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

learn more… | top users | synonyms (1)

0
votes
0answers
19 views

What is 0^0??? Is it 0 or 1? This is a SPECIAL CASE… [duplicate]

$0^0$... what is it? Patterns say this is $1$. The Binomial Theorem $(1 + -1) ^0$ says that this is 1. $0^1 \cdot 0^{-1} = 0^0 =$ undefined... I'm confused. Is it undefined or 1?
0
votes
2answers
19 views

Raising a negative number to an odd negative fractional exponent

Perhaps I am overthinking this but $(-4)^{(-5/2)}$ is not a valid equation, am I correct? Working through the problem gives me $1/(-4^{5/2})$ which then works out to $1/\sqrt{-4^5}$ which leaves a ...
-2
votes
0answers
29 views

Conversion of exponential terms

How do I convert $$ e^{\pi} + e^{2\pi} + e^{3\pi} +1 $$ into $$\frac{e^{4\pi} -1}{e^{\pi} -1}$$ I have no idea how to proceed further.
1
vote
0answers
34 views

Simplify $e^{x \cdot \log{y}}$ where $x, y \in R^N$

I'm looking to simplify the following expression (or to determine if it's even possible). Given two vectors $x, y \in R^N$, simplify $e^{x \cdot \log{y}}$. I found it in some m-code for an infinite ...
0
votes
2answers
64 views

Why is Euler's formula a definition?

Even though there are proofs for Euler's formula for complex exponentials (see wikipedia for instance), it is mentioned as a "definition" in most textbooks. Why is that? My understanding is that a ...
0
votes
0answers
35 views

Definition of an exponential polynomial

From Wikipedia: For formal exponential polynomials over a field $K$ we proceed as follows. Let $W$ be a finitely generated $Z$-submodule of $K$ and consider finite sums of the form ...
3
votes
3answers
92 views

Why does $e^{-x}$ approach $0$ as $x$ gets large? [on hold]

Why is it that $$\lim_{x \to -\infty} e^x = 0?$$ Context: College has started back up again and I like to understand the reasons why things do the things they do, rather than just memorizing. I'm ...
4
votes
2answers
51 views

Limiting value of $\frac{x^n e^x}{n!}$ as $n\to\infty$

For the Taylor Series the remainder is of the form $$R_n = \frac{(x-a)^n}{n!} f^{(n)}(\xi) $$ with $a \leq \xi \leq x$ For the series of $e^x$ about $0$ (that is, the Maclaurin series) the remainder ...
4
votes
5answers
90 views

Solve: $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$

It is asked to solve the ODE $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$ for $x(0)=10, \; x'(0)=0$ It is equivalent to the first order system in two variables $$\begin{bmatrix} x' \\ y' \end{bmatrix} = ...
1
vote
1answer
65 views

Matrix exponential of $\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$

It is asked to evaluate the matrix exponential of $$A=\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$$ It is not hard to do this, since this matrix have 3 ...
0
votes
0answers
34 views

Solving an integral that includes an exponential function and the error function

This question contains all the values needed to compute an equation. My question is, do you get the same result I get? Or do you get the result in the paper I've linked to? I'm trying to decipher ...
0
votes
2answers
43 views

Sketch the graph $x=e^{-t}\sin t$,$ t\ge 0$

My graph is always negative though, and that doesn't make sense cause $t$ is supposed to be positive. I substituted $x$ as $y$ and $t$ as $x$.
0
votes
0answers
41 views

A question on function [on hold]

So,I took this question from my friend assignment which he got from his teacher,I had no idea about the question as it use natural log in function.The question goes: Given $g(y) = \ln y$ and $f(y) = ...
0
votes
0answers
26 views

Is this function commonly known or has some name?

I have used this function for fitting in my research, and I wonder if there is a name for it, or is it commonly known in some reduced form? $f(x)=\alpha\frac{e^{-\gamma x}}{x^\beta}+\delta$ Actual ...
1
vote
2answers
41 views

How to approximate $x^y$ using a quadratic function

I need to build an algorithm that finds the approximately $x^y$ where $x = [0, 1]$ and $y = [0, 0.4)$. This is for a computer algorithm (the standard library is too slow). I thought about making a ...
17
votes
1answer
417 views
+50

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
0
votes
0answers
36 views

Why does $\frac{x^n}{n^x}$ stop growing at the approximate value of $\pi (n)$?

