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

learn more… | top users | synonyms (1)

0
votes
6answers
40 views

How to prove an exponential equation.

Is there any law of exponents that applies to this equation? How can show that the LHS gets converted into the RHS $(e^x −e^{−x})^2 =(e^x +e^{−x})^2 −4$
0
votes
2answers
42 views

Inverse of a function $xe^x$

How should I proceed about finding the inverse of the function $xe^x$? I have been wondering about it for a long time and can't think of anything to do.
1
vote
1answer
102 views

Integral of $xe^{-ax^2-bx^{-1}}$

I am currently facing an integral I have no clue how to solve it. I believe it is rather exoctic, but I hope you might have some good advice: $$\int_0^{\infty} x e^{-ax^2-bx^{-1}} \, \mathrm{d}x$$ ...
6
votes
1answer
50 views

Is there a constructive discontinuous exponential function? [duplicate]

It is well-known that the only continuous functions $f\colon\mathbb R\to\mathbb R^+$ satisfying $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbb R$ are the familiar exponential functions. (Prove ...
0
votes
0answers
13 views

Optimal Space-Travel Departure Time (Issues deriving and solving complex expressions).

Problem This problem aims to determine the optimal time to depart for an intergalactic destination, taking into account the fact that in a number of years technology back on the planet you left may ...
3
votes
2answers
44 views

How to precisely define $C^\infty$ in $f(x) \in C^\infty$

In single variable calculus, a common way to denote a function that is continuous for all derivatives is to write $f(x) \in C^\infty$ i.e. $f(x) = \exp(x)$ Is there a more rigorous way to define ...
0
votes
1answer
35 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
0
votes
0answers
12 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
1
vote
1answer
31 views

Translate exponential distribution into normal distribution

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the ...
2
votes
2answers
77 views

Solve matrix equation $e^A=e^B$ for nilpotent $A, B$.

I need to solve equaton $e^A=e^B$ for nilpotent matrices A and B over field $\mathbb C$, where $B$ is fixed. I solved equation $e^X=E$ for all matrices. The solution is any semisimple (in case ...
6
votes
2answers
68 views

Can I interpret the exponential of the derivative operator, $e^D$, as infinite shift operators each shifting “infinitesimally”?

To better explain what I mean, an example can be very useful. Consider $e^{i\theta}$. We could express this using the series definition or the limit definition of $e^x$ instead: $$e^{i\theta} = ...
4
votes
0answers
78 views

Solution to following functional equation

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
2
votes
0answers
26 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i) \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
0
votes
1answer
20 views

Values of $p$ for which equation $p3^x+2\cdot 3^{-x}=1$ has a unique solution

$p3^x+2\cdot 3^{-x}=1$ I got this down to a quadratic equation by marking $3^x$ as $t$ and I fiddled with the stuff and got some solutions that apparently don't fit the real one in the textbook was. ...
1
vote
0answers
19 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
5
votes
3answers
122 views

How can I calculate $\lim_{n \to \infty} (1 + \frac{1}{n!})^n$ and $\lim_{n \to \infty} (1 + \frac{1}{n!})^{n^n}$?

How do you calculate the following limits? $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^n$$ $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^{n^n}.$$ I really don't have any clue ...
3
votes
2answers
95 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
6
votes
3answers
201 views

What is the $\lim_{n\rightarrow \infty }(1+\frac{1}{n})^{n^n}$

What is the $$\lim_{n\rightarrow \infty }\left(1+\frac{1}{n}\right)^{n^n}$$ I know that the $\lim_{n\rightarrow \infty }\left(1+\frac{1}{n}\right)^{n}=e$, so I wanted to find the limit by the same ...
0
votes
0answers
35 views

Algebra Integral simplification

Let some equation problem final result is like this $0\leq1\leq s\leq t\leq u\leq v$ \begin{align} M=\mathrm{exp}\bigg\{-\pi\lambda v^2+\pi\lambda v^2\bigg(\displaystyle\int_o^s ...
1
vote
3answers
50 views

Explicit form of this series expansion?

