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

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0
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3answers
26 views

What constrains the following functional equation of exponents?

If I am not incorrect,the standard (Is it he standard?) form of an exponential equation is $$y=ab^{x-h}+k$$ What are the constraints on this equation, or in other words, how do each of the variables ...
2
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2answers
60 views

Is it possible to rewrite floor functions applied to a fraction using only the addition, multiplication, and exponentiation operators?

Let's restate this question in using mathematical notation. Let $n,k \in \mathbb{N}$. Let $f(n)=\left\lfloor{\frac{n}{k}}\right\rfloor$. Is it possible to rewrite this using the addition, ...
2
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3answers
35 views

A good site documenting approximations of irrationals

I'm thinking of Sloane here but I believe that only takes sequences/series into account. Basically I've derived an interesting, appealing formula for e and want to know if it's already been ...
25
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3answers
820 views

If there are entire $G_k$s such that $f=\exp\circ\exp\circ\cdots \circ\exp\circ G_k$ ($k$ times), must $f$ be constant?

I am a French guest and I hope that my English isn't too bad... So here is my issue: I consider an entire function $f$ which satisfies the following property for each complex number $z\in ...
3
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2answers
98 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
1
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3answers
108 views

Is x^x an exponential function?

I know that functions of the form $c^x$ are called exponential when $c$ is a constant. How about the function $x^x$? It seems somewhere in between exponential and double exponential to me. Is there a ...
1
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1answer
58 views

Should I use Taylor Expansion or the expansion of $e^x$ to express a second order differential (in this case)

I've been given this equation: $(1+x^2)\dfrac{d^2y}{dx^2} + 4x\dfrac{dy}{dx} + 2y = 0$ I've also been told that: $y=1, \dfrac{dy}{dx} = 1$, at $x=-1$ I've been asked to find a series solution of ...
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0answers
54 views

Inverse Fourier transform of two variable function $F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}$

I am trying to find the inverse Fourier transform of: $$ F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}, $$ where $k^2 = k_x^2 +k_y^2 +k_z^2 = k_\rho ^2 +k_z^2 $ is a constant. I am getting confused as to ...
2
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4answers
549 views

Which has a higher order of growth, n! or n^n? [duplicate]

In our algorithms class, my professor insists that n! has a higher order of growth than n^n. This doesn't make sense to me, when I work through what each expression means. ...
0
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1answer
49 views

Variations on the limit definition of the exponential function

I'd like to know how this might be proven, or why it's true: $\begin{aligned} \lim\limits_{n\to\infty}{\left( 1+\frac{1}{n^k}\right)^n} = \infty, \text{when } k<1\\ = e, \text{ when } k=1\\ =1, ...
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1answer
19 views

Difference Between Generalized and Alternative Compounded Interest Equations

I am currently studying a chapter called "An Economic Interpretation of e" in my Economics class and we are finding amounts of compounded interest. I am not actually looking for help on the problems ...
3
votes
4answers
145 views

Computing a large exp(x) in a numerically robust way.

I'm trying to compute $\lfloor e^x \rfloor$, where x is a 64-bit integer. The problem is that the result of the computation may be close to 2^64. In this range, 64-bit floating point numbers will be ...
1
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0answers
97 views

Maximum Likelihood Estimator of the exponential function parameter based on Order Statistics

Let $X_1, \ldots, X_n$ be a random sample from the exponential distribution $\exp(\lambda)$. Let $M_n=\max\{X_1, \ldots, X_n\}$ with probability density function $$g_{M_n}(x)=n\lambda e^{-\lambda ...
0
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3answers
50 views

Algorithm to calculate price based on number of units

I'm trying to come up with a pricing algorithm for my product. I've already set some prices at low intervals, but I need the algorithm to calculate a reasonable for very large orders. Here are the ...
0
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1answer
43 views

Solution of an equation with polynomial and exponential terms

Can anyone solve for $t$ the equation: $$ e^t=\frac{1-nt}{1-t} $$ with $n \in \mathbb N$ (known) and $t>0$. Online solvers give an answer only for specific values of $n$, but I need a general ...
0
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2answers
75 views

A Complex Variable ODE

suppose $f$ is a holomorphic function on some domain $D$ satisfying $f'(z)=af(z) $ for some >constant a. show that $f(z)=Ce^{az}$, for some constant $C$
0
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6answers
112 views

How to find all solutions of $4^x-3^x=1$?

