For questions on elementary functions, functions of one variable built from a finite number of exponentials, logarithms, constants, and $n$th roots through composition and combinations using the four elementary operations $(+, –, ×, ÷)$.

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2
votes
1answer
33 views

Set builder notation with $\land$?

Is it possible to rewrite set builder notation with conjunction $\land$? For example, $$y\in f(A)=\{f(x) \mid x\in A\} \\ ​​\iff \exists\,y, y=f(x)\land x\in A$$
0
votes
4answers
22 views

A surjection from three countably infinite subsets of the natural numbers?

I am trying to construct such a surjection. More specifically, given $f: \mathbb{N} \rightarrow A$ $g: \mathbb{N} \rightarrow B$ $h: \mathbb{N} \rightarrow C$ as surjections, where $A,B,C \subset ...
0
votes
2answers
17 views

How to show that: $\log_a (x^{a}-x)-\log_a \Big(\dfrac{x^{a}-x}{a}\Big)=1$, where $a$ and $x$ are positive integers.

I was studying Fermat's Little Theorem and Logarithm to see if there is any interesting result or correlation exist between the two. So I came up with this equation. I know few basic logarithmic ...
1
vote
1answer
41 views

Prove $n < 2^n$ for all $n \geq 0$ using induction.

Please verify for me? Base case: $n = 0 $ $0 < 2^0$ $0 < 1$. This is true. Inductive step: Suppose $n \geq 0$. Assume $P(k)$ is true if $k = n$. We must deduce that $P$ holds for $k+1$. $n ...
0
votes
1answer
17 views

How do i calculate a general solution for when its better to increase the base number or the % increase in this game i am playing

So essentialy i have a base number which starts at 1 and a percent multiplier that increases this base number. And i can spend 1 skill point to increase either the base number by 1 or the percent ...
1
vote
1answer
21 views

Prove: the cardinality of the set (A-B) is less than or equal to the cardinality of A

Hi this is my first question so please bear with me. My question is this. If A and B are sets, is $ \#(A-B) \leq \#(A) $ True? I drew some Venn diagrams and intuitively this seems to be true, $ ...
0
votes
1answer
34 views

Elementary Functions, Differentiation, Integration [duplicate]

Why is it that differentiation of a function that is a composition of elementary functions (such as $\sin \:2^x$ or $\ln(\mathrm{arcsec}\: x^3)$ or $x^{1/x}$) always produces a composition of ...
0
votes
3answers
46 views

Any smoother version of the exponential function?

Often one needs to express some quantity of interest in a scale other than its original one. One can use the exponential function to map $(-\infty,0)\to(0,1)$ and $(0,+\infty)\to(1,+\infty)$, but ...
1
vote
4answers
61 views

Under certain conditions $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}\Rightarrow \{a,b,c\}=\{a',b',c'\}$

Let $a,b,c,a',b',c'\in \mathbb{Z}_{\geq 1}$ be such that $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1,\quad \frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}<1. $$ Suppose $$ ...
4
votes
1answer
26 views

Methods to find $f$, given the functions $f \circ g$ and $g$

There is one way, which is to use the fact that $ \ f(g(g^{-1}(x)))=f(x)$. But this method only works if $g$ has a right inverse. There are other heuristic methods, which is to "guess the shape" of $ ...
0
votes
1answer
21 views

Finding equivalence classes in a set of functions with a condition

I can't seem to work around this problem... Let $n$ and $m$ be two positive integers. $F$ is the set of functions from {1,..,n} to {1,...,m}. We define the relation $R$ as: $f R g$ if and only if ...
14
votes
1answer
116 views

Are there some techniques which can be used to show that a sum “does not have a closed form”?

I am aware that there are some techniques which can be used to show that some function does not have an antiderivative expressible using elementary functions, such as Liouville's theorem. (More ...
0
votes
1answer
31 views

My brain doesn't work right now: What's the formula for the $n$th vertex of a discretized sine wave?

So far I have: $$ A \sin(2\pi f ? + \phi) $$ where $f$ is cycles per second, and $\phi$ is in seconds. If I'd like to approximate the sine wave with $N$ points per cycle, and I want to draw $C$ ...
4
votes
1answer
64 views

Can a change of variable result in the evaluation of an integral in terms of elementary functions, whereas before the c.o.v. this was not possible?