I noticed while playing around with these functions that $n^x$ will start slow and then speed up really fast in its growth rate. While $x^n$ grows more slowly, but faster than $n^x$ at the start. ...
1
vote
1answer
61 views

Exponential of 4x4 matrix

It is asked to calculate $e^{tA}$, where $$A=\begin{pmatrix} 0&1 & 0&0 \\ 3\omega ^2&0 &0 &2 \omega \\ 0& 0 & 0 &1 \\ 0& -2 \omega &0 ...
-3
votes
0answers
26 views

clarification domain and range of functions [closed]

Want a clarification on the domains and ranges of addition, subtraction, multiplication, division and enhancement of functions, if possible with an example with roots and absolute value for a ...
-1
votes
0answers
97 views

Finding the value of $(x+y)$ when $x^x+y^y=31$

As the title suggests...the question is to find the value of $(x+y)$ when $x^x+y^y=31$.Is it possible to solve this question without trial-and-error method when only this much information given.Using ...
2
votes
6answers
74 views

Derivation for the derivative of $a^{t}$ from The Equation

In Calculus, the Equation is known as: $$f'(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ This equation allow us to find the derivatives of functions. Let's try this with the exponential ...
0
votes
3answers
24 views

Comparison of two values

I have to figure out the relation between the quantity $(0.9/1.1)^2 +(1.1/0.9)^2$ and 2. How can i do this without explicitly calculating the first value, by using some laws of exponents?
0
votes
0answers
6 views

How to test if an number generated from an exponentially distributed random variables is statistically different from zero at some confidence level?

For instance, let the generated number be 0.3 and the mean of the exponential distribution where this number comes from be 0.03. At a significance level of 0.10 or 0.05, is it 0.3 statistically ...
2
votes
1answer
39 views

Function of single variable $f(x)$, $f(x+y)=f(xy)$ and the exponential.

For a function of a single variable $f(x)$ which has the property that $f(x+y)=f(x)f(y)$ we first set $x=y=0$ and then develop an ODE to show that $f(x)=\exp\{-\beta x\}$. I do not understand how ...
1
vote
2answers
115 views

Solving $x^{2n} = \frac{1}{2^n}$ for $x$

What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation $$x^{2n} = \frac{1}{2^n}$$ for ...
2
votes
2answers
39 views

One integral involving integrals exponential and logarithmic function

Is there a closed-form solution for the integral $$ \int_{0}^{\infty}\log_{2}(1+ax)\cdot e^{-bx} \; \mathrm dx $$ with $a, b \geq 0$? If there is no closed-form solution, whether there is an ...
-1
votes
1answer
22 views

Math - Exponential Function (Help) [closed]

I'm getting lost with all the working out and my results don't seem to fit the criteria. I need to 'Simplify, and express in terms of positive indices' the expression $$\frac {\sqrt{81x^3} - ...
-2
votes
1answer
38 views

Simplifying an expression with exponents [on hold]

I've found this exercise in my work book and I'm scratching my head trying to figure this thing out, a step by step guide and answer would be really helpful! :-) Simplify, and express in terms of ...
0
votes
0answers
25 views

An incorrect proof that the Lie algebra matrix exponential is always injective. What's wrong?

Suppose we have two square complex matrices $X,Y $ in the lie algebra $\mathcal{G}$ of matrix Lie group $\mathbb G$ such that $e^X = e^Y$. Then $e^{tX}$ and $e^{tY}$ define the same one-parameter ...
1
vote
1answer
52 views

Why is $\frac{1}{x} \sum_{n=1}^x \ln (n) \sim \ln(x) - \gamma$

I was playing with some functions and decided I wanted to see at which point the factorial of $x$ became bigger than $e^x$. I set them equal to each other and after doing some algebra I ended up with ...
1
vote
0answers
41 views

solve $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$

Is it possible to find the analytical solution of $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$? Is that a transcendental equation?
1
vote
2answers
32 views

Does the matrix logarithm always converge for exponential matrices?

We have defined functions on square matrices $X$: $$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$ and $$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$ The exponential converges for ...
1
vote
1answer
47 views

Is the following solvable for x?

I have the following equation and I was wondering if I can solve for x given that it appears both as an exponent and a base: $[\frac{1}{\sqrt {2\pi}.S}.e^{-\frac{(x-M)^2}{2S^2}}-0.5\frac{1}{\sqrt ...
1
vote
2answers
28 views

Function that fulfils the equation

Prove that for all $x\in\mathbb{R}$ there exists only one $y=y(x)$ that fulfils the equation: $$3x+e^x=y+e^y$$ I am completely lost with that. What should i do?
2
votes
0answers
38 views

Solving $-1=e^a-2e^{av}$ as part of a equation system

Problem Given $f_2(x)=e^{ax-b}+c$ with $x \in \left(0,1\right)$, I am trying to calculate the parameters $a,b,c$ in respect to the following constraints: $$ \begin{align} f_2(0) &= 0 \\ ...
0
votes
0answers
29 views

How to prove $x^m = O(e^x)$ for any $m \gt 0$?

