\cos x+i\sin x$, but I ...
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0answers
18 views

Solving a complex exponential equality for all solutions

Since I learned how to take complex exponents and write the solution in $a+bi$ form, I've wanted to solve the following problem: $$x^n=y^m$$ Where we have $x,n,y,m\in\mathbb{C}$. Rewriting, this ...
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3answers
57 views

Does the logistic function have a relation with arctan(x)

The logistic function is: $$f(x)=\frac{L}{1+e^{-k(x-x_0)}}+B$$ It's plot looks similar to the plot of $arctan(x)$. Therefore I was wondering whether there is a relationship between these two ...
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2answers
131 views

Show that $f'(0)= \lim_{\Delta x \to 0}\frac{f(\Delta x)-1}{\Delta x} = 1$

This question is related to another question I asked here. Specifically, using the definition of $e$ I gave in that question: There exists a unique complex function $f$ such that $f(z)$ ...
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3answers
112 views

For a fixed complex number $z$, if $z_{n}=\left( 1+\frac{z}{n}\right)^{n}$. Find $\lim_{n \to \infty}|z_{n}|$

This is a step on the way to proving $\lim_{n \to \infty}\left(1 + \frac{z}{n}\right)^{n} = e^{z}$. Please do not mark this question as a duplicate. I am not asking the same thing other people are ...
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1answer
84 views

Proof of existence and uniqueness of the exponential function using ODEs

In our lecture notes for our complex analysis class, we were given the following theorem: Theorem: There exists a unique complex function $f$ such that $f(z)$ is a single valued function ...
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1answer
31 views

calculating logarithmic equation

I have an equation , which is like this $64n\log n < 8n^2$ . (the base of logarithm is 2) I know how to solve the logarithmic equations . I am a programmer , so I wrote a simple program and ...
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1answer
37 views

Waiting Times for Next Person in Line

I'm working on a problem and came up with two different solutions! I'm not sure which is correct. Problem: Two clerks with service time $exp(1)$ are helping 2 customers while a third waits. What is ...
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1answer
17 views

Mutidimensional integration with cosine rule in exponent

I have been attempting to find a way to simplify the following multi-dimensional integration form: $\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\pi} ...
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0answers
575 views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ ...
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4answers
144 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
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1answer
21 views

Exponential equation with same, unknown bases

I have the following equation: $(x-3)^{(x^2-x)} = (x-3)^2$ The book says solutions are: $x_1 = -1, x_2 = 2, x_3 = 3, x_4 = 4$ I was only able to get -1, 2, 4 by doing this: $(x-3)^{(x^2-x)} = ...
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1answer
51 views

How to manipulate the knee of the curve of an exponential function?

I want to be able to manipulate the point of inflection of an exponential curve equation:$$a\exp\{xb\}.$$ could somebody tell me which parameter I may introduce in such a formula in order to make the ...
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5answers
62 views

Proof problem: show that $n^a < a^n$ for all sufficiently large n

I would like to show that $n^a < a^n$ for all sufficiently large $n$, where $a$ is a finite constant. This is clearly true by intuition/graphing, but I am looking for a rigorous proof. Can ...
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1answer
28 views

How to interpret b in $y=x^{e^{bz}}$ in nonlinear regression?

What is the correct way to interpret b in this nonlinear equation $y=x^{e^{bz}}$? I've estimated the model and b seems to be the percent change in y with a unit change in z, but I am unsure how to ...
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1answer
14 views

Find the value of $k$ in the exponent

I am trying to calculate the $k$ value in this equation: $\dfrac1{n^c} \le \left(1 - \dfrac2{n(n-1)} \right)^k$ by using the logarithm, I am getting for $k$: $\log_{1- 2/n(n-1)} n^{-c} \le k$ is ...
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1answer
66 views

Show that $ \exp \left(SL(2,R)\right)$ is the set of all matrices with positive trace $\geq -2$

Using the fact that every matrix in $SL(2,\mathbb{R})$ is conjugate in $SL(2,\mathbb{R})$ to one of the following matrices: $$ \left(\begin{array}{rr} a & 0\\ 0 & \frac{1}{a} ...
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4answers
52 views

Finding the limit of the sequence $x(n) = (1+2/n)^n$ [duplicate]

What we are allowed to use - 1) The fact that limit of $(1+1/n)^n$ exists and assumed to be some real number $e$ 2) Subsequencial properties of limits of sequences 3) Basic properties of limit In the ...
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2answers
38 views

Solve equation with exponentials

I'm trying to obtain $x$ in the following equation: $$ 1= ae^{bx}+ce^{dx} $$ with a,b,c,d known. What I did was to take the ln of all the equation, so I have: $$ \log\frac{1/ac}{b+d} = x $$ ...
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1answer
38 views

Gradually and eventually slow and exponential [closed]

I am not a math expert. Thats exactly why I am here. Please help. I need an equation for the below use case. Consider a lottery. Winner always gets 1$ (constant). No more. As the number of ...
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3answers
120 views

General formula for the higher order derivatives of composition with exponential function

Suppose I have a function $x:\mathbb{R} \to \mathbb{R}$ and consider: $$g(t) = e^{x(t)}$$ When I start differentiating with respect to $t$ I obtain: \begin{align} g'&=e^xx'\\ ...
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1answer
52 views

How to prove the equality or inequality

Can anybody prove that the following equation is right or wrong? $$\int_0^te^{-t}(1-e^{-2x})^ke^x dx=\int_0^t2k(e^{-2x}-e^{-t-x})(1-e^{-2x})^{k-1}dx$$ where $t>0$ and $k$ is and integer. My small ...
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0answers
49 views

Proof that $|1 - e^{i \theta}| \geq \frac{2|\theta|}{\pi}$ for $-\pi \leq \theta \leq \pi$?

