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

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2
votes
1answer
572 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 ...
0
votes
1answer
25 views

Find the half life using exponential expression

a) For a particular radioactive substance, the mass $m$ (in grams) at a time $t$ in years is given by $m = m_0e^{-0.02t}$, where $m_0$ is the original mass. If the original mass is $500$g, ...
9
votes
5answers
144 views

How to show $\frac{19}{7}<e$

How can I show $\dfrac{19}{7}<e$ without using a calculator and without knowing any digits of $e$? Using a calculator, it is easy to see that $\frac{19}{7}=2.7142857...$ and $e=2.71828...$ ...
0
votes
1answer
27 views

Rewriting $n(2^{2^{n-1}}-1)-2^{2^n}+2^{2^{n-1}+1}$

Could you help me to write in a better way the following expression? (by better I mean for example simplifying if I can) $$ n(2^{2^{n-1}}-2)-2^{2^n}+2^{2^{n-1}+1} $$
0
votes
2answers
27 views

Find base of exp function within the range of summation

I got a sequence where the relation between elements of the sequence is given by: \begin{align} y_1 &= b \\ y_{i+1} &= 2 y_i + b \quad (i \in \mathbb{N}) \end{align} where $b$ is called base, ...
0
votes
0answers
13 views

Make a formula based on a data table (Exponential function)

I always, since high school never found a good trick to do these kind of questions. Lets say you've got a table (x and y) X: 1 2 3 4 5 Y: 1 3 7 15 31 How can I make a function out of it? I ...
0
votes
3answers
75 views

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$?

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$? So far I have seen the following results: If $f\colon U\to\mathbb C$ has an antiderivative $F$ on $U$ then ...
2
votes
0answers
37 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! ...
4
votes
4answers
101 views

Is the natural logarithm actually unique as a multiplier?

The Wikipedia page on the natural logarithm says: 'Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from ...
1
vote
1answer
18 views

Find the maximum value of the function

So I was just messing around with finding the maximum and minimum values of functions, and I came across this: $$ \text{Find the maximum value of} \,\, f(x)=\frac1{x^{2x^2}}.$$ Any ideas?
0
votes
1answer
29 views

Solving equation in Matlab [on hold]

I want to solve this equation for several values of y y=4 exp(0,034x) y= [60 14 63 34 21 12 11]; Thanks
1
vote
1answer
20 views

Is the set $\bigsqcup_{p\in M} \{v\in T_pM: |v|_g< r_p\}$open in $TM$?(where $r_p$ the injectivity radius at $p$)

Let $(M,g)$ is a Riemannian manifold. (1)If $D_p$ is the largest domain on which $\exp_p$ can be a diffeomorphism, then is the set $$D=\bigsqcup_{p\in M} D_p$$ open in $TM$? (2)Likewise, if we denote ...
1
vote
1answer
27 views

Evaluating a tricky exponential function integral

I am trying to evaluate the following integral $$ I = \int_0^t s^{2\alpha - 1} \exp\left(\frac{i \sqrt{2} \left(t^{2 \alpha + 1} - s^{2 \alpha + 1}\right)}{ 2 \alpha + 1}\right)\mbox{d}{s} $$ where ...
10
votes
3answers
406 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 ...
0
votes
1answer
20 views

Random Walk with overshoot, step sizes $+1, -2$. Solve the polynomial in $e^λ$ [on hold]

If the moment generating function is $$mS(θ) = E(e^{θS}) = pe^θ + qe^{−2θ} = 1$$ Show that setting $$mS(\lambda) = 1$$ yields the unique positive solution: $$ \lambda = \log { \frac{q + \sqrt{4pq + ...
2
votes
1answer
29 views

Help in evaluating an integral of exponential function

I am trying to evaluate the following integral $$ I = \int_{0}^{t}s^{-b-1}e^{-\frac{1}{2} a^2 s^{-2 b}} ds$$ where $a > 0$ and $ 0 \le b \le 1$. I am not quite sure how to solve this. Any help ...
2
votes
3answers
32 views

Minor flaw in understanding of the proof of the derivative of exponential functions

I understand the majority of the proof of the derivative formula for exponential functions of the form: (full proof at bottom of post) $\frac{d}{dx}a^x$ but I have a little trouble with the last ...
0
votes
2answers
65 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: How to integrate $1/x$? After the algebra I said that 'This step of course gives the argument of $ln()$ the value $e$ and note that so far we ...
1
vote
1answer
32 views

Functions of the form $f(x) = k^x - x^k$

Let $f: \mathbb{R} \rightarrow \mathbb{R},\ f(x) = k^x - x^k$ where $k \in \mathbb{R}$ is a given constant. Currently I am thinking of positive $k$ and positive $x$ because there would be complex ...
5
votes
3answers
486 views

How much proof is needed in such paper (Maths related)?

