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

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0
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2answers
61 views

No. of real solutions of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $

How many real solutions are there of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $? Please illustrate.
3
votes
1answer
68 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
1
vote
3answers
61 views

Can you help me to get the integration of $\frac{1}{x} \exp(-x)$?

I need the solution of integral like $$\int^\infty_a \frac{1}{x} e^{-x} \,dx.$$ Thank you
0
votes
1answer
44 views

No. of real solutions of the equation $\big(\!\frac{9}{10}\!\big)^x = - 3 + x - x^2$

How many real solutions are there of the equation $\left(\dfrac{9}{10}\right)^x=-3+x-x^2$ ? Please illustrate.
0
votes
0answers
16 views

Scaling model output to be between 0 and 1

I have fitted Cox model and the output is generated as: $e^{\beta x}$, where $\beta$ is the coefficient. Now, I would like to have the model output ranging between $0$ and $1$. I'm currently using ...
2
votes
0answers
31 views

Integral Evaluation: Exponential of and Hyperbolic Function

I'm trying to evaluate $$G^{\pm} = \frac{-i}{8\pi^2 X} \partial_X \int_{-\infty}^\infty d\phi e^{i m \left[X \sinh \phi \pm T \cosh \phi \right]}$$ for $T = \pm X$. Where $T, X, m \in \mathbb{R}$ ...
2
votes
0answers
21 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
4
votes
2answers
87 views

Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.

Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$. I demonstrated the reciprocal: $\Leftarrow )$ The two equations are ...
7
votes
2answers
92 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
8
votes
7answers
251 views

How to prove continuity of $e^x$.

I simply want a proof that $e^x$ is continuous. I have never really been able to find something satisfying these points: $e$ is defined to be the limit $\lim_{n\to\infty}\left(1+{1\over ...
4
votes
4answers
102 views

How to calculate $\exp\left(t\begin{bmatrix}0 & z\\z^* & 0\end{bmatrix}\right)$?

or, in a more general case: $e^{\begin{bmatrix}0 & v\\w & 0\end{bmatrix}}$, where: $v, w \in \mathbb{C}$
0
votes
1answer
18 views

finding limits as function approaches zero

Set $n=1+\epsilon$ and let $\epsilon$ tend to zero. $$ \begin{align} c_1 &= \frac1{2\pi} \left[\frac{e^{i\pi (1+\epsilon)}-e^{-i\pi(1+\epsilon)}}{(1+\epsilon)^2-1}\right]\\ &= ...
1
vote
1answer
44 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
1
vote
2answers
101 views

Fourier Transform the following exponential and cosine function: $f(x) = e^{-a^{2}x^{2}}cos(bx)$

I have a previous exam here for my course (Provided by the professor) that requires me to do a Fourier Transform of the following equation. Here is the function: $f(x) = e^{-a^{2}x^{2}}cos{(bx)}$ ...
2
votes
0answers
24 views

Area of intersection of polynomial and exponential functions

I was inspired to explore this by a recent post on the math subreddit, which to my knowledge went nowhere. Consider the families of functions $x^y$ and $y^x$. Given some $y \in \Bbb R$, the roots ...
5
votes
3answers
107 views

Integral definition of e

I know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...
1
vote
3answers
107 views

On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?

I was seeing the proof that $\exp(A)\exp(B)=\exp(A+B)$ on link Show that $ e^{A+B}=e^A e^B$ where uses the hypothesis $AB=BA$? Thanks!
0
votes
2answers
83 views

Is it possible to convert $\sigma = \int_0^\infty e^{-x^2} dx$ to an integral problem over $(0,1)$? [closed]

Is it possible obtain a transformation to convert $\theta=\displaystyle\int_0^\infty e^{-x^2}\, dx$ to an integral problem over $(0,1)$?
0
votes
1answer
45 views

exponential regression fit error problem

I have the following data and im trying to get an exponential fit. Ive tried a variety of different tools for this, which all seem to give quite a large error margin at the top of the curve. Plotting ...
2
votes
4answers
73 views

Limit of a rational function to the power of x

Ok so I have been trying for days already to find a solution to this all around the web and in math books but to no success. The problem is to evaluate a limit of a function composed by polynomial ...
0
votes
2answers
32 views

What is the 'growth constant'?

I'm looking into the formula of growth, namely $$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'. What is the growth constant and how do I find it? I'm looking at a bug that has on average 1,67 ...
1
vote
1answer
36 views

division by sum of exponentials of large negative numbers

I need to evaluate the following numerically: $$ f = \frac{\exp(a)}{\exp(a)+\exp(b)+\exp(c) + \exp(d)} $$ $a,b,c$ and $d$ are large negative numbers, they are smaller than -1000. Numerically ...
1
vote
3answers
53 views

Proof that $b^{\log_b(x)} = x$

I understand that the exponential functions are inverses, and would therefore map $x$ when formed as a composition, but I cannot find any formal mathmatical proofs. My thought process is: ...
0
votes
2answers
24 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
0
votes
1answer
44 views

Proof of simple interest formula

Can someone please prove to me that $I = PRT$, where $P$ is the principal, $R$ is the interest rate, and $T$ is the number of years/time. I have seen $I = P(1+TR) = P+PTR$ which does not equal $PRT$, ...
2
votes
0answers
37 views

Lie group - exponential Diffeomorphism

Let $G$ be a nilpotent, connected simply connected Lie group and $\mathfrak{g}$ its Lie algebra. It is known that the exponential map $\exp$ is a diffeomorphism. Now let $\mathfrak{g}_0$ be a Lie ...
2
votes
3answers
36 views

Programming Help - Solving for e(n)

I've been wrestling with this issue for a week and I just need some guidance on the math part of it. If I could just understand the math behind it I could piece together the functions to make it ...
1
vote
3answers
134 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
0
votes
0answers
22 views

What's the MLE of $\frac1\lambda$ for $f(x)=\frac1\lambda\exp − \frac x\lambda$?

