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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0
votes
2answers
39 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
47 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
102 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
44 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 ...
36
votes
10answers
952 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
33 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
28 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
32 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) = ...
3
votes
1answer
122 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
264 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
31 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
93 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
40 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
98 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
87 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
97 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.
0
votes
0answers
26 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
43 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
38 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
56 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
40 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
235 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
70 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
19 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
94 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
61 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
188 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
86 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 ...
2
votes
1answer
205 views

Is this elementary proof of fundamental theorem of algebra correct?

There is an elementary proof which considers the polynomial $ p(z) $, $z \in \mathbb{C}$ as a function of $ (r,\theta) $ where $ z= r e^{-i \theta} $, $r\ge 0$ . There are two assumptions- Assumption ...
2
votes
2answers
82 views

Is it possible to find such a $f$?

I search a continuous function $f : [0,+\infty[ \to \mathbb{R}$ such as : $\lim \limits_{x\to +\infty} \frac{1}{x} \int \limits_{0}^{x} f(t)\mathrm{d}t=\pm \infty$ and ($\lim \limits_{x\to ...
7
votes
1answer
94 views

Why is it that the Lambert W relation cannot be expressed in terms of elementary functions?

According to this Wikipedia page, the Lambert W relation cannot be expressed in terms of elementary functions. However, it does not explain why this is the case. An elementary function is "a ...
2
votes
2answers
98 views

What do we mean by 'defining a function'?

First I will start by quote from Wikipedia about function's defining methods : There are many other ways of defining functions. Examples include piecewise definitions, induction or recursion, ...
0
votes
0answers
18 views

Judge whether a function is $\mathcal{EF}$ or $\mathcal{PRF}$

All of the functions discussed below are total number-theoretic functions. Define two functions: $$ f:\mathbb{N}\to\mathbb{N},f(n)=\lfloor n\cdot e \rfloor \\ g:\mathbb{N}\to\mathbb{N},g(n)=\lfloor ...
3
votes
0answers
50 views

Is $\sum_{k=0}^\infty \frac{1}{k!!} x^{k!}$ non-elementary?

Is $z \mapsto \sum_{k=0}^\infty \frac{1}{(k!)!} z^{k!}$ an elementary function? I designed it to be analytic but analogous to Liouville's constant, but don't know how to search for this function.
11
votes
5answers
2k views

Why do un-integrable funcitons exist?

By un-integrable I mean functions whose antiderivative can not be expressed in terms of elementary functions. I recently learnt that any differentiable function can be expanded using the Taylor ...
-1
votes
1answer
44 views

Is $\log(x)|1-x/n|^n$ bounded?

I have to find if the function $\log(x)|1-x/n|^n$ is bounded but I cannot. I have tried Bernoulli's inequality and a few others things. Any hints please? Thanks.
4
votes
5answers
105 views

Derivation of the Exponential Nature of $e^x$

Presumably, the transcendental number $e$ was first found by taking the power series solution to the (arguably most fundamental) differential equation $f'(x)=f(x)$, with the initial condition $f(0)=1$ ...
0
votes
2answers
47 views

Finding inverses of a function which maps ordered pairs of positive integers onto the positive integers.

The function $f(x,y) = \frac{(x+y-1)(x+y-2)}{2} + y $ is a bijection which maps ordered pairs of positive integers onto the positive integers. I would like to find the functions $g$ and $h$ such that ...
3
votes
2answers
61 views

Can $a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$ be an integer if $x$, $y$, and $z$ are not squares?

Let $\gcd(x,y,z)=1$.Can we find 3 non-perfect squares $x,y,z\in \mathbb{Z},$ such that $a \in \mathbb{Z} \geq 2$ $$a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$$ ...
6
votes
2answers
76 views

Solve $(2+\sqrt{3})^{x/2}+(2-\sqrt{3})^{x/2}=2^x$.

How to solve $(2+\sqrt{3})^{x/2}+(2-\sqrt{3})^{x/2}=2^x$ for $x$?
4
votes
2answers
389 views

Is the gamma function expressible as a proper integral?

Is the gamma function expressible as a proper integral of elementary functions? You're also allowed to compose it with however many elementary functions. But strictly no limits. [edit] So far the ...
3
votes
0answers
100 views

Implementing the Risch algorithm to integrate $\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}$

Following the work of Andreas Wurfl i am trying to implement the Risch algorithm on $\int{\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}dx}$ following his method for extensions that are purely logarithmic, we ...
2
votes
1answer
67 views

$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th symmetric polynomial of $n$-variable. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...
1
vote
0answers
25 views

Definition of an Elementary Number

I am currently writing my dissertation on elementary functions and elementary numbers (in the sense of ritt and liouville etc) The definition of an elementary number i am using is as follows; We say ...
2
votes
3answers
104 views

How could this be true $n=\log(e^n)$?

I am learning elementary logarithms. How could this be true $n=\log(e^n)$? I searched online and couldn't find any info on this, could anyone give me some clue?
2
votes
2answers
48 views

A simple question:

let $a,b,c,d$ be all positive integers such that $a-bc \neq 0$,and $\gcd(a,b)=1$. Under what conditions, $(a-bc)$|$(a-b^d)$? In other words, does it exist any integer $k \neq 1 $ such ...
0
votes
1answer
28 views

If $x = \operatorname{argmin}_{x \in X} \lvert Ax - y\rvert^2$ does it mean that $Ax = \operatorname{Proj}_X(Ay)$?

Suppose that $A$ is an invertible matrix and $$x = \operatorname{argmin}_{x \in X}\lvert Ax - y\rvert^2,$$ then does it mean that $Ax = \operatorname{Proj}_X(y)$ like in the definition of ...