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

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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 ...
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2answers
73 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 ...
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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 ...
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44 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 ...
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8 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 ...
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1answer
14 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 ...
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64 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 ...
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1answer
39 views

Proving that: $e^{-x(1/\tau - i\xi)} \to 0$ as $x \to \infty$.

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.
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0answers
37 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 ...
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2answers
74 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, X_2,\...
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1answer
47 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 =(\frac{e^{(ik+1)x}}{ik+1}-K\frac{e^{ikx}}{ik})|_{x=...
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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 ...
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1answer
45 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 $x$
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1answer
22 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 ...
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1answer
27 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 ...
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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 ...
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3answers
2k views

Why is Euler's number used as a base for logarithms? [duplicate]

Is there some special property of '$e$' which makes it suitable to be used as a base for logarithms? Moreover, does the natural logarithm possess some advantage over the common logarithm? I don't ...
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3answers
70 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 ...
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2answers
60 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?
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3answers
34 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}. \...
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1answer
38 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 $$T_1\left(e^x+e^{ix}+e^{-x}+e^{-ix}\right)=e^x-ie^{ix}-e^{-x}+ie^{-ix}$$ $$T_2\left(e^x+e^{ix}+e^{-x}+e^{-ix}\...
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2answers
52 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 $$\exp( - A(x))$$ ...
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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 ...
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2answers
79 views

How do I integrate this expression involving exponential and polynomial

I tried a few ways (integral by parts, expanding), but I'm unable to compute this integral. $$\int_0^\infty\frac{\lambda^{n}e^{-\lambda}}{(\lambda + b)^2}\, \text{d}\lambda$$ n >= 0, b >= 0
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1answer
58 views

Prove the Inequality on sequence [closed]

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

Finding the hourly growth rate

A species of bacteria doubles in population every 6.5 hours. There were 100 bacteria to start with. What is the hourly growth rate of the bacteria? How many bacteria will there be after a day and a ...
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1answer
45 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} - \left(\frac{2}{3}\right)^...
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6answers
70 views

How to deduce the limit relation $\lim_{x\to0} \frac{e^{cx}-1}{x}=c$

Let $f(x) = e^{cx}$ where $c$ is constant. Show that $f'(0)=c$ and use this to deduce the limit relation $$\lim_{x\to0} \frac{e^{cx}-1}{x}=c$$ Proving $f'(0)=c$ is easy but I'm not sure how the limit ...
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0answers
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Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
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0answers
50 views

Solving exp integral in closed form?

I am trying to solve the following integrals: 1) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 m^2})} dxdy $ 2) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 n^2})} dxdy $ 3) $\int \int ...
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1answer
35 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 + \...
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4answers
144 views

Dubious “proof” of $e^x$ derivative?

The proof to which I am referring is amply discussed here: Derivative of exponential function proof, but I remain unconvinced by the answers that pertain to the specific proof discovered by ...
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4answers
132 views

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$? [duplicate]

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$. Is it possible to write the function $f(x)=x^n$ and since we know $\frac{n}{1+n}\to 1$, so $f(\frac{n}{1+n})\to 1^n=1$. So the limit it $1$. ...
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4answers
50 views

Proving uniform convergence of $(1+\frac{x}{n})^n$ to $e^x$ on compact intervals in the real numbers

My goal is to prove that if $b> a > 0$ are real numbers, then: $\lim_{n \rightarrow \infty} \int_a^b (1 + x/n)^n e^{-x} dx = b-a$. I think the best way to do this is to show that $(1+x/n)^n$ ...
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1answer
74 views

Show by series definition of exponential function that $\exp(-x) \rightarrow 0 $ for $x \rightarrow \infty.$

There are many arguments I have seen using $\ln-$ arguments and other properties of the exponential function to show the existence of this limit $\exp(-x) \rightarrow 0 $ for $x \rightarrow \infty$. ...
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1answer
28 views

Is it possible to represent pieces of two functions with one equation?