What's the MLE of $\frac{1}{\lambda}$ for $f(x)=\frac{1}{\lambda}\exp \left(− \frac{x}{\lambda}\right)$? Is it equal to the mean of $x$ or the inverse of mean of $x$? Thanks,
-3
votes
1answer
31 views

Can I have an exponential function such that if x = infinity, y = 100?

I tried the most basic y = 100*constant^(1/x) assuming that 1/x = 0 when x is infinity, but it doesn't seem to work. This gives me a function that starts with a higher value of y and goes down till ...
1
vote
3answers
71 views

Prove that $e^x \gt 0$ for $x \in \mathbb{R}$ [duplicate]

This is a consequence of the exponential rule, but how do I actually prove it to be true?
0
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0answers
31 views

Continuous exponential functions

In my book, it makes it appear that any continuous exponential function, such as those regarding money, do not follow the traditional formula of $$\text{growth} = (1+\text{return})^x $$ Rather, it ...
0
votes
0answers
144 views

Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
1
vote
1answer
54 views

What's the MLE of lambda for $f(x)= \frac{1}{λ}\exp{\frac{−x}{λ}}$?

What's the MLE of $\lambda$ for $$f(x)= \frac{1}{λ}\exp\left({\frac{−x}{λ}}\right)$$ Values of x are 5,7,9,3,6,8 Is it just the mean of $x$? Thanks.
1
vote
0answers
38 views

What is the product of bessel functions of first and second kind when their arguments are same and tends to zero?

As we know, $\lim_{x \to 0} J_m(x)=0$ where $m\geq 1$ and $\lim_{x \to 0} Y_m(x)=\infty$ then what would be $\lim_{x \to 0}J_m(x)Y_m(x)$. Matlab shows the product is finite and $< 1$. What should I ...
1
vote
1answer
38 views

problem about population growth

At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of ...
0
votes
1answer
33 views

Find $t$ in $N = b \times g^t$.

The problem is the following: Find the value of $t$ in $N = b × g^t$. So for example "$512.000 = 2000 × 2^t$" I'm not really a mathematician so their may be a simple way or it could be hard.
0
votes
0answers
69 views

A question about exponential matrices

So here is my question, I would like to prove, If $R,S\in \mathcal M_{n\times n}(\mathbb R)$ are matrices such that, $$e^{t(R+S)}=e^{tR}e^{tS},\;\forall t\in\mathbb R$$ Then, $$RS=SR$$ And here is ...
1
vote
2answers
21 views

How do I find the inverse of this exponential function?

$x=-3(3^{-x})+9$ I know the steps up until a certain point. $x=-3(3^{-y})+9$ $x-9=-3(3^{-y})$ $\frac{(x-9)}{-3} = 3^y$ $ln (\frac{x-9}{-3}) = -y * ln 3$ Not sure what to do from here. I know I ...
0
votes
0answers
25 views

Generic Exponential curve base derivation

Alrighty so I am working on a computer program that forms ADSR envelopes including exponential curves for the attack, decay, and release segments. It uses the following equation for the exponential ...
-1
votes
2answers
31 views

Limit of n * ln(1+x/n)

How can you compute with the most primitive tools that: $$ \lim_{\stackrel{n \to \infty}{n > -x}}n \:\ln (1+\frac{x}{n})=x $$ Using l'hospital verifies this. However we hadn't proofed ...
1
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2answers
52 views

An exponential/polynomial inequality

Prove that there is at least $1$ real number $a>0$ with the property $$a^x\ge x^a $$ for any $x>0$.
0
votes
1answer
35 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
2
votes
2answers
42 views

why is $\lim_{t \to \infty}t^m e^{-\alpha t} = 0$ for every $m \in \mathbb{N}$ fixed and $\alpha \in \mathbb{C}$ with $Re(\alpha) \gt 0$?

I'm having trouble trying proving this fact: $\lim_{t \to \infty}t^m e^{-\alpha t} = 0$ for every $m \in \mathbb{N}$ fixed and $\alpha \in \mathbb{C}$ with $Re(\alpha) \gt 0$ I tried to use ...
0
votes
1answer
45 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
votes
1answer
17 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
1
vote
0answers
32 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???
2
votes
5answers
412 views

Which is actually exponential?

I've heard the term "exponential" applied to two sorts of functions: $$n^x\text{, where $n$ is a constant (e.g., $2^x$)}$$ and $$x^2$$ Which is really exponential, and what do I call the other ...
0
votes
1answer
35 views

First order ODE with $f'(x) = 810(10)^x$

I'm trying to find an explicit form of the series $f(0) = 89.1,f(1) = 899.1,f(2) = 8999.1, \cdots$. My first though was to take the derivative and integrate it, which I've done before with a fair ...
0
votes
3answers
70 views

Calculate exponential decay

I know this has been asked a few times before, but I'm struggling to apply it to my scenario... I have some known values in a table: $$ \begin{array}{|c|c|} \hline \text{Degrees} & ...