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

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3answers
47 views

Proving that $ 1-u = e^{-u - \,u^2/2 - \,u^3/3 -…}$

How can one see that for $-1 < u < 1$ we have the following equality $$ 1-u = e^{-u - \,u^2/2 - \,u^3/3 -...} \,\,\,\,?$$ It's probably easy to prove, however I've tried a couple of things so ...
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0answers
18 views

Normalizing a probability density function

I need to find a normalization term $N(\alpha,\beta)$ for the probability density function: $$PDF(\alpha,\beta)=(x-x_1)^{\alpha}e^{-\beta(x-x_1)}$$ In other words, solve the following equation: ...
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0answers
7 views

Poisson distributed graphs

I am currently reading a paper about poisson distributed graphs and came across the following formula. Apparently the degrees of the graph are distributed binomially through the following ...
0
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0answers
46 views

How do you solve this differential equation? $\tfrac{dx}{dz} = i (M x)$

How do you solve this differential equation : $\tfrac{dx}{ dz} = i (M x)$ where $M$ is a tridiagonal matrix with elements $100$. That is, $M$ is an array with $100$ elements in triagonal form, ...
2
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2answers
53 views

What do you get when you differentiate a $e^{f(x)}$-like function

I need help with exponential functions. I know that the derivative of $e^x$ is $e^x$, but wolfram alpha shows a different answer to my function below. If you, for example, take the derivative of ...
2
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0answers
35 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
2
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0answers
25 views

Parameterizing an implicit curve

I have to parameterize this curve: $$F(x,y)=y-x^2+x-e^{-yx^2}=0$$ But I don´t know how to do it. thanks
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1answer
37 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
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0answers
71 views

How to approximate large sum of exponential variables

Is there any way to approximate the following sum: $$ \sum_{i_1=1}^N \sum_{i_2=1}^N \cdots \sum_{i_k=1}^N \cdots \sum_{i_N=1}^N \exp(-r_{i_1} - r_{i_{k+1}} - r_{i_{2k+1}} - r_{i_{3k+1}} \cdots - ...
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2answers
28 views

evaluate exponential using Euler identity

let us consider following exponential $e^{-j*\pi*k/2}$ and $e^{j*\pi*k/2}$ we can decompose it as $cos(\pi*k/2)-j*sin(\pi*k/2)$ and second one same with plus sign ...
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0answers
17 views

Sufficient condition for a indefinite integral to be an elementary function

I would like to find a sufficient condition on two polynomials $P(s)$ and $Q(s)$, such that the function $s \mapsto Q(s)e^{P(s)} $ has a primitive integral of the form $s \mapsto R(s)e^{P(s)} $ (with ...
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1answer
26 views

Need help creating a special function

I'm creating a special function in a game and needed some help with the maths end of it. Essentially, I need a programmable, non-linear function so that $f(100) = 0$, and $f(0) = 100$ (or some other ...
2
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0answers
38 views

Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
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1answer
29 views

Don't understand answer from exponential growth question

"A biology student finds a large glass bottle which can be used to grow a bacterial culture. She has a bacterial culture that doubles in size every minute, and with the amount she currently has, she ...
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2answers
85 views

How to Show Polynomial Growth < Exponential Growth (Without L'Hopital!)

Can anyone offer me a way to show that exponential growth trumps polynomial growth, without using L'Hopital's Rule? When I learned function growth speeds in high school, the closest thing to a proof I ...
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2answers
32 views

Probability Random Variable question Need Help Please

You have a set of ten light bulbs - the lifetime of each of them being given by an exponential RV with mean 1000 hrs. Find the probability that.... (a) at least 7 of the bulbs function for 1500 or ...
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2answers
77 views

how to convert 1e+11 into number?

What will be 1e+11 in number? I know e2 means * 10^2 but i am confuse with this above question. what will be its value? I know how to use exponential function when ...
1
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1answer
54 views

How to find the parametric equation of $x^y=y^x$ without Lambert W function?