I'm trying to create a rudimentary weighting system for evaluating how close two numbers are to each other. (This corresponds to string lengths - coding project for work... happy to explain in more ...
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3answers
45 views

Calculating exponential limit

I've been breaking my mind over this one. Find the limit. $\lim\limits_{n \to \infty} (\frac{n^2+3}{n^2+5 n-4})^{2n} $ I know it equals $\frac{1}{e^{10}} $ but can't figure out how to find it. Help?
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5answers
109 views

Maclaurin Expansion for $e^{e^{z}}$ at $z=0$

I need to find terms up to degree $5$ of $e^{e^{z}}$ at $z=0$. I tried letting $\omega = e^{z} \approx 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$, and then substituting these first few terms ...
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1answer
38 views

Acceleration: If I know distance, time, and initial velocity, what's acceleration and final velocity?

So I know the Initial Velocity ($V_i$), Time ($t$), and Distance ($d$). I know that $$d = V_it + \frac{1}{2} at^2$$ If I rearrange this, would acceleration $a = \dfrac{2(d - V_it)}{t^2}$ ? Then ...
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1answer
43 views

Finding probability a particle will appear after t seconds (exponential r.v)

Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density with $\lambda=1$ The probability $P(0,T)$ that a particle will appear in the next T ...
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2answers
54 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
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1answer
100 views

Find monotonic functions going from $0$ to $+\infty$ for $x \in (-\infty,+\infty)$ (similar to $e^x$)

How can we find functions on $\mathbb{R}$ with exponential-like properties, namely: $f(x)$ is infinitely differentiable; $f(x)$ and all its derivatives are monotonic; $f(x)$ and all its derivatives ...
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2answers
33 views

Solving exponential equation (quadratic type)

I fail trying to solve the following equation: $9^x-6^x-2^{2x+1}=0$ Trying to write it as a quadratic equation makes my constant term exponential $(3^x)^2-2^x3^x-2^{2x+1}=0$ How can I solve this ...
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2answers
26 views

Clarification Needed Regarding $\sinh^{-1}(-3)$

As the definition of $\sinh^{-1}(x)$ goes : $\sinh^{-1}(x)=\ln\left(x+\sqrt{x^{2}+1}\right)$ So what I expect to get is $\sinh^{-1}(-3)=\ln\left(-3+\sqrt{10}\right)$ The value inside of the ...
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1answer
26 views

Using exponential decay function to predict outcome

Let's say I have a graph that follows the function $y= ae^{-bx}$ , and I'm trying to predict the chlorine residue left in a pool after a certain amount of time. So for $2$ hours, the chlorine residue ...
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0answers
51 views

Is there a way to solve the exponential equation $a^x + b^x + c^x = d$ analytically?

So I came across this equation. $$a^x + b^x + c^x = d$$ where $a, b, c$ and $d$ are all constants. And I just wondered, is there any way to solve for x analytically?
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3answers
34 views

Is there a non-exponential function whose limit at infinity is a real, irrational number?

$e$, for example, can be calculated through a non-polynomial function $(1+1/x)^x$, but I cant think of an example for a non-exponential function (or rational function) where the limit to infinity ...
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1answer
63 views

Prove that $n! = O(n^n)$

I thought $n^n$ was greater than $n!$. How would I go about proving this? I have this so far: Assume that $P$($n$) is true $n!$ = O($n^n$) Assume that $P$($n+1$) is also true $(n+1)! ...
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1answer
65 views

Two problems with Exponents

How to solve following problems on exponents: $$\frac1{1+p^{a-b}+p^{a-c}}+\frac1{1+p^{b-c}+p^{b-a}}+\frac1{1+p^{c-a}+p^{c-b}}=?$$ and If $a^2bc^2=5^3$ and $ab^2=5^6$, what is $abc$? Please ...
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1answer
50 views

How to solve this difficult one variable equation analytically?

Would anybody like to explain me clearly how to solve analytically this equation? $$5.56=\frac{1-e^{-5.5x}}{1-e^{-x}}$$ I have already solved it with Mathematica and it gives $x=-0.004809$. However, ...