I have problem with equation: $4^x-3^x=1$. So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other ...
1
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2answers
310 views

Derivative of exponential function proof

I'm looking for a straight forward proof using the definition of a derivative applied to the exponential function and substitution of one of the limit definitions of $e$, starting with $e = ...
0
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2answers
113 views

The problem of x = ln(x)

I am trying to find x values for points along the normal distribution curve, and I ended up with a problem that goes back to the method of solving $x = \ln x$. Right now, I have $\ln(a \mu) - \ln(10) ...
0
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2answers
37 views

Proportionality to find spent years for price drop

Well, the title's kinda messy, but this is a concrete example of what I'm trying to find out: Lets say there is a price of 40.000 USD, if the price drops at half, how many years does it take for the ...
0
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1answer
53 views

Naming and meaning of exponential power functions

I apologize if something like this has already been asked, but I don't have any ideas on search terms for this family of functions. I'd like to know two things: 1) Is there a name for the family of ...
3
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2answers
121 views

Why does $\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$?

Here is a standard identity: $$\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$$ Why does it hold true?
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2answers
88 views

Solving an infinite summation involving exponential and factorial

I'm trying to understand an equality that I found in this biology article. $$\sum_{i=0}^\infty\frac{e^{-x}x^i(1-y)^i}{i!} = e^{-x\cdot y}$$ Can you help me proving this equation holds true?
3
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1answer
60 views

A question on Exponential Equation

I came across the following question a few week ago (Exponential equation+derivative): Solve $3^x+28^x=8^x+27^x$. The answer for the above question is 0 and 2. I generalized the question, as ...
1
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1answer
223 views

Proving functions to be Big Oh

How do I determine if there exists a function $f$, such that \begin{equation} f(n) = {\mathcal O}(\log n), \end{equation} but \begin{equation} 2^{f(n)} ≠ {\mathcal O}(n). \end{equation} Is ...
3
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2answers
79 views

Quick Fact Check if $A$ and $B$ Commute, $\exp((A+B)t)= \exp(At)\cdot\exp(Bt)$?

If $A$ and $B$ Commute, $\exp((A+B)t)= \exp(At)\cdot\exp(Bt)$? is this statement true?
10
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1answer
187 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
0
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0answers
49 views

Are the solutions to this integral known?

Mathematica knows that the real part of $y$ in this integral: $$\int_0^{\infty } \frac{1}{x^{1/y}+2} \, dx$$ is: $$0<\Re(y)<1$$ Therefore I am wondering if the solutions to this integral known? ...
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1answer
47 views

If$ a+b-1=1+\frac{ln(2^a-1)}{ln4}+\frac{ln(2^b-1)}{ln4}$ then $a=b$?

If $$a+b-1=1+\frac{ln(2^a-1)}{ln4}+\frac{ln(2^b-1)}{ln4}$$ where $a,b>0$ are real numbers and ln is $log_e$, then is a=b?
9
votes
2answers
849 views

Integral of matrix exponential

Let us be given a square $n \times n$ matrix $A$. For a system \begin{align*} \dot{x}(t) = A x(t), \hspace{0.3 cm} x(0) = x_0 \end{align*} the solution is given by $x(t) = e^{At} x_0$. I am ...
0
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1answer
115 views

probability of maximum of two independent random variable

Suppose $X$ and $Y$ are two independant random variable with exponential distribution with paramet $\lambda=1$ and $M=$max{$X$,$Y$}. Then $P(M \ge 4)$ is equal to : Answer: 0.036 how do i come to ...
3
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1answer
143 views

Solve the differential equation $y'=\mathrm{e}^{-y^2}-1$

Consider the initial value problem $$ \frac{dy}{dx} = \mathrm{e}^{-y^2} - 1\ \ \text{for}\ \ x\in [0,1],$$ with initial data $$ y(0)=0.$$ Then: $$\int \frac{dy}{(e^{-y^2} - 1)} = x\ \ +\ \ c.$$ I ...
1
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1answer
43 views

Find $\exp(D)$ where $D = \begin{bmatrix}5& -6 \\ 3 & -4\end{bmatrix}. $

The question is Find $\exp(D)$ where $D = \begin{bmatrix}5& -6 \\ 3 & -4\end{bmatrix}. $ I am wondering does finding the $\exp(D)$ requires looking for the canonical form... Could ...
2
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1answer
112 views

Solving Integral that contain exponential and Power

I have an integral of this form: $$\int_0^\infty e^{-\frac{x}{a}-\frac{z^2}{bx}-\frac{z}{bx}}\left(\frac{c}{c+x+z}\right)^K~dx$$ where $K$ is a positive integer. $a$ , $b$ and $c$ are reals and ...
0
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1answer
110 views

