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

learn more… | top users | synonyms (1)

-1
votes
2answers
45 views

Solving exponential-linear equation

How to solve the next exponential linear equation? $y=1.6^x+3x$ I need to find $x$ in terms of $y$
2
votes
3answers
66 views

On the integral of $e^{aix}$.

I've been trying a simple trick, but I'm unsure as to why it is failing. Let's say I wanted to compute $ \int^{\pi} _{ - \pi} e^{aix} dx$ for some constant a. Why won't this trick work? $ \int^{\pi} ...
4
votes
1answer
94 views

How do you prove $e^x=\exp x$ for real, non-rational $x$?

Let $\exp x=\sum_{n\geq 0} \frac {x^n}{n!}$. Let $e=\exp 1$. Let $a,x\in \Bbb R$, $a>0$. We define $a^x=\sup \{a^r:r\in \Bbb Q, r<x\}$. I've already proved that for $x=\frac pq \in \Bbb Q$, ...
4
votes
4answers
409 views

Uniqueness of exponential function

To my knowledge, the exponential function is the unique function satisfying $f'=f$ and $f(0)=1$ however, unless I've made a mistake, we have $$\frac{\partial}{\partial x} (ax)^x = x (ax)^{x-1} a = ...
2
votes
1answer
38 views

Exponential matrix decay

I’m working on contractive systems that have a system of ODE equations. I have an exponential matrix multiply by time that is for a given matrix $A$ I’m getting $e^{At}$. I want to know what are the ...
3
votes
1answer
70 views

What type of functional equation is this?

I'm trying to solve the following functional equation $f\left(x\right)=A\mbox{ exp}\left\{ \int\frac{1}{f\left(x\right)x^{2}+Bx}dx\right\}$ where ...
13
votes
3answers
140 views

How to express $f(n\alpha)$ in terms of $f(\alpha)$

Original question: Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\dfrac{a^x-a^{-x}}{2}$, where $a>0$ and $a\ne 1$, and $\alpha$ be a real number such that $f(\alpha)=1$. Find ...
3
votes
1answer
89 views

What is the evaluation of $\sum_{k=1}^{\infty}\frac{k^n}{k!}$? [duplicate]

I stumbled upon a similar problem and really liked the answers there, so I wondered if there were a general solution for $$\sum_{k=1}^{\infty}\frac{k^n}{k!}=?$$ Sadly, when I try to apply some of ...
1
vote
1answer
31 views

What function describes this problem of every possible breeding of a set of dogs?

If I have n dogs [a, b, c, ...], and I want to breed them in every possible combination (every possible binary tree made of ...
6
votes
0answers
108 views

An integral for $2\pi+e-9$

Motivation Lucian asked about the almost-integer $2\pi+e\approx9$ in a comment to a partially answered why question about $e\approx H_8$. This is more involved than approximations to $\pi$ and ...
1
vote
4answers
119 views

Show $\sum_{k=1}^\infty \frac{k^2}{k!} = 2\mathrm{e}$ [duplicate]

I stumpled upon the equation $$\sum_{k=1}^\infty \frac{k^2}{k!} = 2\mathrm{e}$$ and was just curious how to deduce the right hand side of the eqution - which identities could be of use here? Trying ...
1
vote
1answer
54 views

What are the solutions for $2^x=x^2$? [duplicate]

What are the solutions for $2^x=x^2$? I noticed there were 2 roots: $2,4$. Are there any other roots, and how do you calculate them?
0
votes
1answer
29 views

Solving an expression containing two added exponential functions

I have a problem solving the below equation with respect to $x$: $0.6\cdot \exp(\frac{-40}{x})+0.4 \cdot \exp(\frac{10}{x})=1$ My problem is that I have two exponential functions which are added ...
2
votes
3answers
96 views

Limit of $(1+3/n)^{4n}$ as $n$ goes to infinity

This afternoon I was trying to evaluate $$\lim\limits_{n \to \infty} \biggl( 1 + \frac{3}{n}\biggr)^{4n}$$ but was having some difficulty in doing so. I know the answer to be $e^{12}$, and can ...
0
votes
2answers
49 views

Simplify $\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$

How do you make the jump from: $$\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$$ To: $$\frac{25^{21}-4^{21}}{25^{21}-4(25^{20})}$$
4
votes
3answers
87 views

Prove Exponential Function Inequality: $e^x \le \frac{1}{1-x}$

Prove that $e^x \le \dfrac{1}{1-x}, x\lt 1.$ I find that if we set $f(x)=e^x(1-x)$ then $f(0)=1 $ and $f'(x)<0, x\in(0,1]$ proving the inequality for $x\in[0,1]$ but I don't see how to prove it ...
0
votes
3answers
47 views

how tell if a series of power numbers is bigger then others

I trying to order a list of mathematical expressions in string format as: "2*2" "4^1" "4^2^5" so far, so good for non exponential operations (^). I could compute ...
0
votes
1answer
67 views

How can one properly understand the fact that $e^x$ can be differentiated an infinite amount of times?

