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

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1answer
46 views

Exponential function given two points

I am trying to find an exponential function satisfying two points (having base "exp"). After some search, I couldn't find something relative (the most relevant was that ...
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1answer
51 views

Exponential Equation $4\cdot7^{x+2}=9^{2x-3}$

Let $4\cdot7^{x+2}=9^{2x-3}.$ I do not know how to solve for $x$. Progress Took logarithms, got $$4(x+2\log7)=(2x-3)\log9$$ $$(x+2)\log7=[(2x-3)\log9]/4$$
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1answer
35 views

Integral exponential and fraction of powers

I am trying to solve the following integral $$ \int_0^y \frac{x^{m-1}}{(1+x)^{m+k}} \exp\left(-\frac{m}{\gamma} x \right) dx. $$ I tried to look into different books such as Gradshteyn and Prudnikov ...
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3answers
61 views

Adding complex exponentials

Can somebody please explain $$e^{-\frac{3}{4}\pi i}+e^{-\frac{9}{4}\pi i}+e^{-\frac{15}{4}\pi i}+e^{-\frac{21}{4}\pi i}=0$$ WolframAlpha Computation.
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2answers
47 views

what to do with: logarithmic, trigonometric and exponential inequalities with variable outside

After encountering this inequality: $$ e^{x/2}=2x+1 $$ that leads me to: $$ x=2\ln(2x+1) $$ I realized that I don't know how to solve it. But this lack of knowledge expands also to $\cos(x)=x$ or ...
0
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0answers
24 views

Exponential convergence of controlled variables

I am reading a paper and I don't understand why, after some math they say that the controlled variables $$ \dot{\psi}_{13} $$ and $$ \dot{\psi}_{23} $$ converge exponentially. This is the paragraph ...
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0answers
53 views

Study $f_{\lambda}(x) = \lambda e^x + x^2 + 2x +2$ for any $\lambda \in \mathbb{R}$

This time I have the following questions: Consider $$f_\lambda: x \longmapsto \lambda\exp(x)+x^2 +2x +2$$ for any real $\lambda.$ 1) Compute $f'_\lambda$ (the derivative of $f_\lambda$). Show ...
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2answers
60 views

Calculating $ \lim_{n\to \infty} (1+\sin({1}/{n}))^{n}$ without L'Hopital or series expansions [duplicate]

I am trying to calculate the following limit, without using the L'Hopital rule or series expansions: lim (1+sin(1/n))^(n), n->infinity I now that it is the ...
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0answers
23 views

Prove convergence of $(1-\frac xk)^k$ as $k\to\infty$ using arithmetic-geometric mean

Define $f(x):=x^{t-1}e^{-x}$. For $k=1,2,\dots$ let $$f_k(x)=\begin{cases}x^{t-1}\left(1-\frac xk\right)^k & 0<x<k\\0&k\le x\le \infty\end{cases}$$ Show that $f_k(x)\to f(x)$ and ...
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1answer
34 views

Rewrite formula using exponential generating functions

In the equation below I want to extract $b_k$. $$\frac{a_n}{n!} = \sum_{k=0}^{n}\frac{b_k}{k!(n-k)!}$$ For all other exercises in the book I had to use $e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$ or ...
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1answer
51 views

Transformation Ricker equation

The classical Ricker equation for modelling density-dependent population growth is: $N_{t+1} = N_t * e^{r * \left(1-\frac{N_t}{k}\right)}$ where $N_t$ is the initial number of individuals (starting ...
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1answer
9 views

Basic Variable Isolation

I'm trying to Isolate DR in the function below. Was wondering if I got it correct. $(1 + DR)^y$ = $(1 + N/C)^C$ My answer $$Dr = e^{\ln(1 + N/C)^C \over y}$$ Sorry about that last line. ...
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1answer
86 views

How to continue solving? Perfect Cuboid

I am doing research on perfect cuboids, and I'm looking for values $a,b,c$ such that the following is integer, and I'm not sure how to continue this. Any suggestions are appreciated! $PED$ is a very ...
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0answers
17 views

How do you interpret this 3D function: Z = EXPX (a,b) * EXPY (1,c)

I have fitted a curve to my data using TableCurve3D software. The best graph which fits my data almost perfectly is Z = EXPX (a,b) * EXPY (1,c). Note that "a", "b", and "c" are constants. The problem ...
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1answer
85 views

Proof that $e^x$ can be expressed in a series of ascending powers of $x$

In a pure maths textbook I have, they prove that $e^x$ can be expressed as $1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots+\frac{x^n}{n!}+\ldots$ However, before they prove this, they say they ...
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5answers
207 views

Solving $2^{2x+1} - 2^{x+4} = 2^3 - 2^x$

$$2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$ How can I solve an exponential equation that has many terms as the one above. Include more than one method if available.
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4answers
61 views

