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

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34 views

Prove a matrix is non-negative.

Let $\textbf{r}_1$ and $\textbf{r}_2$ be $n \times n$ symmetric, diagonally dominant, Metzler matrix with eigenvalue $\max(|\lambda_i|)<1$ for both $\textbf{r}_1$ and $\textbf{r}_2$. Let ...
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2answers
34 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
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1answer
61 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, ...
5
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1answer
40 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 ...
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0answers
8 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
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2answers
40 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
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1answer
31 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 ...
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0answers
9 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. ...
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1answer
18 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 ...
1
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1answer
54 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 ...
4
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1answer
27 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} = ...
3
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0answers
62 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
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2answers
45 views

How to solve this equation to find a closed-from for x? [on hold]

I want to find the value of $x$, i.e., $x=$ $$Ax+10^{-Bx}=C$$ Any suggestions?
-3
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0answers
20 views

Exponential problem for phase calculation. Find periodic t [on hold]

Given $e^{2At{\pi}i} = - e^{2{\pi}(A-149)ti}\text{, where }A = 42.58\cdot10^6.$ Find periodic $t$.
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0answers
19 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
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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
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0answers
18 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
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3answers
117 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
90 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^+, ...
0
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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
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3answers
49 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
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0answers
26 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
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0answers
27 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
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2answers
28 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
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0answers
15 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 ...
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0answers
13 views

Expected value of exponential random variable [closed]

If an exponential random variable, X, has failure rate λ, what is E[X|X<λ]? I'm not sure how to start here. I know that E[X] = 1 / λ for an exponential random variable. Is the probability that X ...
5
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6answers
175 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
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0answers
46 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 ...
2
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3answers
52 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
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3answers
27 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
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1answer
59 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.
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0answers
22 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
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2answers
50 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
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2answers
48 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
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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
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0answers
39 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 $\sum_{j=1}^m a_j = 1$ ...
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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
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4answers
194 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
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4answers
75 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.
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1answer
73 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
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2answers
57 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 an ...
0
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1answer
23 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
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1answer
36 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 ...
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0answers
43 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 ...
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2answers
47 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 $c$ ...
0
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0answers
11 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
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1answer
23 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
39 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 = ...
12
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0answers
145 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) ...