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

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3
votes
1answer
102 views

How can we describe the graph of $x^x$ for negative values?

We usually only see the graph $y=x^x$ for $x>0$, because $x^x$ is a complex number for most negative values of $x$. Yet here is a full graph of $y=x^x$ on the real line: This graph may seem like ...
2
votes
1answer
32 views

How to simplify $\cot(\sec^{-1}(e^x))$

I've been trying to simplify $\cot(\sec^{-1}(e^x))$. I thought substitution might the way to go about it so I said: let $u = \sec^{-1}({e^x})$ I'm therefore trying to find $\cot(u)$ From $u = ...
2
votes
1answer
46 views

Second order linear ODE not making sense…

I am given: $y''-3y'+2y=0$ $y(0)=1$ $y'(0)=2$ I know that $r_1=2$ and $r_2=1$ The solution therefore is: $y(x)=C_1e^x+C_2e^{2x}$ Solving for initial values, I have: $y(0)=C_1+C_2=1$ ...
0
votes
1answer
53 views

How to solve this system of ODE's?

I'm not sure how to proceed to solve this system of ODE's; $$ \begin{bmatrix}\dot{x}_1 \\\dot{x}_2\end{bmatrix}=\begin{bmatrix} \cos t & -\sin t\\ \sin t & \cos t ...
2
votes
3answers
333 views

How to solve exponential function inequality?

How do I solve the exponential equations like $2^\frac{x}{8}<x$? I can solve this by plotting into graph. But is there any way to do it mathematically? or like $2^x < 100x^2$ . I am trying to ...
0
votes
2answers
45 views

$\sup\{a^{r}\mid r<x; r\in\mathbb{Q}\}=\inf\{a^{s}\mid x<s; s\in\mathbb{Q}\}$ How to prove it?

This proposition is a lemma related to another stage for defining exponential function $a^{x}$, in this case for reals, taking into account it is defined for rationals. Proposition Let $a>1$ and ...
0
votes
0answers
15 views

relationship between two set of variable

i am trying to determine what kind of mathematical modeling could be applied following two variables,let us call them $x$ and $y$ ,namely change one variable has effect second on,i have several ...
0
votes
2answers
54 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
0
votes
1answer
56 views

Solve exponential equation $(1−ax^2)∗e^{bx^2}=c$ for x. transcendental algebraic equation?

is it possible to solve an equation with the given form analytically? $$ (1-ax^2)*e^{bx^2}=c $$ $$ e^{bx^2}-ax^2e^{bx^2}=c $$ I've already tried it using a logarithmic function but I cannot manage ...
2
votes
3answers
91 views

Exponential Regression: how to get a formula from a given pattern

I'm trying to code a computer script (in Java) that returns an array of numbers that follows a certain pattern. The numbers should be: ...
0
votes
1answer
33 views

Creating an exponential scale

Good morning, I am trying to create an exponential scale for attributing values in a scoring model. Here is the function I was thinking of using: y = z^x Where: y = Score X = Risk assessment ...
0
votes
1answer
24 views

Expected Value of Exponential

I want to calculate $\log E[\exp(-\sqrt{d} S \epsilon)]$, where $\epsilon \sim N(0,1)$ and everything else is deterministic. The result should be $\frac{d}{2}||S||^2$ but why?
1
vote
1answer
86 views

The transformations on the nome and Landen's transformation

Could someone please explain how to transform the nome $q = e^{-\pi K'/K}$ from $q^2$ to $q$ and then to $-q$? In other words, how does changing $q^2$ to $q$ and then $q$ to $-q$ affect $k$ and $K$. ...
0
votes
2answers
32 views

Prove that $(1-\frac{1}{k})^d \le e^{-\frac{d}{k}} $

Prove that $(1-\frac{1}{k})^d \le e^{-\frac{d}{k}} $ for $d,k \ge 0$ I know that $(1+\frac{1}{n})^n \le e$ but does that help? Actually, I don't really 'know' this, but I've heard it's true at least ...
0
votes
2answers
56 views

finite sum over a Gaussian

I have a sum of the form: $$\sum_{n,m=-N}^N e^{-\alpha (n-m)^2}$$ where $\alpha > 0$ is some constant, and $n,m$ take the integer values: $-N,..,N$. I know there is a possibility of exchanging ...
0
votes
1answer
37 views

show that $(1+ \frac {x}{n})^n < e^x$ and $e^x < (1- \frac{x}{n})^{-n}$ if $x<n$

If $n$ is a positive integer and if $x>0$,show that $(1+ \frac {x}{n})^n < e^x \quad$ and that $\quad e^x < (1- \frac{x}{n})^{-n} \quad $ if $x<n$ I proved the first one by the ...
0
votes
3answers
119 views