Simply put if I follow the rule derived by the simple proof denoting $e^x$ to be the derivative of $e^x$ then it follows that it should have an infinite number of derivatives. Is this a conceptual ...
10
votes
3answers
459 views

Number system with $e^x = 0$ for some $x$

It is well known that $e^x \ne 0$ for all $x \in \mathbb{R}$ as well as $x \in \mathbb{C}$. Upon reading this article and doing a bit of research I have found that this also applies to the ...
1
vote
1answer
28 views

Limit of a sequence of functions recursively defined by integrals

$f_n:[0,\infty)\to\mathbb{R}$ is defined recursively by $f_1:=0$ and $$f_{n+1}(x)=e^{-2x}+\int_0^xf_n(t)e^{-2t}dt,\qquad n\ge 1$$ I need to show that the limit $f(x):=\lim_{n\to\infty} f_n(x)$ exists ...
3
votes
3answers
75 views

Square root of $e^{ix}$

Is this $$\sqrt{e^{ix}}=e^{\frac{ix}{2}}=\cos\frac{x}{2}+i\sin\frac{x}{2}$$ true? Or $$\sqrt{e^{ix}}=\sqrt{\cos x+i\sin x}$$ How do I express square root of $e^{ix}$ as a non-square root expression? ...
7
votes
3answers
78 views

Limit similar to $\lim_{n \to \infty} \left(1-\frac{1}{n} \right)^n = \text{e}^{-1}$

I want to show that $$ \lim_{n \to \infty} \left(1-\frac{n}{n^2} \right) \left(1-\frac{n}{n^2-1} \right) \cdot \ldots \cdot \left(1-\frac{n}{n^2-n+1} \right) = \lim_{n \to \infty} \prod_{k=0}^{n-1} ...
2
votes
2answers
59 views

Proof that $\lim_{x\to\infty} b^x=0 \iff 0 \leq b<1$

Are there any errors in the following attempt to prove the above? $(\Leftarrow)$ Let $f(x)=b^x$, with $0 \leq b<1$. Then, for all $x$, $f(x)>0$ and $f'(x)=b^x \ln(b)<0$. This means that $f$ ...
1
vote
4answers
77 views

Complex Finite Product $\prod_{k=0}^{n-1} (1-\zeta^k z)$

I am working on a review for a graduate level Complex Analysis course. The following problem is on the review: Let $\zeta= e^{\frac{2\pi i}{n}}$ $(n\in \mathbb{N})$; show that ...
0
votes
2answers
63 views

Show that $\exp(-\lambda x) \cdot\exp(\lambda x)=1$ using the power series

Let $A$ be a commutative Banach algebra. Consider the exponential function $$\exp(\lambda x) = \sum_{n=1}^\infty\frac{(\lambda x)^n}{n!},$$ where $x \in A$ and $\lambda \in \mathbb C$. We can easily ...
6
votes
7answers
513 views

How to solve the following equation involving an exponential function

How do you solve $10^x = x$? I'm not sure how to solve this algebraically. Using log functions wasn't enough.
1
vote
3answers
42 views

Provided that $f(u)$ is holomorphic, prove that $u$ is constant

Let $u: \mathbb{C} \rightarrow \mathbb {R}$ be a real valued function, so that $f(z)=\cos \left( u(z) \right) +i \cdot \sin \left( u(z) \right)$ is holomorphic in $\mathbb{C}$. Show that $u$ is ...
1
vote
2answers
24 views

Can't solve exponential equation using logs?

I can't figure out why my method isn't working. I know it is possible to solve this using a substitution but I don't know when to use the substitution. In general when are you supposed to substitute ...
0
votes
1answer
25 views

Integral Euler's formula equals integral $\frac{sin(t)}{t}$ dt

For real $c$ we should have that \begin{align} \int_{-T}^{T} \frac{e^{itc}}{2it} dt = \int_{0}^{T} \frac{\text{sin}(tc)} {t}dt. \end{align} However, for me this is not directly clear. I know that ...
2
votes
3answers
49 views

How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$

How can we show that: $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ for large N.
0
votes
2answers
27 views

More than one way to solve a exponential equation?

What techniques can you use to solve the following equation: $$5 \times 2^x = 2 \times 3^x$$ I know we can use logarithms, but I don't have a lot of confidence solving exponential equations in ...
0
votes
1answer
27 views

equivalent characterization of the complex exponential

I want to prove the following statement: Define the complex exponential function $\exp$ by $\exp(z)= \sum_{n = 0}^\infty \frac{z^n}{n!}$. Then $\exp$ can be characterized by $$\frac{\mathrm ...
0
votes
1answer
34 views

A simple question about limits.