$k'=k$ only for $e^x$ [duplicate]

How can one prove without using anything but differentiation, that $e^x$ is the only function with $f'=f$? Clearly I can prove that $(e^x)'=e^x$, and $0'=0$, but how can one show that no other ...
0
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1answer
33 views

Surjectivity of the complex exponential without using $π$

I want to prove that the exponential $\exp\colon ℂ → ℂ^×$ is surjective without using polar coordinates and without even using a definition of $π$. Is there such a conceptional argument? Say, I ...
6
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0answers
84 views

Is there a function $f(x)$ such that $f(f(x))=e^x$? [duplicate]

Is there a function which would be a "functional square root" of the exponential function? I.e., function $f(x)$ such that: \begin{equation} f(f(x))\equiv f^2(x)=e^x \end{equation} If not, what is ...
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2answers
31 views

derivative of $\frac{d}{dn}(1+\epsilon/2n)^n.$

I need to show that derivative of $\frac{d}{dn}(1+\frac{\epsilon}{2n})^n > 0.$ I use formula $(a^x)' = a^x\ln x.$ For now i have: $\frac{d}{dn}(1+\frac{\epsilon}{2n})^n = ...
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2answers
69 views

How to solve for nth term in series

I am making a game, and controlling a character's velocity. The game works by updating the character's velocity 60 times per second. At each frame, I do this: "set new velocity to current velocity ...
6
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2answers
101 views

Show that $\bigl| e^x + e^{-x}-2-x^2\bigr| \le {x^4 \over 6} $ for $|x| \leq 1$

My try at it $$ \left| e^x + e^{-x}-2-x^2\right| \iff | f(x) - p_2(x)| = |R_3(x)| $$ where $ f(x) = e^x + e^{-x} $ and $ |x| \le 1 $ This gets me $$ |R_3(x)| \le (e-e^{-1}) {x^3 \over 6} $$ This ...
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0answers
34 views

Simplifying a complex exponential equation

Can Someone please explain which identities are required to show that Thank you
5
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1answer
124 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 ...
3
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0answers
106 views

Is there a way to exploit local redundancy in a function to speed up Monte Carlo integration?

In every Monte Carlo method I've ever seen, $f$ must be recomputed from scratch for each point that is (somehow randomly) selected to contribute to the overall integral. However, most functions have ...
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4answers
51 views

help understanding how $\ln$ and $e$ cancel.

I realise cancel may be the wrong term and inverse may be more appropriate but these is one situation I really don't get…or rather haven't found a suitable explanation. Most sources I have come across ...
3
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3answers
61 views

Prove that $\frac{e^{2x}-1}{e^{2x}+1}i=\tan{ix}$

I have a doubt in complex numbers which I am unable to solve. The question is Prove that $$\left(\frac{e^{2x}-1}{e^{2x}+1}\right)i=\tan{ix}$$ I tried using hyperbolic sin and cosines but failed. Can ...
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2answers
97 views

Integral of an exponential

I have the following: $$ I(a,b) \equiv\int_{-\infty}^\infty e^{\frac{-1}{2}\left(ax^2+\frac{b}{x^2}\right)}dx$$ where $a,b>0$. And I have the following substitution as a hint: ...
0
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0answers
14 views

Analyze functions ($\exp(l) E_1(l)$ and $l\exp(l-1) E_1(l-1)$) that contain an exponential integral

Let $f_1(l)= \exp(l) E_1(l)$ and $f_2(l)= l\exp(l-1) E_1(l-1)$, where $E_1(.)$ is the exponential integral function. When I plot these 2 functions, I notice that $f_1$ and $f_2$ are 2 decreasing ...
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1answer
12 views

Exponential Price Growth Help

I am in the process of developing an online game. Unfortunately, I've run into an issue. I cannot figure out how to make the price of a 'level' increase at a proper rate. I am trying to make a ...
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0answers
24 views

exponentially decaying Fourier transform

Assume you have a real-valued function $f(x)$ defined over whole $\mathbb{R}$ and $f \in L^2(\mathbb{R})\cap L^1(\mathbb{R})$. What additional characteristics should this function have in order that ...
2
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1answer
82 views

$\lim_{h \to 0} \frac{\text{e}^h -1}{h}=1$

I want to show that $$\lim_{h \to 0} \frac{\text{e}^h -1}{h}=1$$ by using the Squeeze theorem. Is it possible to prove this with the Squeeze theorem? Maybe the two inequalities $$ \forall \, h ...
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0answers
27 views

Properties of log map on matrices in $SE(3)$

I am learning about the log map on $SE(3)$ and I want to check my understanding of properties for use in solving an equation. Are the following true, for A, B, C as elements of $SE(3)$? $$ \log(ABC) ...
3
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2answers
52 views