I would like to prove (geometrically if possible) the above result. Could someone help?
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4answers
95 views

Why is $\frac{(e^x+e^{-x})}{2}$ less than $e^\frac{x^2}{2}$?

I have read somewhere that this equality holds for all $x \in \mathbb {R}$. Is it true, and if so, why is that? $$\frac{(e^x+e^{-x})}{2} \leq e^\frac{x^2}{2}$$
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1answer
104 views

Find the limit of the sequence $(1-1/n)^n$

All that we have proven so far is that limit $(1+1/n)^n$ exists and considered to be a number 'e' which belongs to $(2,3)$ We haven't proven that 'e' is irrational or that lim $(1+(x/n))^n) = e^x$ ...
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1answer
28 views

exponentiating a matrix and sum of elements

$$ M= \begin{bmatrix} 1&1&0\\0&1&1\\0&0&1\\ \end{bmatrix} $$ Then the sum of all entries of $e^{M}$ i just don't know how to calculate this sum as this would be an infinite ...
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1answer
42 views

Bounding a function of norms on the unit cube

For a vector $v \in [0,1]^n$ and $p > 1$ we denote the p-norm of $v$ as: $||v||_p = (\sum_iv_i^p)^{\frac{1}{p}}$. where $v_i$ are the entries of $v$. Define the following (weird looking) function ...
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2answers
36 views

Find the original function by using convolution theorem

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. ...
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5answers
687 views

Is there a way to prove this exponential inequality?

I came across this proposition while trying to prove that a function was injective: if $a>b$ then $a^a>b^b$, where $a$ and $b$ are real numbers bigger than 1 . Intuitively it (somehow) makes ...
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2answers
39 views

prove limit of exponential function without concept of logarithm

The question is, prove that if a real number $x>1$, then $\lim_{n\to\infty}x^n = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure ...
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1answer
37 views

An easy way to define $\exp(x)$ - does it work?

$\exp(x)$ is usually defined in three different ways: 1) By its Taylor series: $\exp(x)=\sum_{k=0}^{\infty} \frac{x^k}{k!}$ 2) By its derivative: $\exp(x)'=\exp(x)$ 3) By the limit $\exp(x)=\lim_{N ...
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1answer
131 views

Solving $e^{\sin(z)}=1$ in the complex plane

I am trying to solve $e^{\sin(z)}=1$ in the complex plane. I know that this means that $\sin(z)=2k\pi i$ for some integer $k$. This is equivalent to saying that $$\frac{e^{iz}- e^{-iz}}{2i}=2k \pi ...
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2answers
36 views

Bounds on function $\exp(-\frac{1}{2}x^2)$

I have the following function : $$f(x)=\exp(-\frac{1}{2}x^2),$$ where $x >0$. I am looking for some tight bounds (upper bound and lower bounds) on $f(x)$. Any idea ? P.S.: The problem arises when ...
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4answers
51 views

Past exponential functions?

We have been taught that linear functions, usually expressed in the form $y=mx+b$, when given a input of 0,1,2,3, etc..., you can get from one output to the next by adding some constant (in this case, ...
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1answer
15 views

Volume Exponential Function

I should find the Volume received by rotating the region bounded by: $y = e^x $, $ y = 0 $,$ x = 0 $, $ x = 1 $ rotated around the x axis. I know how to find it by using the disc method but I could ...
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1answer
74 views

Solving an exponential equation with different bases

Solve the equation $2^x + 5^x = 3^x + 4^x$. I can figure out two special solutions $x=0$ and $x=1$, and I try to prove that they are the only two solutions. However, I find it hard to do so because I ...
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0answers
39 views

Expectation of an exponent of a random variable

Suppose that $X \geq 0$ is distributed according to some distribution $F$. What can be said about $E[e^{-r X}]$? I.e. is there a way to express this expectation only in terms of some characteristics ...
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2answers
27 views

Confusing result obtained taking second derivative of ye^y

I was doing my calculus homework, and one of the questions asked for the first and second derivative of $ye^y=x$, I did the computations and arrived at $-(x+1)^{-2}$, which was a lot neater and ...
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0answers
36 views

Need help with $\int\mathrm{exp}[-C(\frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1)^S ] \mathrm{dx}$ for a statistical mechanics problem

Can someone help solving this integral?: $$\int\limits_{-\frac{1}{2}}^{0} \mathrm{exp}\left[-C \left( \frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1 \right)^S \right] \mathrm{dx} = ...
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1answer
15 views

Generalizing exponential moving average to n samples

Assume that we have a moving average like this: $E_t = a*S_{t-1}+(1-a)*E_{t-1}$ where $E$ would be an estimate we are interested in, and $S$ is a sample we take at each point in time. Now, if we ...
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1answer
40 views

Simplify exponential equation

I really need your help to solve this exponential equation. It looks so simple, but I haven't been able to find a solution so far: $$ {A_1 + A_2 \over 2} = A_1 \exp\left({-x^2 \over c_1^2}\right) + ...
0
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1answer
43 views

Infinite Sum Defined by $\int \frac{e^x}{x}dx$ vs. Exponential Function Taylor Series

Recently, when fiddling around with integration by parts, I noticed that it is possible to define infinite series that led to an integral. My calculus teacher noticed this, and told me to find $$ ...
0
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1answer
43 views

Supremum of $\cot(\pi z)$ where $z$ is on circle with radius $n+1/2$

I try to estimate the supremum of $|\cot(\pi z)|$ and where $z=(n+1/2) e^{i t}$, $n\in\mathbb N$ and $t\in[0,2\pi)$. I should be a constant. So far I did by wiriting it in exponential form and ...