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

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0
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3answers
93 views

Complex number problem- separating into real and imaginary parts!

Please help with a question that I am working on just now...:) If $z=2e^{i\theta}$ where $0<\theta<\pi$, how can I find the real and imaginary parts of $w=(z-2)/(z+2)$? Hence, how can I ...
0
votes
1answer
65 views

Prove $\lim_{n \to \infty}$ $(1+\frac xn-o(\frac 1n))^n=e^x$ [duplicate]

We know that $\lim_{n \to \infty}$ $(1+\frac xn)^n=e^x$. How to prove that $\lim_{n \to \infty}$ $(1+\frac xn-o(\frac 1n))^n=e^x$? Attempt of the proof: Let $\epsilon>0$ $\exists n_0$ such that ...
1
vote
3answers
74 views

Why is $ \overline{e^z} = e^\overline{z} $?

How can you conjugate an entire function? $ \overline{exp(z)} $ I need an equivalent. I thought this is only possible with complex numbers. What is the proof for $ \overline{e^z} = e^\overline{z} $ ...
1
vote
1answer
114 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
47 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
75 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
vote
1answer
61 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
vote
2answers
67 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
34 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
votes
1answer
35 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
votes
4answers
881 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
vote
1answer
52 views

Absolute value in exponential, signal energy?

How can this give this result? Isn't the absolute of $(e^(-2*t))$ always 1?
2
votes
7answers
167 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
votes
0answers
74 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
76 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
98 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 ...
1
vote
1answer
55 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
votes
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
44 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
68 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
53 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
votes
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
67 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
vote
0answers
29 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
102 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
votes
0answers
21 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
94 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
228 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
85 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
votes
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
76 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
votes
0answers
35 views

Simplifying a complex exponential equation

Can Someone please explain which identities are required to show that Thank you
5
votes
2answers
189 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
68 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
106 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
15 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
13 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
33 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
87 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
35 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
55 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 ...