How to find the sum of this power series $\sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}$

How to prove that $$ \sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}= \frac{2}{5} e^{-\cos \left( 1/5\,\pi \right) x}\cos \left( \sin \left( 1/5\,\pi \right) x \right) +\frac{2}{5}\, e^{\cos ...
0
votes
1answer
30 views

Just one step away. Exponential formula

The data points are. I have worked out that. y =.032(2.5)^x is correct where my table is x = 0, y = .032 x = 1, y = .08 x = 2, y = .2 How do I get my formula to reflect y=-3, x =.0320 ...
1
vote
0answers
19 views

Sigmoid Function Question

Ive been trying for well over a week to try to understand how to use a simple sigmoid or logistic function works. Specifically I'm trying to understand how to build proper polynomia parameters for ...
1
vote
2answers
41 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
4
votes
1answer
120 views

Convolute exponential with a gaussian

I have data measuring an exponential decay that is convoluted by a gaussian response function. I have the measured shape of the gaussian, and want an analytical expression for the exponential ...
1
vote
4answers
107 views

Solve equation: $5^x = -2x + 7$

How to solve that equation: $$5^x = -2x + 7$$ I already have the answer $x=1$. Can anyone please explain to me?
3
votes
0answers
49 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
1
vote
3answers
66 views

How is the Logarithm derived from the exponential function? (aren't they inverses?)

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series ...
18
votes
4answers
287 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
3
votes
1answer
55 views

How to show that $e^{x+y} = e^x e^y$ by series expansion [duplicate]

I know that $e^xe^y=e^{x+y}$ but I want to show it by expanding the exponentials in MacLaurin Series. $$ \left(\sum_{n=0}^{\infty} \frac{x^n}{n!}\right) \left(\sum_{m=0}^{\infty} ...
3
votes
1answer
51 views

Help calculating this integral

Prove this for every $n>1$ (belongs to $\mathbb{N}$ ) $$\displaystyle \int_{0}^{1}\left( \frac{x^{2n+3} - x^{2n+1}}{1+x} \right) \, \mathrm{d}x =\frac{1}{2n+3} - \frac{1}{2n+2}$$ I don't see ...
1
vote
0answers
17 views

The variance of an arrival process with shifted exponential interval

Here we have a arrival process. The inter-arrival time follows a shifted negative exponential distribution as: $f(t)=e^{−\lambda(t−\theta)}$ How to derive the variance of the number of arrivals in ...
0
votes
2answers
29 views

Get N values between 0 and 1 AND control spacing of results.

So I need to get N number of values between 0 and 1. The values should be evenly spaced. Thats easy... However, I also want another variable X that will shift the resulting values closer to 0 or 1. ...
0
votes
2answers
11 views

Calculating how many iterations to make a list that doubles each iteration, n elements long

If I have a list of numbers starting with four numbers, the list doubles in size after each iteration, how would I calculate how take to have a list of exactly n elements long? Thanks
0
votes
1answer
35 views

Simplify this expression?

I have the following expression $$\frac 12 x_0e^{-\beta t}\left[\left(\frac {\beta}{i \sqrt{\omega ^2-\beta ^2}}+1\right)e^{i \sqrt{\omega ^2 - \beta ^2}t}+\left(\frac {- \beta}{i \sqrt{\omega ^2 - ...
0
votes
3answers
24 views

What constrains the following functional equation of exponents?

If I am not incorrect,the standard (Is it he standard?) form of an exponential equation is $$y=ab^{x-h}+k$$ What are the constraints on this equation, or in other words, how do each of the variables ...
2
votes
2answers
39 views

Is it possible to rewrite floor functions applied to a fraction using only the addition, multiplication, and exponentiation operators?