I am considering the following series expansion: $$f(k):=\sum_{n\geq 1} e^{-k n^2}$$ with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at ...
2
votes
0answers
27 views

Fourier transform of exponential of a function

I am wondering what $\mathcal{F}[\exp(f)]$ is in terms of $\mathcal{F}[f]$. The farthest I have got is using the series expansion of $\exp$, such that I end up with $\mathcal{F}[\exp(f))] = ...
2
votes
0answers
34 views

Invertibility of infinite order matrix

how the matrix $[e^{-(x_j-x_k)^2}]$ is invertible where $\{x_j\}$ be any real sequence such that $(x_{j+1}-x_j) >0$ for all $j \in Z$ where $Z$ denotes the set of integers.
2
votes
2answers
29 views

Tweaking formulas to increase scoring

I am building a model for SVM classification. However, the confidence score that i have would be from negative to positive. This is the formula i am using to normalize the confidence score ...
0
votes
0answers
16 views

Finding probability of being in a certain state in a CTMC.

There are two transatlantic cables each of which can handle one telegraph message at a time. The time to breakdown for each has the same exponential distribution with parameter λ. The time to repair ...
5
votes
6answers
194 views

Are the any **non-trivial** functions where $f(x)=f'(x)$ not of the form $Ae^x$

This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is ...
3
votes
0answers
51 views

What is $\int \frac{e^{a x}}{1+x^2} dx $?

In my answer to another question (here: Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$), I came up with this integral: $\int \frac{e^{a x}}{1+x^2} dx $. I ...
3
votes
3answers
72 views

Apex of an Exponential Function

Is there a way of calculating where the apex of an exponential lies? There's probably a deeper / more mathematical way of explaining what this is exactly. The image hopefully demonstrates what I mean. ...
0
votes
3answers
28 views

Show that $ f(x) = A.exp(2x) $ if $ f'(x) = 2f(x) $

Show that $ f(x) = A.exp(2x) $ if $ f'(x) = 2f(x) $ for some $ A \in \mathbb{R} $ Is it sufficient to say that if the derivative of a function contains itself, then it must be the exponential ...
2
votes
1answer
60 views

Evaluate the limit $\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $

It seems reasonable to assume that $$\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $$ goes to zero but I can't figure out how to prove it.
0
votes
0answers
25 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
0
votes
2answers
52 views

L'Hôpital's rule exercise with sqrt(x) as exponent

I'm a bit stuck trying to find the limit of the following function: $$\lim_{x \to 0^+}\,\,{x^{\sqrt{x}}} $$ We are expected to use L'Hôpital's rule, and thus far I've managed to resolve the equation ...
2
votes
2answers
50 views

Proving $\log n < \sqrt n$

I am trying to prove $\exists n_0 > 0: \forall n > n_0: \log n < \sqrt n$. My attempt uses the series representation of the exponential function, but it does not seem to accomplish the proof: ...
0
votes
1answer
11 views

Distribution of exponential(X/c)

Suppose $X \sim Exponential(\lambda)$. That is, the PDF for $X$ is $f_X(x)=\lambda \cdot e^{-\lambda x}$, $x\ge 0$, and the CDF of $X$ is $F_X (x)=\int_{-\infty}^x f_X(x)=1-e^{-\lambda x}$, $x\ge ...
0
votes
0answers
43 views

Solving a ratio of summation

I have to solve the equation \begin{equation*} y = \frac{\sum_{j=1}^m a_j x'_j}{\sum_{j=1}^m a_j x_j} \end{equation*} We have $\sum_{j=1}^m a_j \frac{x'_j}{x_j} = 1$ and $\sum_{j=1}^m a_j = 1$ ...
1
vote
2answers
36 views

exponential functions.

I am confused of solving expnential functions they look easy but cant solve it. 1: $$\large e^{8\cdot\ln(b^{1/4})}$$ and this one solving for x: 1: $$\ln(6x-2) = 5$$ FYI : Its not an assignment. i ...
3
votes
4answers
212 views

Solving equations with exponentials and a non-exponential term.

I know how to solve exponential equations. Just use logarithms, e.g., $$ 2^x-3=0 \\ 2^x=3 \\ x=log_23 \\ $$ But on a recent math test I found an equation of the form: $$ 2^{n-3}=\frac {20}{n} $$ ...
5
votes
4answers
107 views

Root of $(x+a)^{x+a}=x^{x+2a}$ and $e$

Let us denote solution to the equation $$(x+a)^{x+a}=x^{x+2a}$$ with $X_a$. ($a$ is a non-zero real number) Prove that: $$\lim_ {a \to 0} X_a = e$$ This is something that ...
2
votes
4answers
81 views

How can I evaluate the infinite series $\sum_{n=0}^\infty\frac{ n^2}{n!} $?

