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

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1
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1answer
142 views

Uniform Convergence to the Exponential Function over a Compact Interval

I'm trying to show that the sequence of functions $f_n(x)=(1+(x/n))^n$ converges uniformly to $f(x)=e^x$ over any compact interval of the real line. We're assuming that it converges pointwise. Here is ...
2
votes
2answers
50 views

yet another simple Laplace transform

what is $ℒ(t^2e^{3t})$ I have got this far so far: $=\int_{0}^\infty (t^2e^{t(3-s)})$ Integration by parts using: $u = t^2$ and $du = 2t$ $v = \frac{e^{t(3-2)}}{3-s}$ and $dv = e^{t(3-s)}$ Which ...
0
votes
1answer
79 views

Solve exponential equation $6\times3^{2x}-13\times 6^x +6\times 2^{2x}=0$

I have tried solving the following equation by using exponential properties and logarithms, but can not find some link between all of the terms: $$6\times3^{2x}-13 \times6^x +6\times 2^{2x}=0$$ ...
1
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1answer
165 views

Exponential Growth and Decay / compound interest

This is the question: "If you want to have $\$75,000$ after $35$ years in your account that pays $12\%$ annual interest compounded quarterly, how much should you put in as your original investment?" ...
1
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2answers
68 views

Integral $\int_0^b \frac{1-\exp\{-x\}}{x}\text{d}x\qquad 0<b<\infty$

Is there any closed form expression for the definite integral $$\int_0^b \frac{1-\exp\{-x\}}{x}\text{d}x\qquad 0<b<\infty$$ as I could not find one in Gradshteyn and Ryzhik Table of Integrals?
0
votes
1answer
35 views

Point of intersection between two exponentials with a constant term

Is there any way to solve algebraically for $x$: $a^x - b^x = C$ If not, is there a commonly used function that can be used to represent its solution? e.g., the Lambert W function for $a^x - bx = C$ ...
4
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1answer
39 views

Exponential Growth and Decay : $y = a (1+r)^t$

I know this is a really basic question for this website, but I can't find it anywhere else. This is the question: "If you deposit $\$3,750$ in an account that pays $6\%$ annual interest compounded ...
10
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4answers
891 views

Confused about complex numbers

I am confused about something: \begin{eqnarray} (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray} e^{2 i \pi}=1~ and~ 1^{0.5}=1 \end{eqnarray} ...
1
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1answer
58 views

Absolute value in exponential, signal energy?

How can this give this result? Isn't the absolute of $(e^(-2*t))$ always 1?
2
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7answers
184 views

How to estimate the value of $e$. [closed]

I am currently studying how to estimate $e$. To solve this problem I use these methods discuss below: Method 1: We know that $e^x = 1 + \dfrac{x}{1!} + \dfrac{x}{2!}+ \cdots $ So if we consider a ...
2
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0answers
75 views

Solve for $x: \ln(x+4)+\ln(x-2)=5$

Solve for x: $\ln(x+4)+\ln(x-2)=5$ Where do I go from here? If there weren't four terms in the equation I would use the quadratic formula. How can I solve for x? EDIT 1: Is this correct? ...
4
votes
1answer
78 views

Solve the integral [closed]

Can anyone solve these two integrals . $$ \int_{0}^{ \infty } \frac{x^2 e^{-x^2/2 \sigma ^2}}{(x-a)^2+b^2} dx $$ and $$ \int_{0}^{ \infty } \frac{e^{-(\ln x - \mu )^2/2 \sigma ...
-1
votes
3answers
104 views

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

Recently I asked a question on Maths SE Proof that at most one of $e\pi$ and $e+\pi$ can be rational after solving this one one I was thinking whether $e^\pi$ is greater or $\pi^e$ ? On calculating ...
2
votes
1answer
60 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 ...
0
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2answers
82 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 $$4(x+2\log7)=(2x-3)\log9$$ $$(x+2)\log7=[(2x-3)\log9]/4$$
0
votes
1answer
46 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 ...
0
votes
3answers
70 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.
0
votes
2answers
56 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
votes
0answers
32 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 ...
0
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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 ...
0
votes
2answers
69 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 ...
1
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0answers
34 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 ...
1
vote
1answer
37 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 ...
0
votes
1answer
58 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 ...
1
vote
1answer
11 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. ...
0
votes
1answer
107 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 ...
0
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0answers
22 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 ...
1
vote
1answer
95 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 ...
8
votes
5answers
234 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.
0
votes
4answers
62 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
votes
1answer
36 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
votes
0answers
86 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 ...
0
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2answers
34 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 = ...
0
votes
2answers
85 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
votes
2answers
105 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 ...
0
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0answers
35 views

Simplifying a complex exponential equation

Can Someone please explain which identities are required to show that Thank you
5
votes
2answers
191 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
votes
0answers
107 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 ...
1
vote
4answers
75 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
votes
3answers
62 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 ...
4
votes
2answers
108 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
votes
0answers
16 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 ...
1
vote
1answer
14 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 ...
2
votes
0answers
40 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
votes
1answer
88 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 ...
0
votes
0answers
38 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
votes
2answers
58 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
votes
2answers
43 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 ...
6
votes
1answer
107 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 ...
10
votes
2answers
103 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 ...