Solving the exponential equation $x^2 = e^{-mx}\cdot k$

I just had this problem come up at work, as part of a simulation where I had to solve the equation mentioned above (where $m$ and $k$ are constants). I googled solving exponential equations and I got ...
0
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2answers
38 views

Get $a_1$, $a_2$, $b_1$, $b_2$ from $a_1 \times \exp{b_1 \times x} - a_2 \times \exp{b_2 \times x}$

I have experimental data which follow the function below. $$f(x) = a_1 e^{-b_1 x} - a_2 e^{-b_2 x} + \epsilon$$ ($a_1$, $b_1$, $a_2$, $b_1$ are all positive real numbers. $\epsilon$ represents ...
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1answer
99 views

How to prove $x^ax^b = x^{a+b}$

I am looking for a proof of one of the exponent combination laws, namely the sum of powers. Here $x, a, b \in \mathbb R$ and $x > 0$. I thought about induction but since a,b are not only positive ...
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2answers
88 views

Why is the ratio of the number of terms needed to achieve successive integer values in the harmonic series approximately $e$?

Consider the harmonic series: $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5} + \cdots .$$ It takes $1$ term to achieve a partial sum of $1$, since $1$ is the first number. It takes $4$ terms to ...
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3answers
84 views

Computing $\lim_{x \to \infty} \frac{|x^n|}{e^x}$

How to prove that $e^x$ goes faster to infinity than any polynomial of $x$ without using the Taylor expansion of $e^x$ or L'hopital rule? in other words, the proof that: $$\lim_{x \to \infty} ...
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2answers
26 views

Nash Bargaining Equilibrium with exponential utilities

I'm trying to derive the answer to the following question: Two players play the classic divide-the-dollar game, which is an imperfect information version of the ultimatum class of games. Utility ...
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1answer
27 views

When are the functions $x_1=Ae^{iωt} + Be^{-iωt}$ and $x_2=Ae^{-iωt} + Be^{iωt}$ identically equal?

Suppose we have two equation $x_1=Ae^{iωt} + Be^{-iωt}$ and $x_2=Ae^{-iωt} + Be^{iωt}$. Where $A$ and $B$ are complex number and $A^*$ and $B^*$ are their conjugate correspondingly. Now if we want to ...
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0answers
61 views

Inequality involving integral of $\Gamma(x)$

The graph of $\frac{e^x}{\Gamma(x+1)}$ is somewhat bell-shaped. I think the proof of the following requires an understanding of the integral of this function that I can't glean from either Mathematica ...
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1answer
46 views

Modulus of exponential function with real and complex arguments

Can anyone please explain why $$|e^{\frac12 \sin(2x) }|\le e^{1/2}$$ for all real $x$, while $$|e^{-i\sin(x)^{2}}|=1$$ for all real $x$?
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2answers
41 views

How to prove the gaussian functions are linear independent?

Assume that I have N Gaussian functions with different means $\mu_i$ and variances $\beta_i$, How to prove $e^{-\beta_i(x-u_i)^2}$ are linear independent? 1$\le$i$\le$N
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1answer
34 views

Calculating a bound on the norm of a matrix exponential

The problem is this: Let A be a square $n \times n$ matrix, and define $$e^A=\sum_{k=0}^\infty \frac{1}{k!}A^k$$ Find a bound for $\lvert e^A \rvert$ in terms of $\lvert A \rvert$ and $n$. I was ...
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votes
8answers
322 views

Why is the differentiation of $e^x$ is $e^x$?

$$\frac{d}{dx} e^x=e^x$$ Please explain simply as I haven't studied the first principle of differentiation yet, but I know the basics of differentiation.
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0answers
35 views

Transforming a logarithmic expression?

Do you know any nice way to rewrite $\log(1-e^{A})$?
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1answer
49 views

Solving an equation of the type $axe^{qx} + be^{rx} + cx + d = 0$

I need to solve an equation of the type, $axe^{qx} + be^{rx} + cx + d = 0$ I tried but couldn't solve it. Does anyone have an idea how to solve this(for x)? Thanks
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4answers
1k views

Exponential growth of cow populations in Minecraft

Minecraft is a computer game where you can do many things including farm cows. When fed wheat, cows in Minecraft breed with each other in pairs and produce one baby per pair. After about 20 minutes ...
1
vote
1answer
56 views

Find $(1+i)^i$ in simpler terms, without imaginary exponents. [duplicate]

I was asked to find $(1+i)^i$, I don't know what to do when there is an imaginary component in the exponent. since $1+i=\sqrt{2}e^{-\frac{1}{4}i \pi}$ then $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ ...
0
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0answers
22 views

Integral of a Poisson-Exponential Joint Distribution

I'm considering a joint distribution of Exponential variable $X$ and Poisson variable $Y$. the thing is, I can't figure out a way to derive the $pdf$ for such a distribution. I know that ...