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

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2
votes
2answers
26 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
30 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 ...
1
vote
1answer
15 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 ...
0
votes
0answers
8 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
95 views

how to solve this limit with $e^{x}$

I was trying to solve the derivative of $e^{x}$ the traditional way with the definition of the derivative: $$ \lim_{h\rightarrow 0}\frac{e^{x+h}-e^{x}}{h} $$ so I solved like this: ...
1
vote
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
votes
1answer
26 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
votes
0answers
56 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) ...
6
votes
2answers
220 views

“Bizarre” continued fraction of Ramanujan! But where's the proof?

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$ "Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued ...
1
vote
0answers
18 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 ...
-2
votes
2answers
44 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?
0
votes
3answers
112 views

What is the limit of $\mathrm{e}^{At}$ when $t\to \infty$, for $A$ a square matrix? [on hold]

I was doing a matrix calculation and need to find $$\lim_{t\to \infty} \mathrm{e}^{At}=?$$ What is the limit of $\mathrm{e}^{At}$ when $t\to \infty$, for $A$ a matrix?
-3
votes
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 $t$.
12
votes
0answers
144 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) ...
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 ...
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. ...
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^+, ...
5
votes
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 ...
1
vote
0answers
17 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
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 ...
1
vote
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
votes
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
votes
3answers
51 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. ...
2
votes
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 ...
2
votes
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.
0
votes
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 ...
-2
votes
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 ...
3
votes
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 ...
0
votes
3answers
26 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
58 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
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 ...
0
votes
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 ...
2
votes
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
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 ...
-2
votes
2answers
86 views
0
votes
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 $\sum_{j=1}^m a_j = 1$ ...
1
vote
1answer
72 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 ...
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
191 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 ...
4
votes
4answers
99 views

Prove that $a^x$ is continuous

I'm having trouble with proving the following: Let $a > 0$ be a positive real number. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) := a^x$ is continuous. I'm a ...
1
vote
2answers
55 views

least squares using exponential model

I'm trying to fit values from this model $$y(x)=ae^{−bx}+c$$ where a, b and c are 3 different parameters that I want to find with least squares. So using least squares I want to find the value of a, b ...
4
votes
5answers
272 views

Finding: $\displaystyle\lim_{x\to\infty}x\big((1+\tfrac1x)^x-\mathrm{e}\big) $

Find: $$\displaystyle\lim_{x\to\infty}x\Big(\big(1+\tfrac1x\big)^x-\mathrm{e}\Big) $$ EDIT: so we have here a $\infty\cdot0$ so I'll try LHR, Edit2: I don't think LHR will get me anywhere since ...
2
votes
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.
0
votes
1answer
21 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
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 ...
5
votes
2answers
84 views

the value of $e$ and the method of getting it

We define e to be a number which satisfies the following condition $$\lim _{a \to 0} \frac{e^a-1}{a}=1. $$ How did we arrive to the following from above equation $$e=\lim _{n \to \infty} ...
0
votes
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 ...
1
vote
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 ...