Suppose we have an integral $$I=\int f(x)dx,$$ for some function $f$ which can not be evaluated in terms of elementary functions, e.g. in terms of exponentials, a finite number of logs, and algebraic ...
1
vote
0answers
35 views

Undecidable problems involving elementary functions

I am reading the article "Some undecidable problems involving elementary functions of a rial variable" by Daniel Richardson and have some problems with understanding Lemma Three : Let $h(w)=w\sin w, ...
1
vote
2answers
33 views

Elementary Set Theory: Surjective Functions

I have some trouble trying to proof the following: Let $U: \Omega \rightarrow A$ and $V: \Omega \rightarrow B$ be surjective functions. Suppose that we have the condition that for all $\omega, ...
1
vote
2answers
41 views

Show that $f(x)=\sin(x)$ is continuous at 0 using an assumption

Problem: Assume that $\lim_{x\to0} \frac{\sin(x)}{x}=1$ Show that the function $f(x)=\sin(x)$ is continuous at 0. Attempt: My intuition is that since the $\lim_{x\to0} \frac{\sin(x)}{x}=1$, then ...
5
votes
3answers
63 views

Is $\sin(x)$ =$-\sin(180^o+x)$?

I figured out that $\sin(x)$ should equal $-\sin(180+x)$ like in this picture But when I type on Wolfram $$\sin(a\mathrm{deg})=-\sin(180+a \mathrm{deg})$$ it says it's false. Why? I've tested it ...
2
votes
3answers
54 views

Number of real roots of the equation $2\cos(x) =(2^x+2^{-x})/2$ [closed]

What's the number of real roots of the equation $2\cos(x) =(2^x+2^{-x})/2$? If any questions ask about real root what is the main thing to check first, and most important? Any particular way to ...
0
votes
2answers
43 views

Partial fraction in two variable problem

How to write partial fraction of $$\frac{12m-n-3mn+7}{5m-2n-2mn+5}$$ I just write first and second denominator: $5-2n$ and $m+1$.
0
votes
1answer
56 views

when $\int x^m (a+bx^n)^p dx$ is elementary function

In one of the answers of the question Integration of sqrt Sin x dx, I saw something similiar to that: $m,n,p \neq 0 \in \mathbb{Q}$ $\int x^m (a+bx^n)^p dx$ is elementary function $\implies$ ...
5
votes
6answers
108 views

Find solutions to $|x|<x$

Find the solution set of $$|x|<x.$$ I know that the solution set is $\emptyset$. But I am stuck in the case when $x<0$. Shall I intersect the condition with the obtained result? I mean the ...
1
vote
1answer
20 views

is $G(x) = |x-2|$ one to one OR onto? [closed]

Please help me determine if $G(x) = |x-2|$ is one to one or onto.
1
vote
1answer
45 views

How to make functions of three cylinders?

Three-cylinder with height $4$ $m$ and radii of the base $5,3,1$ $m$ are going to put (in this order). Give an explicit formula for the following functions, you examine the functions on continuity and ...
38
votes
10answers
1k views

What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the ...
2
votes
1answer
34 views

An equality of maxima

Let $0<x_1<x_2<\ldots<x_N<1$ be $N$ points of $[0,1]$. I was reading something in which it says that the following two expressions are clearly equal: $$\max_{1\leq i\leq ...
0
votes
1answer
32 views

Power series for non elementary functions

Since the function $f(x)=e^{-x^2}$ cannot be integrated using elementary functions, how could one find a power series for $F$, where $F$ is an elementary function such that $F'(x)=e^{-x^2}$?
0
votes
1answer
39 views

Is $f([a]_{mn}) = ([a]_m,[a]_n)$ a bijection?

Given, $f : Z/mnZ → Z/mZ × Z/nZ$, is $f([a]_{mn}) = ([a]_m,[a]_n)$ a bijection? I have already done the work to prove that this function is well-defined. Can I say that this is bijective though, ...
1
vote
1answer
41 views

How to write this summation as a function of $x$?

I have a function $$Z(x) = \sum_{n=1}^x \frac{1}{\log \left( \frac{n}{x+1} \right)}$$ Can this summation be written as an elementary function? The first few values I have computed are $Z(1) = ...
4
votes
1answer
135 views

Is there a proof that the Harmonic numbers are not an elementary function? [duplicate]

The Harmonic numbers $H_x = \sum_{n=1}^x 1/n$ are the sum of the reciprocals of the natural numbers up to a given number. The first few are $0, 1, 3/2, 11/6, \ldots$. $H_x$ can be defined for ...
7
votes
0answers
290 views

Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids using complex numbers in the proof. Existence of complex roots will ...
0
votes
1answer
34 views

What is the partial sum?

Let $0<\alpha<1, a_n= \frac{n!}{\alpha(\alpha+1)(\alpha+2)..(\alpha+n-1)}$. Is it possible to write the partial sum $\sum_{k=0}^{n}a_k$ in a compact form? Thanks in advance.
1
vote
2answers
95 views

How to find the range of $1 / (1+x^2)^{1/2}$?