I'm writing a paper (report) regarding Euler's Number $\space e \space$ (even though he didn't discover it). Within this paper, I show that: $${d\over dx} {e^x} = {e^x}$$ **NOTE: ** This is not ...
0
votes
1answer
14 views

The upper bound of a sum of exponential function

Could someone help me to find the upper bound of the following function: $f(x) = \sqrt{\sum_{n=i}^{N} e^{-\alpha_{i}\cdot x}}$, where $x > 0$, the $i^{th}$ coefficient $\alpha_{i} > 0$. I got ...
15
votes
1answer
644 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
0
votes
0answers
8 views

Transforming sigmoid function to concave function [on hold]

Can someone please tell me of a function that transforms a sigmoid function into a pure concave function?
1
vote
0answers
22 views

Formal Power Series Composition with Exponential

I have seen formal power series expressed as $$B(z) = \sum_{i=1}^{\infty} b_i{z}^i,$$ but then also as $$B(z) = \sum_{i=1}^{\infty} \frac{b_i}{i!}{z}^i.$$ Is there a significant difference between ...
1
vote
1answer
23 views

Exponential Decay of a radioactive substance

If $375$ mg of a radioactive substance decays to $300$ mg in $72$ hours, find the half-life of the element. I first used the mathematical formula of $$A = A_0e^{kt}$$ or exponential decay. After ...
1
vote
0answers
41 views

A variant of the exponential integral

Consider the following integral (for $x,y\in \mathbb{R}_{>0}$) $$E(x,y) = \int_0^1 \frac{\mathrm{e}^{-x/s-ys}}{s}\,\mathrm{d}s,$$ which is a variant of the usual exponential integral $E_1(x)$ to ...
0
votes
0answers
7 views

Name for this type of function form.

I use the function form of $$f(x)=ln^{2}(1+e^{(Ax)})$$ to denote the different between operating regimes of the MOSFET. This form let's me glue the diffusion and drift current of the MOSFET I-V ...
0
votes
0answers
60 views

Very difficult functions to prove with O notation

I am trying to prove some O notations as is it one of the tasks for my assignment in my course in algorithms and data structures. First of all I'd like to be sure that I got the "recipe" right. I use ...
0
votes
1answer
13 views

Finding the modulus of complex functions

Let $\gamma$ be the path$$\gamma:\left[0,1\right]\rightarrow\mathbb{C}, t\rightarrow\exp\left(t+it\right)$$ I have found that $$\gamma'\left(t\right)=\left(1+i\right)\exp\left(t+it\right)$$ To find ...
0
votes
1answer
44 views

exponential equation system without log [duplicate]

How should I solve this equation system without using logarythms,using just a simple method? (E.g. turning it into a quadratic one using t) $$\left(\frac{3}{2}\right)^{x-y} - ...
3
votes
4answers
118 views

Quick way to solve the system $\displaystyle \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} = \frac{65}{36}$, $xy-x+y=118$.

Consider the system $$\begin{aligned} \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} & = \frac{65}{36}, \\ xy -x +y & = 118. \end{aligned}$$ I have solved it by ...
0
votes
1answer
34 views

Solving Equation for $x$

Solve $(a + \sqrt {a^2 - 1})^{x^2 - 2x} + (a - \sqrt {a^2 - 1})^{x^2 - 2x} - a = 0$ for $x$ , where $a>1$ . My approach is as follows : $(a + \sqrt {a^2 - 1}) (a - \sqrt {a^2 - 1})=1 $ Let $(a + ...
4
votes
4answers
148 views

How can a complex exponential represent a real world quantity?

Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex ...
0
votes
1answer
40 views

Complex exponential with 2 pi

I wonder why is it wrong to do the following: $e^{i2\pi x}=(e^{i2\pi})^x=1^x=1$ for a real $x$ but not for an integer $x$
1
vote
1answer
33 views

Proof that: $e^{-x(1/\tau - i\xi)}$ evaluated at $x=\infty$ is zero

I remember my friend showing me how sandwich theorem can be applied here. Unfortunately, I can't find his solution anymore and I am not familiar with sandwich theory.
1
vote
0answers
29 views

Newtons Law Of Cooling (And Heating)

Rule is: $D= A.e^{-kt}$, Where: $k,a$ are elements of real numbers, $D$ is the difference between the temperature of the item and the surrounding air, and t is the time in hours since the object ...
11
votes
5answers
1k views

Exponential Function as an Infinite Product

Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e. $$\mathrm ...
2
votes
1answer
52 views

$n$ complex numbers with modulus $1$

The problem: Let $z_1$,$z_2$,...$z_n$ $(n \geq 3)$ be complex numbers such that $\left| z_1 \right|=\left| z_2 \right|=\ldots=\left| z_n \right|=1$. Then show that the following statements are ...
1
vote
7answers
3k views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
0
votes
1answer
45 views

How to calculate a definite integral with complex numbers involved?

I'm trying to calculate this integral, and I find it difficult when coping with complex numbers. $$ f(k) = \int_{lnK}^{\infty} e^{ikx} (e^{x}-K) dx ...
1
vote
0answers
48 views

Question about the proof of Central Limit Theorem

My instructor proved the central limit theorem using the characteristic function. I think the proof is a standard one because I found basically the same proof in wikipedia. So for i.i.d. ${X_1, ...
0
votes
1answer
21 views

finding the growth rate for exponential growth

I have this question, Determine the initial population of a bacterial culture whose growth is exponential if, after $7$ days, the population is $10$ million, and the number triples every in three ...
1
vote
1answer
19 views

Exponential form of a log

I'm a bit confused on the wording of this question: An equation is shown below x = log(20) What is the exponential form of this equation? So my answer is $10^x$=20. But I am not sure if that is ...
0
votes
1answer
18 views

Explanation of homogenous function

Is there someone, who can explain why the function $g(s)=f(e^s,e^s)$ is not homogeneous when it can be written as $\frac{9}{4}e^{s/2}s$. I got the function $f(x,y)=\sqrt x +2\sqrt y +\frac{3y}{\sqrt ...
-1
votes
3answers
68 views

Find a solution to $z+e^{-z}=a$ where $a>1$.

Find a solution to $z+e^{-z}=a$ where $a>1$. I have tried many manipulations with little success. I don't see how I can solve this for $z$. Any solutions or hints are greatly appreciated. I think ...
1
vote
2answers
58 views

What's the point of Euler's number in exponents? [closed]

I want to know why we use $(1+e^{\text{something}})^{-1}$ for artificial intelligence. I know $e$ is just $2.7$. So what? Why $2.7$ and not $3$? Does it have a special property?
0
votes
3answers
25 views

Understanding exponential decay

Say I have a variable $x$ that decays over time $t$ as follows: $$ \frac{dx}{dt} = \frac{-x}{\tau}. $$ Solving for $x$, I get \begin{align} x &= \frac{-1}{\tau}\int x dt\\ &=e^{-t/\tau}. ...
1
vote
1answer
36 views

Finding a Transformation for a Sum of Exponentials

I am looking to see if it is possible to find a transformation $T_i(f(x))$ such that $$T_1\left(e^x+e^{ix}+e^{-x}+e^{-ix}\right)=e^x-ie^{ix}-e^{-x}+ie^{-ix}$$ ...
0
votes
2answers
44 views

Matrix exponentials

Let $A(x)$ be a real valued matrix with $x$-dependent coefficients where $x\in \mathbb{R}$. What is the necessary and sufficient condition on $A(x)$ such that the matrix exponential $$\exp( - A(x))$$ ...
0
votes
1answer
52 views

Prove the Inequality on sequence

$a_n=(1+\frac{1}{n})^n$ , $b_n=\sum_{k=0}^n \frac{1}{k!}$. Show that $b_n-\frac{3}{2n} < a_n < b_n$.