Finding the Y axis value on a exponential trendline (MS Office Excel)

Hello I'm having some problems finding out the correct equation, or if there is one, to find the value on the Y axis. I have the following values: ...
5
votes
2answers
254 views

Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
0
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0answers
30 views

how to decrease the norm of a matrix

I have a time marching equation as follows: $$u^{n+1} =\frac{\text{source}^n}{2}+ \exp(M\cdot \mathrm{d}t) \cdot(\text{source}^{n}+u^n).$$ Now the time step of this equation depends on norm of $M$ ...
0
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0answers
21 views

Determine whether the set is uniqueness set

We say that $\Lambda$ is a uniqueness set for the Paley-Wiener space $PW_{\pi}$ if $$(F \in PW_{\pi} \wedge F|_{\Lambda}\equiv 0) \rightarrow F\equiv 0.$$ For example, $\Lambda =\mathbb Z$ is a ...
2
votes
6answers
333 views

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\ldots+a_n<\frac{1}{2}$, prove that $(1+a_1)(1+a_2)\ldots(1+a_n)<2$.

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\cdots+a_n<\frac{1}{2}$, prove that $$(1+a_1)(1+a_2)\cdots(1+a_n)<2$$ I've tried Hölder's inequality (the same result can easily be derived using ...
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0answers
22 views

Decay Function with some additional features

I'm writing a computer function to model data which seems to slope down exponentially until it gets to an optimal point. At that point it steadily grows and then stabilizes. What type of function ...
0
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2answers
57 views

Linear system of ODEs

Given is the ODE system $y'=\left(\begin{matrix}1\\1\\0\\ \end{matrix}\right)+\left(\begin{matrix}0&0&0\\0&k&0\\0&-k&k\\ \end{matrix} \right)y$ with boundary conditions ...
2
votes
3answers
139 views

Find $\exp(D)$ where $D = \begin{bmatrix}2& -1 \\ 1 & 2\end{bmatrix}. $

$$C = \begin{bmatrix}2& -1 \\ 0 & 2\end{bmatrix}\quad $$ I break it down into two matrices $$A = \begin{bmatrix}2& 0 \\ 0 & 2\end{bmatrix}\quad \text{and}\quad B =\begin{bmatrix}0 ...
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1answer
186 views

Find square matrices $A, B$, such that $\exp(A + B) \neq \exp(A) \exp(B)$.

The question is as the Title stated: I picked a very easy example. However, I am afraid, I am missing something. The two matrices that I picked are $$A = \begin{bmatrix}1& 0 \\ 0 & ...
4
votes
1answer
101 views

Proof of $e^{\ln(x)\ln(2)}$, which natural logarithm do I bring down?

I'm currently stumped with the proof for the following problem: $$F(x) = 2^{\ln(x)}$$ $$\Rightarrow F(x) = y$$ $$y = 2^{\ln(x)}$$ $$\ln(y) = \ln(2^{\ln(x)})$$ $$\ln(y) = \ln(x)\cdot\ln(2)$$ $$y = ...
0
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1answer
61 views

Understanding e when there is continuous compounding at less than 100%

I am not a math professional but I know the formula for continuous compounding and also (finally) just studied its derivation using the limit when n --> infinity; how r, n, t all play together in the ...
0
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1answer
20 views

Backsolving power curve with uplift

We have a curve $cx ^ b$. Let's say I sum $x$ 1 through to 52 using that curve and it gives me a number, say 10. I want to increase number 10 by 10% and we know that's 11. I can increase 10% very ...
1
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1answer
55 views

Graph of $y=x^x$ for $x<0$

I have been wondering about the graph of $y=x^x$. Most graphing calculators will quite happily graph it up to $0$, but after that they don't do anything else. Basic calculation suggests that, while ...
1
vote
3answers
66 views

exponential equation with different bases

We have $3^x-5^\frac{x}{2}=4$ My question is what we can do here ? Can we solved it algebraically or we need to notice that $x=2$ and then show that for $x \neq 2$ there aren't any other solutions?
0
votes
2answers
156 views

Solve equation $\exp(ax)+\exp(bx)=1$

The equation is $$ \exp\left(ax\right)+\exp\left(bx\right)=1, $$ where $a$ and $b$ are known real constants, $x$ is unknown. I would like to have the solution in form of relatively known special ...
1
vote
1answer
47 views

Outputting inequality with $e^x$

I many books I can find inequality which estimates $e$: $$\left(1+\frac{1}{n}\right)^n \lt e \lt \left(1+\frac{1}{n}\right)^{n+1}$$ I am wondering if correct is also to write: ...