This is sort of a follow-up to my previous question. I've done basic conversions of parametric to to cartesian and back as part of my A-level, but never anything more advanced than a sin/cos ...
1
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1answer
43 views

$\mathrm{Ei}(x)$, the exponential function, some question.

I have a question involving with $\mathrm{Ei}(x)$, define as $\int_{-x}^{\infty}e^u \cdot u^{-1} \mathrm{d}u$. My question is, when I have a expression say $\exp(x) \cdot \mathrm{Ei}(x)+1$. I want ...
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5answers
224 views

Why $e^x$ never equal $x$?

Je veux savoir pourquoi $x=e^x$ n'a aucune solution dans $\Bbb R$. Lorsque j'ai essayé de tracer le graphe de la fonction $e^x$, j'ai trouvé en fait qu'elle est une fonction strictement croissante ...
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1answer
11 views

Consumption change calculation

I want to calculate yearly consumption change according to the following formula: $$C_{t+1}=C_{t}e^{x_{t}}$$ I need to calculate ${x_{t}}$. I have the consumption data $C_{t+1}$ and $C_{t}$.
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1answer
70 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
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1answer
40 views

Something that grows faster than NP class of problem does

I have a theoretical question. F.ex. we have a NP-class of problem, i.e. which do need exponential time on deterministic Turing machine. Is there anything that is growing faster than exponent does. ...
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0answers
27 views

Can this be solved analytically?

I have a sum of two Gaussian type functions, $g_1(x) = C_1 Exp[-\alpha (X_1-x)^2]$ and $g_2(x) = C_2 Exp[-\beta (X_2-x)^2]$ and have found that the derivative w.r.t. $x$ is $f(x) = 2 C_1 (X_1 - x) ...
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2answers
295 views

How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
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2answers
75 views

Contour integral $\int^\pi_{-\pi}(a-\cos\theta)^b\exp(c\cos\theta)d\theta$ assuming $a>1$, $b>0$, $c>0$

Under the condition $a>1$, $b>0$, $c>0$, is there any good function to express the following integral? $$ \int^\pi_{-\pi}\left(a-\cos\theta\right)^b\exp\left(c\cos\theta\right)d\theta $$ I ...
5
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0answers
141 views

$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.

Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
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1answer
20 views

Question about calculating exponent of polynomial

$V=R_{3}[X] $ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$ I need to find $e^{T(1+x+x^{2}-x^{3})}$ I don't understand how to do it? what does it mean to calculate exponent ...
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1answer
69 views

How can I solve this equation: $e^{2x^3 - 6x^2 + 3} = 0$ [closed]

I don't remember what I supposed to do in this situation...I know that it's necessary transform both sides of the equation in the same base. However, what I need to do when i have a 0? My equation: ...
5
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6answers
83 views

Need explanation for simple differential equation

I can't figure out this really simple linear equation: $$x'=x$$ I know that the result should be an exponential function with $t$ in the exponent, but I can't really say why. I tried integrating ...
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0answers
15 views

Exponential Integral Function representation by sum

I have an expression for the Exponential Integral Function as the followings: $$ E_{L+1}(x) $$ where L is a positive integer larger than zero; and x is real number larger than zero. Now I have this ...
3
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0answers
54 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ ...
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1answer
13 views

find exponential line going through 3 points

I have 3 points:(0,0);(55,64);(137,200) How could i get the formula going through those 3 points? They line up in an exponential line like this one:
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2answers
147 views

Which is greater, $e^{\pi}$ or $\pi^e$? [duplicate]

I'm familiar with a simple method of demonstrating that $e^\pi$ is greater: $f(x) = \ln|x|/x$ $f'(x) = (1 - (\ln|x|))/(x^2)$ so f's max is at $(e, 1/e)$ so $1/e > \ln(\pi)/\pi$ and $e^{\pi} > ...
3
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2answers
60 views

What is the domain of $x^x$

I'm trying to figure out the domain of the function $y=x^x$. When I graph it, it appears to be defined on $[0, \infty)$, but then when I plug in individual negative numbers, for some of them I get ...
3
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2answers
50 views