Can someone help me to evaluate $$\sum_{n=0}^\infty\frac{n^2}{n!}?$$ It can be written as $$\sum_{n=1}^\infty\frac{n}{(n-1)!},$$ but I am unable to analyze this.
0
votes
1answer
81 views

Integral of $\int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy$

Is it difficult to compute or find a good computable lower bound on the integral \begin{align*} \int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy \end{align*} where $c$ and $M$ are ...
0
votes
2answers
91 views

Show$\lim\limits_{x \to \infty} \frac{2^{2x}}{5^{x-1}} = 0$

Exactly as the title says. $$\lim\limits_{x \to \infty} \frac{2^{2x}}{5^{x-1}} = 0$$ I am at a loss for how to show this one. At first I thought of using L'Hopital's rule on the numerator and the ...
0
votes
1answer
28 views

fitting by linear combination of exponential functions

Suppose that we have a set of points $(x_1,y_1), \ldots (x_n,y_n)$, and we want to fit a function of the form $f(x) = ae^{2x} + be^x + c$ to those points. If we make $z=e^x$, then our function becomes ...
0
votes
1answer
39 views

How to compute $\int_{-\infty}^{\infty} | e^{-(y-a)^2/2}-e^{-(y+a)^2/2}| dy$

I am looking on how to compute or a table of integral that has solution to \begin{align*} \int_{-\infty}^{\infty} | e^{-(y-a)^2/2}-e^{-(y+a)^2/2}| dy \end{align*} Using Wolfram-alpha I found it to be ...
1
vote
0answers
46 views

Integration on an exponential function

I am struggling to solve this expression. I want to show that, $$\frac{1}{p}\nabla_{j}\int e^{ipR\cos(\theta)} dT=i\int \hat{p_{j}} e^{ipR\cos(\theta)} dT$$ here, $dT=d(\cos(\theta))d\phi$ I tried ...
0
votes
2answers
54 views

How to compute $\sum_{n=1}^N e^{-( n-c)^2}$

I have to compute or at least find good upper and lower bounds on \begin{align*} \sum_{n=1}^N e^{-( n-c)^2/b} \end{align*} and \begin{align*} \sum_{n=1}^N ne^{-( n-c)^2/b} \end{align*} where $c$ ...
5
votes
4answers
49 views

Evaluating natural limit $\lim_{n \to \infty} \left( e^{2n} - 1\right) ^\frac{1}{n}$

Any idea evaluating this $$ \lim_{n \to \infty} \left( e^{2n} - 1\right) ^\frac{1}{n} $$ after I raise all to e like so $$ \exp\left( \frac{\ln\left(e^{2n}-1\right)}{n}\right) $$ and Hopital's it I ...
0
votes
0answers
13 views

Finding a tunable exponential function between two points (cost is less based on quantity)

I would like to figure out, or well, to remember how to find the corresponding exponential function between two points based on number of items bought up front i.e. how much discount to give based on ...
0
votes
1answer
25 views

Rules of powers of exponents

Why is $e^x \times e^{\ln 2} = e^{x \times \ln 2}$ Not correct? I thought that if you had something to the power, you could split them E.g $e^4 = e^2.e^2$ Sorry for the lack of latex I find it very ...
0
votes
2answers
40 views

Implicit Differentiation problem (Exponential Derivatives) Please help!

Use the process of implicit differentiation to find $dy/dx$ given that: $$x^2e^y − y^2e^x=0 $$ I am trying first to find $y$, $$y^2e^x = x^2e^y$$ $$y^2 = (x^2e^y)/e^x$$ $$y = ...
14
votes
0answers
186 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$: (Mathematica code) ...
1
vote
3answers
68 views

How to prove $A^{n\times n}=I_n\Rightarrow A^n=A^{f(n\times n)}$?

Let $A\in M_2(\mathbb{Z})$ s.t. there is a positive integer $n$ satisfying $A^n=I_2$. Show that $A^{12}=I_2$. I have no idea where to start. Suggestions?