How to find the range of $$\frac{1}{\sqrt{1+x^2}}$$? Ok. I've revised the (easy theory). I would like to complete the exercise finding the derivative of f(x) and setting equal to zero. I do it ...
0
votes
2answers
43 views

How to graph elementary functions?

Could you point me out some clear and extensive sources in this regard, please? I haven't found an interesting and extensive document so far. I have knowledge about parent graphs of functions and ...
8
votes
1answer
104 views

Calculating $\sqrt{-1}$

. . . mod $p$, of course, for $p$ prime and equal to 1 mod 4. For any prime $p$ which is 1 mod 4, $-1$ has a square root in $\mathbb{Z}/p\mathbb{Z}$. But it quickly gets frustrating to find the ...
-1
votes
2answers
88 views

Proving $x^{\log_2 n} = n^{\log_2 x}$ [duplicate]

How could one go about proving that $x^{\log_2 n} = n^{\log_2 x}$? I'm not really sure how to get started.
0
votes
4answers
98 views

Proving $x^{\lg n} = n^{\lg x}$ [closed]

How could one go about proving that $x^{\lg n} = n^{\lg x}$? I'm not really sure how to get started.
1
vote
0answers
31 views

Why there's no chain rule for integrals of elementary functions which are expressible in terms of elementary functions?

The derivative of every elementary function is elementary; this is owing to the existence of the chain rule for differentiation. On the other hand, the integral of an elementary function may turn ...
0
votes
0answers
49 views

Inquiry on prime counting function

One of my close friends and I have been working towards an exact prime counting function. The approach we have came up accurately produces the number of composite numbers that occur before a given ...
3
votes
1answer
39 views

Find $k \in \mathbb{N}$ such that $\frac{k^2}{(1+10^{-3})^k}$ is maximum.

The problem as in the title. What I tried so far: I investigated a function $f(x) = \frac{x^2}{(1+10^{-3})^x}$ for $x \geq 0$ So what I found out is that it's maximum is $x = \frac{2}{ln(1+10^{-3})}$ ...
0
votes
4answers
58 views

Inverting $f(x)=\frac{a^x-1}{a^x +1}$

This is the problem: $$f(x)= \frac{a^x-1}{a^x+1}, \quad a > 0, \quad a \ne 1.$$ What I can get, but I don't think it is right: $$f^{-1}(x) = \frac{-x-1}{x \ln a-\ln a}.$$ So this is what I ...
1
vote
1answer
43 views

An analogous in $\mathbb R[x]$ of the property $p=a^2+b^2$ in $\mathbb N$

It is known that all prime number $p\space (>0)$ of the form $4n+1$ is a sum of two squares of integers $a, b \ge 0$. Prove the following analogous property for positive polynomials in $\mathbb ...
6
votes
3answers
238 views

How to solve $B = x^c - (1 - x)^c$

How to solve for x ? Where we are interested in the range $0 < x < 1$ and $C \neq 0$. $$ B = x^c - (1 - x)^c.$$ The only thing I could come up with is to substitute $$ x = \sin^2y ...
6
votes
4answers
73 views

Is $\mp a$ actually different than $\pm a$?

So, the way I understand $\pm a$ as a general concept is basically as follows: $\pm a$ is really just two numbers, functions, or whatever $a$ represents, but the catch is that one of the $a$'s is ...
0
votes
0answers
13 views

Minimum value some expression

What is minimum value of $2^{\frac{(\log n)}{\big(\frac{t}2\big)^m}}{(\log n)^{2^mt}}$ where $n\in\Bbb N$? With $m=1$ it is at $t=c\log\log n$ where $c\in\Bbb R$ is fixed.
0
votes
1answer
23 views

Simple rearrangement

I am not good at identifying simple things like the fact that this: $\frac{n(n+1)+2(n+1)}2$ can be rearranged to $\frac{(n+1)(n+2)}2$ But why and how? Sorry for this completely basic question but ...
3
votes
1answer
99 views

Solving linear differential equations

Find the general solution for the following equation: $$\frac{dy}{dt}+2ty=\sin(t)e^{-t^2}$$ Find a solution for which $y(0)=0$ First I found the integrating factor which is $e^{t^2}$ ...
0
votes
1answer
96 views

Differentiation of Rodrigues' formula

Iam trying to differentiate Rodrigues' formula m times with respect to x. to attain the form Any help ?
12
votes
1answer
207 views

Proof that the factorial is nonelementary

Is there a proof that the factorial function $!:\mathbb N\to\mathbb N$ is nonelementary? If it were equal to an elementary function (call it $P(n)$), then it would extend the factorial function to ...
3
votes
3answers
89 views

Simple confusion in complex analysis

I've been learning Complex Analysis from George Cain's website: https://people.math.gatech.edu/~cain/winter99/complex.html In chapter 3, Elementary Functions, it claims that the complex logarithm ...