This may seem like a simple question, but I feel as if it is wrong but I am unsure why. Is it possible to evaluate a limit in two stages for example: say you know that $x(1- a)\rightarrow b$ as ...
0
votes
4answers
64 views

Prove that $\lim\limits_{n \to \infty} (1-\frac{1}{2n+1})^{3n} = \frac1{e\sqrt{e}}$

I have this problem that I cannot seem to solve. I tried splitting it into two factors $$\lim\limits_{n \to \infty} (1-\frac{1}{2n+1})^{2n}\times \lim\limits_{n \to \infty} (1-\frac{1}{2n+1})^{n}$$ ...
0
votes
2answers
46 views

Sequence solutions of $ax=e^x$

This question comes from my answer to: Solving $4x = e^x$ without graphing and looking for intersection Here I've used a sequence of nested exponentials constructed from $$ x=\frac{1}{a}e^x $$ and a ...
1
vote
2answers
92 views

Solving $4x = e^x$ without graphing and looking for intersection

If I want to solve the equation $4x = e^x$, is there a way to solve for $x$ without graphing and looking for intersection?
1
vote
1answer
73 views

Determining parameters of $y=ab^x+c$ given 3 points

We can find the parameters for the equation of a parabola through $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ by solving the system \begin{cases} ax_1^2 + bx_1 + c = y_1 \\ ax_2^2 + bx_2 + c = y_2 \\ ...
0
votes
2answers
37 views

Growth of debt: exponential, logarithmic, or linear? [closed]

If I have increasing debt that I don't intent to pay off for a really long time, how would I prefer to have it grow? Exponentially, logarithmically, or linearly?
1
vote
1answer
35 views

exponential growth rate

Let's suppose I have $3$ flowers in a field initially and that the number of flowers doubles every month. I can then write that $$N=3(1+0.5/12)^{12t}$$ where $t$ is the time in years. Right? But then ...
1
vote
1answer
27 views

Evaluate if series with exponential diverges or converges

The task is to evaluate for what values of $a \in \Bbb R_+$ does the series $$\sum_{n=1}^\infty \frac{a^n \times n!}{n^n}$$ converge. I've already checked with the ratio test that it converges for $ a ...
0
votes
2answers
69 views

How to solve exponential equations?

How to solve the equation: $$2\cdot 3^x +2^{2x}+5^{2x-1}-13^x+10=0$$ Well the answer can be found by trial & error to be $x=2$. But I am not able to proceed in a systematic way. I cannot see ...
4
votes
1answer
73 views

If $\theta$ is a rational number, is $e^{i\pi\theta}$ algebraic?

I want to know if $\theta$ is a rational number, is $e^{i\pi\theta}$ an algebraic number or not? For the first step I tried to write it $(e^{i\pi})^\theta$, that equals $(-1)^\theta$, but I think ...
9
votes
1answer
65 views

if $f(x + y) = f(x)f(y)$ is continuous, then it has to be injective.

Let $f$: $\Bbb R$ $\rightarrow$ $\Bbb R$ be a non-constant function such that $f(a + b) = f(a)f(b)$ for all real numbers $a$ and $b$. Prove that if $f(x + y) = f(x)f(y)$ is continuous, then it has ...
1
vote
2answers
26 views

Simple e equation

$$e^{-x}-x+1=0$$ $$\frac{1}{e^x}=x-1$$ $$e^x(x-1) = 1$$ $$\therefore e^x = 1, x-1 = 1$$ Where $$x=0, x=2$$ Or, $$e^x = -1, x-1 = -1$$ Where $$x=nil,x=0 $$ Therefore, there is no solution to ...
0
votes
1answer
13 views

How to Find Variables of Exponential Function Based on Other Information

Given the exercise in the screenshot below, I don't understand why, in order to find the value of the constant 'r', we need to equate r2 to 0.55 (as they did in the screenshot), when we actually need ...
0
votes
0answers
14 views

How can I “map” a parameter with range $[0,∞]$ to a “ratio parameter” such as probability?

Newbie in the house! On one hand, I have this sense that there exists one non-arbitrary, a priori or 'natural' function to map $[0,∞]$ into ratio parameter, natural in the sense $e^x$ is natural. On ...
0
votes
1answer
18 views

Population decline.

I'm looking at a question here and I'm a bit confused on how I'm supposed to solve it. A population of 460 decreases at 5% monthly. How many years will it take for there to be 100 left on the island? ...
1
vote
1answer
64 views

A basis for the algebra $\mathbb{C}\{z^{\alpha}(1-z)^{\beta}\}$?

Let us consider the domain $$ \Omega=\mathbb{C}\setminus (]-\infty, 0]\,\cup\,[1,+\infty[) $$ (the doubly cleft plane). On it, we have the functions, $z^{\alpha}(1-z)^{\beta}$ for $\alpha,\beta\in ...
0
votes
2answers
37 views

How can I get Maclaurin series for $\frac{x^2 + 3e^x}{e^{2x}}$?

The answer for it is $$3 + \sum_{k=1}^n (3+k(k-1)2^{k-2})\frac{(-1)^k}{k!} x^k + o(x^n)$$ Well, I've tried to change every $e^x$ to $1 + x + \frac{x}{2!} + ... + o(x^n)$ and got nothing useful. I know ...
0
votes
0answers
26 views

Simplifying a probability distribution function using an exponential function

I have a pdf for a variable $r$ given two other variables $m, \kappa$ defined as follows: \begin{align} p(r|m,\kappa)=\frac{I_0(\kappa r)}{I_0(\kappa)^m}r\psi_m(r), \end{align} where $\psi_m(r)$ is ...