Let's restate this question in using mathematical notation. Let $n,k \in \mathbb{N}$. Let $f(n)=\left\lfloor{\frac{n}{k}}\right\rfloor$. Is it possible to rewrite this using the addition, ...
2
votes
3answers
33 views

A good site documenting approximations of irrationals

I'm thinking of Sloane here but I believe that only takes sequences/series into account. Basically I've derived an interesting, appealing formula for e and want to know if it's already been ...
17
votes
2answers
483 views

Is this function a constant?

I am a french guest and I hope that my english isn't too bad... So here is my issue : I consider an entire function $f$ which satisfies the following property for all complex number $z\in \mathbb{C}$ ...
3
votes
2answers
84 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
1
vote
3answers
76 views

Is x^x an exponential function?

I know that functions of the form $c^x$ are called exponential when $c$ is a constant. How about the function $x^x$? It seems somewhere in between exponential and double exponential to me. Is there a ...
1
vote
1answer
38 views

Should I use Taylor Expansion or the expansion of $e^x$ to express a second order differential (in this case)

I've been given this equation: $(1+x^2)\dfrac{d^2y}{dx^2} + 4x\dfrac{dy}{dx} + 2y = 0$ I've also been told that: $y=1, \dfrac{dy}{dx} = 1$, at $x=-1$ I've been asked to find a series solution of ...
2
votes
4answers
385 views

Which has a higher order of growth, n! or n^n? [duplicate]

In our algorithms class, my professor insists that n! has a higher order of growth than n^n. This doesn't make sense to me, when I work through what each expression means. ...
0
votes
1answer
32 views

Variations on the limit definition of the exponential function

I'd like to know how this might be proven, or why it's true: $\begin{aligned} \lim\limits_{n\to\infty}{\left( 1+\frac{1}{n^k}\right)^n} = \infty, \text{when } k<1\\ = e, \text{ when } k=1\\ =1, ...
0
votes
1answer
15 views

Difference Between Generalized and Alternative Compounded Interest Equations

I am currently studying a chapter called "An Economic Interpretation of e" in my Economics class and we are finding amounts of compounded interest. I am not actually looking for help on the problems ...
3
votes
4answers
118 views

Computing a large exp(x) in a numerically robust way.

I'm trying to compute $\lfloor e^x \rfloor$, where x is a 64-bit integer. The problem is that the result of the computation may be close to 2^64. In this range, 64-bit floating point numbers will be ...
1
vote
0answers
70 views

Maximum Likelihood Estimator of the exponential function parameter based on Order Statistics

Let $X_1, \ldots, X_n$ be a random sample from the exponential distribution $\exp(\lambda)$. Let $M_n=\max\{X_1, \ldots, X_n\}$ with probability density function $$g_{M_n}(x)=n\lambda e^{-\lambda ...
0
votes
3answers
33 views

Algorithm to calculate price based on number of units

I'm trying to come up with a pricing algorithm for my product. I've already set some prices at low intervals, but I need the algorithm to calculate a reasonable for very large orders. Here are the ...
0
votes
1answer
36 views

Solution of an equation with polynomial and exponential terms

Can anyone solve for $t$ the equation: $$ e^t=\frac{1-nt}{1-t} $$ with $n \in \mathbb N$ (known) and $t>0$. Online solvers give an answer only for specific values of $n$, but I need a general ...
0
votes
2answers
71 views

A Complex Variable ODE

suppose $f$ is a holomorphic function on some domain $D$ satisfying $f'(z)=af(z) $ for some >constant a. show that $f(z)=Ce^{az}$, for some constant $C$
1
vote
3answers
97 views

Derivative of exponential function proof

I'm looking for a straight forward proof using the definition of a derivative applied to the exponential function and substitution of one of the limit definitions of $e$, starting with $e = ...
0
votes
2answers
92 views

The problem of x = ln(x)

I am trying to find x values for points along the normal distribution curve, and I ended up with a problem that goes back to the method of solving $x = \ln x$. Right now, I have $\ln(a \mu) - \ln(10) ...
0
votes
2answers
35 views

Proportionality to find spent years for price drop

Well, the title's kinda messy, but this is a concrete example of what I'm trying to find out: Lets say there is a price of 40.000 USD, if the price drops at half, how many years does it take for the ...
0
votes
1answer
38 views

Naming and meaning of exponential power functions

I apologize if something like this has already been asked, but I don't have any ideas on search terms for this family of functions. I'd like to know two things: 1) Is there a name for the family of ...