0
votes
2answers
44 views

When $\ln(1+y) = y + o(y)$?

I was reading a proof which utilize the fact that: $\ln(1+y) = y + o(y)$ http://math.stackexchange.com/a/842557/160028 I'm not so sure what is the meaning of $\ln(1+y) = y + o(y)$. When is it ...
2
votes
4answers
121 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
2
votes
2answers
27 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
0
votes
0answers
27 views

Function plotting

I have a function $f(x)=\binom{N}{K} \ln(1-F(x)), x \geq 0$, where $F(x)$ is a cumulative distribution function. Then, $\ln(1-F(x))$ is negative for various values of $x$ as $F(x) \geq 0$. Also, ...
2
votes
0answers
43 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
0
votes
0answers
21 views

I need to find $n$ that $\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$

$\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$ So what I did is this: $(n+1)\ln(n+1) > 1000 \Rightarrow n>190$ When I put it back I see that $\frac{1}{192 \cdot ln(192)} \not < 10^{-4}$. ...
0
votes
2answers
289 views

Help with limit of radical function

$$\lim_{x \to \infty} \frac{\sqrt{4x^{4}+3}}{5x^2+3}$$ $$= \lim_{x \to \infty} \frac{(4x^{4}+3)^{1/2}}{5x^2+3}$$ $$= \lim_{x \to \infty} \frac{(\frac{4x^{4}}{x^{1/2}} +\frac{3}{x^{1/2}})^{1/2} ...
0
votes
1answer
90 views

The shape of a graph of a function with $n$th-roots?

Not just these type of functions: $$\sqrt[3]{x}=x^{1/3} \;\;\;\text{and} \;\;\; \sqrt[8]{x}=x^{1/8}$$ But also more complicated expressions, like expressions that have $n$th roots inside of ...
1
vote
2answers
28 views

On the definition of product of groups

What I'm asking comes from Bosch, Algebra, the first chapter on elementary group theory. 1) Let $X$ be a set and $G$ a group. Then the set $G^X$ of maps from $X$ to $G$ is a group in a natural way, ...
2
votes
1answer
158 views

The domain of fractional exponents

Take the following: $$f(x) = x^{6/4}$$ The domain of this function is all real numbers. This function can be simplified to: $$f(x) = x^{3/2}$$ The domain of this function is all real numbers ...
1
vote
1answer
47 views

Is the statement $a+b < c+ d \implies e^{-a} + e^{-b} > e^{-c} + e^{-d}$ true?

As the title says, I am trying to ascertain whether the following is true: Suppose $a,b,c,d\in \mathbb{R}^+$ are such that $a + b < c + d$, then it is also true that $e^{-a} + e^{-b} > e^{-c} + ...
1
vote
2answers
54 views

Power correct notation

Ok, I know this may sound dumb, but I am trying to understand which is the correct (most beauty) notation for the power function ${\rm pow}(f(x),n)$. This is the correct one: $[f(x)]^n$ From ...
2
votes
1answer
100 views

Name of the $(-1)^n$ function?

Does the function $f\left(n\right)=\left(-1\right)^n, n \in \mathbb{Z}$ used in a lot of mathematical formulas have a special name ? EDIT: The context of this question is that I need a name for this ...
0
votes
0answers
204 views

Approximate a complicated mystery function

Let there exist a mystery function ƒ. ƒ accepts exactly 2 arguments, A & B. As B approaches A, ƒ approaches A, at a simple exponential growth rate E. As B approaches 0, ƒ approaches the mean ...
22
votes
10answers
4k views

How to solve $x^{1/2}-x^{1/3} = 0$

How can I solve the following equation? I really can't figure out how to solve it: $x^{1/2}-x^{1/3} = 0$ Thank you.
1
vote
1answer
114 views
11
votes
4answers
710 views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
0
votes
1answer
107 views

How do I solve this exponential equation?

$$x = 2^{x-3}$$ Does there exist an analytical solution to this equation? If so, how do I find it? What if it is changed to an equality? $$x>2^{x-3}$$
36
votes
11answers
3k views

How is $e^x$ read aloud?

My current research colleague from New Castle told me that I was reading it wrong. I usually read it as e power x. How do you read aloud $e ^ x$? Is it: e raised to x e power x e powered x or e ...
1
vote
1answer
40 views

Formula to scale a series that is being bent with a root / power.

I have a reference number, Rx, and a series of numbers, Sx[], to compare to it. Let's call the output Ox[]. I am using a simple square root to accelerate the apparent difference between the reference ...
3
votes
3answers
231 views

What is the fractional part function of $e^x$?

Given a real positive number $x\in\mathbb{R^+}$. What is the function of the fractional part of $e^x$?
4
votes
1answer
66 views

Find the absolute maxmium of the function $f(x) = x \cdot {e}^{-x}$

How can I find the absolute maximum of this exponential function? $f(x) = x \cdot {e}^{-x}$ I know that the first step is to take the derivative of the function, like so: ${f}^{\prime}(x) ...
5
votes
3answers
1k views

Examples of continuous growth rates greater than exponential

I read on Wikipedia that growth rate of a function can sometimes be greater than exponential. Can you give me some examples of such functions (preferably continuous ones)? Obviously $x^x$ grows ...
1
vote
2answers
318 views

Fixed points of $e^z$

How would one find the fixed points of $e^z$, where $z$ is complex (if there are any)? I feel this problem probably has a really obvious answer, and for some reason, I'm just not getting it. Thanks.
4
votes
3answers
510 views

How to find a Newton-like approximation for that function?

I want to find the complex fixpoint $t=b^t $ for real bases $b> \eta = \exp(\exp(-1))$. added remark: I'm aware that there is a solution using branches of the Lambert-W-function, but I've no ...
2
votes
2answers
165 views

power of a sets

I need to figure out what is the power of the group of functions from R to R that for each x that is not from Q, it's f(x) belongs to {x,x+1} ...