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

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-3
votes
0answers
18 views

Exponential problem for phase calculation. Find periodic t

Given $e^{2At{\pi}i} = - e^{2{\pi}(A-149)ti}\text{, where }A = 42.58\cdot10^6.$ Find periodic $t$.
1
vote
0answers
13 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
0answers
23 views

Can someone please help me to prove if a matrix is non-negative?

Let $\textbf{r}_{1}$ and $\textbf{r}_{2}$ be two symmetric, diagonal dominate, Metlzer matrices. Let $\textbf{F}(m)= (\textbf{I}-e^{m\textbf{r}_1})(\textbf{I}-e^{m(\textbf{r}_1+\textbf{r}_2)})^{-1}$. ...
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
16 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
112 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
86 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
votes
0answers
30 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
48 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
25 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
25 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
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
votes
0answers
14 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 [on hold]

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
votes
6answers
168 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
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
votes
3answers
48 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
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
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
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 ...
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

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$ [closed]

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$... if $x > 0$ and $cos(x)$ $> 0$
0
votes
0answers
44 views

Reduction of trigonometric functions to $x$ power [closed]

$$\huge{(\sqrt{1 - \sin^2x})^{2^{x^\sqrt{1 - \sin^2x}}}}$$ $x > 0$ if the domain of $x$ is between $1$ and $1.5$
0
votes
0answers
37 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
189 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: $$ 2^{n-3}=\frac {20}{n} $$ ...
-3
votes
1answer
29 views

Please help me with this exponential growth problem [closed]

a cell culture contains 5 thousand cells and is growing at a rate of r(t)=7e^(0.01t) thousand cells per hour. Find the total cell count after 5 hours.
5
votes
4answers
106 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
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.
1
vote
1answer
71 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
56 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
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
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 ...
1
vote
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
votes
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
votes
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
votes
0answers
138 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
66 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?
1
vote
0answers
24 views

Growth rate of bacteria involving logarithmic functions

I was trying to solve the following question but I keep getting the wrong answer, could anyone help me out and see why? A bacteria culture starts with 900 bacteria and grows at a rate proportional to ...
0
votes
2answers
31 views

easy exponential population growth problem help?

The question is: Let $C(t)$ be the number of cougars on an island at time t years (where $t > 0$). The number of cougars is increasing at a rate directly proportional to $3500 * C(t)$. Also, $C(0) ...
4
votes
4answers
98 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 ...
5
votes
4answers
189 views

Solving an exponential equation involving e: $e^x-e^{-x}=\frac{3}{2}$

In a previous exam, my professor had the question \begin{equation*} e^x-e^{-x}=\frac{3}{2}. \end{equation*} I attempted to take the natural log of both side to solve it, but evidently that was ...
0
votes
1answer
24 views

Operator - Exponential form

It is well known that for every unitary operator $\hat U$ an exponential of the form $$ \hat U = e^{i\hat H} $$ exists ($\hat H$ is hermitian). But I can only prove it the other way round: $$ ...
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 ...
0
votes
1answer
30 views

spherical wave expansion

In the paper, Sheen, David M., Douglas L. McMakin, and Thomas E. Hall. "Three-dimensional millimeter-wave imaging for concealed weapon detection." Microwave Theory and Techniques, IEEE Transactions ...
0
votes
1answer
17 views

How to derive sigmoid function from e by scaling & translating?

The Sigmoid function is like this: $\frac{1}{1 + e^{-x}}$ Can it be derived by simply scaling and translating the graph of $e^{-x}$ ? It looks to me as thought you could: 1). Translate it up, by 1 ...