My attempt: It's true for $m = 1$ clearly. Now assume true for $m=1\dots M-1$. Then $x = O(e^x)$ and $x^{M-1} = O(e^{M-1})$. Lemma: if $f = O(g)$ and $f' = O(g')$ then $ff' = O(gg')$. Proof: $f = ...
1
vote
1answer
19 views

The Limit of an Integral Containing Exponentials

I am unsure how to show this. Suppose $\delta(s)$ defined on $(-\infty , s_*)$ is increasing and satisfies $\lim _{s\rightarrow s_*} \delta = \lim _{s\rightarrow s_*} \frac{d \delta}{d s} = \infty$ ...
4
votes
1answer
77 views

How to show that $|\exp(z)-1|\le2|z|$ for $|z|\le 1$

How to show that $|\exp(z)-1|\le2|z|$ for $|z|\le 1$ ...
0
votes
0answers
20 views

solve implicit equation with lambertw, exponentials, logarithms and first order polynom

I have a very complicated problem to solve. I am almost sure it's impossible to solve but maybe one of you guys has a miracle solution for me. I am modelling the behaviour of a photovoltaic cell and ...
1
vote
3answers
54 views

Exponential integral representation

According to exponential integral eqn. (8) $\; E_{1}(x) \;$ can be represented by: $$ E_1(x)= - \gamma - \ln(x) - \sum _{n=1}^{\infty } \frac{(-1)^n x^n}{n n!} $$ where $\gamma$ is the ...
0
votes
1answer
47 views

Evaluating the integral $\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt$

It is stated that (for $\lambda>0$) $$\frac{1}{\pi}\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt = 1-\frac{|\mu|}{\lambda}$$ for $ 0\leq|\mu|\leq\lambda$, and zero ...
0
votes
1answer
40 views

Expotential Growth/Decay - Problem Deriving Atmospheric Pressure Formula

I have a problem deriving the following formula: $$\frac{dP}{dh} = k\left(\frac{P}{T}\right)$$ Using the following 'rule': If $\ \dfrac{dA}{dt} = kA\,$ then $\,A = A_0\left(e^{\,kt}\right)\,$ ...
1
vote
1answer
47 views

Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.

We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known ...
0
votes
2answers
22 views

Trying to understand an expansion/limit from geometric sum to exponentials, what kind of rule is at play?

Can someone help me understand what's going on here? This is for a problem involving moment generating functions, which is related to statistics and probability, but I figured it was more of a math ...
2
votes
0answers
50 views

Does every even degree polynomial in the expansion of $\exp x$ have no real roots? [duplicate]

Does every even degree polynomial in the expansion of $\exp x$ have no real roots? I tried the first few values and it seems like it... Is this a known result?
2
votes
0answers
84 views

Tough integral with exp

Can anybody integrate this: $$ \int_0^K e^{i(A\sqrt{\mathstrut k^2+m^2} - Bk)} dk,$$ where $K$, $A$, $B$ and $m$ are real constants? Sorry folks, I didn't realise anybody would be interested in the ...
4
votes
2answers
152 views

Is $e^x=\exp(x)$ and why?

In the comments to this question a discussion came up wether we have $e^x=\exp(x)$ by definition and what the "correct" definition of $\exp(x)$ is. Building on that, I want to line out the problem ...
3
votes
4answers
59 views

Using the $\ln(\cdot)$ for $(1-e^{-x})$

The given function: $$B= A(1-(e^{-x}))$$ Now, I want to 'destroy' the e-function by taking the logarithm of it. First, since $\ln(ab) = \ln(a) + \ln(b)$ we get that $\ln(b) = \ln(a) + ...
2
votes
2answers
62 views

What is the fallacy of this proof?

I recently was working with square roots and came across this- $({\sqrt -1})$$=-1^\frac12$$=-1^\frac24$$=(-1^2)^\frac14$$=1^\frac14$$=1$ I understand that this is not true,but despite repeated ...
0
votes
0answers
26 views

How to solve the integral in this case?

If I have a Kernel from the path integral technique, and I want check if the product rule is valid in 2D, (I know it is) then I need to solve the following integral: \begin{equation} \text{Let} \quad ...