Proof of the derivative of $a^x$ [duplicate]

I've tried for a while myself from first principles and applying various rules, but always end up going in circles. I've gotten as far as $$ y = a^x $$ $$ \frac{dy}{dx} = a^x \left( \lim_{x ...
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0answers
69 views

Only one positive solution

When $(a+b)^2=(x-3a)(x-b)e^x$ has only one positive solution, find the relationship between a and b. Here, a and b are constants and satisfy $a>b>0$. Hint: consider the graph of ...
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1answer
26 views

Calculus exponential function/ slope and equtation

Consider the function $f(x) = 3(1 − e^x)$. Use exact values when answering the following questions: Find the slope of the graph of $f(x)$ at the point where it crosses the $x$-axis. Find the ...
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1answer
16 views

Derivative of exponential function

1) $f(t) = (\ln 5)^t$ what is the $f'(t)$? I tried $t\ln(5)$ but it was wrong. 2) $f(x) = x^{\Large π^6} + (π^4)^x$ This one I did not attempt in it because I find it confusing little bit.
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10answers
1k views

how to see the logarithm as the inverse function of the exponential?

I saw here in math.stackexchange some proofs of how the log and exp functions are related to each other, but I want to get an intuition for that. In layman terms, how would you explain the connection ...
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0answers
34 views

Fractional derivative of exponential function

With the $n$th order derivative ($n$ as a positive integer) of $e^{ax}$ given by $$D^{n}e^{ax}=a^ne^{ax},$$ is the generalized (or fractional) derivative the same? Does it apply for any arbitrary ...
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4answers
58 views

Need an example of piece wise function continuous but not differentiable

I Need an example of piece wise function continuous but not differentiable. One of the functions has to be trigonometric and the other has to be exponential. Please
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0answers
38 views

Derivative of Log of Summation of exponential function (base e)

A financial formula that I am implementing requires that I find the first derivative of a function to find a local maxima, from scratch. Can someone please help me with finding the first derivative of ...
3
votes
1answer
47 views

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

Let $X$ be a complex Hilbert space, and let $A$ be a bounded linear operator on $X$. Define the real part of $A$ to be $\Re(A)=\frac{1}{2}(A^{\star}+A)$, and define ...
3
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1answer
26 views

Skew-symmetric matrix and exp function $e^A$

Let $A_{nXn}(\mathbb{R})$ Skew-symmetric matrix $A=-A^t$ prove that $e^A(e^A)^t=I$ while: $e^A=\sum_{i=0}^{\infty} \frac{A^n}{n!}$ I tried this: $A=-A^t \Rightarrow A$ is Diagonalizable with ...
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1answer
28 views

Exponential Distribution as a density function

I have an important presentation on tuesday about the exponential distribuion as a density function. My question is: What are the advantages of using this function? In order to fulfill my task i have ...
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3answers
151 views

How many different definitions of $e$ are there?

It seems as though, in my analysis and calculus courses, in particular, a common cop-out when asked to prove an identity involving $e$, is the phrase "it's true by definition". So, I'm trying to find ...
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3answers
49 views

Determine all positive numbers $a$ for which the curve $y = a^x$ intersects the line $y = x$ without calculus

The answer is $0 < a < e^{1/e}$ , but how to find it? Is it a system of equations? Which ones? I just need an idea at least, because I'm stuck. If it is impossible without calculus, solve it ...
2
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2answers
69 views

Does e = limit as x tends to negative infinity hold true?

Does $$e=\displaystyle\lim_{x \to -\infty}\left(1+\frac{1}{x}\right)^x\qquad\quad?$$
2
votes
2answers
67 views

$(1-x)^y ≈ e^{-xy}$

Here is an approximation I often see in biology articles but don't really understand: $$(1-x)^y ≈ e^{-xy}$$ I think this $e^{-xy}$ closely approximates $(1-x)^y$ whenever $x$ is small. Can you help ...