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 $(+, –, ×, ÷)$.

learn more… | top users | synonyms

0
votes
2answers
27 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
55 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 $a \in \mathbb{Z} \geq 2$ $$a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$$ ...
0
votes
0answers
44 views

Reduction of trigonometric functions to $x$ power [closed]

$$\huge{(\sqrt{1 - \sin^2x})^{2^{x^\sqrt{1 - \sin^2x}}}}$$ $x > 0$ if the domain of $x$ is between $1$ and $1.5$
7
votes
2answers
70 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 $x$?
0
votes
0answers
18 views

Project 5E Part c - Clean Up The Great Lakes in an Elementary Differential Equations

I am working on Project 5E - Clean Up The Great Lakes in an Elementary Differential Equations course and have done pages of math to get to this equation and I don't know if I am just tired or what but ...
4
votes
2answers
337 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
41 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
49 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
21 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
100 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
46 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
23 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 ...
1
vote
0answers
19 views

Bounding a function by its arguments

Does anyone know any sufficient conditions (and necessary would be great too if possible) on $f$ such that the following is satisfied: $|f(x)-f(y)| \leq C|x-y|$, where $C$ is a constant, $f(x): ...
0
votes
1answer
36 views

Find functions with ''smart'' tangents.

This is a didactic question. Given a differentiable function $y=f(x) \;, x,y \in \mathbb{R}$, I want to construct an exercise in which we have to find a straight line that passes through a point ...
3
votes
2answers
43 views

Proof by induction that $P_n(a) \neq 0$ for $n>3$.

Let $a,b,c$ be 3 non-zero coprime integers and $P_n(a)=a^n+\sum_{k=1}^{n}{{n\choose{k}}a^{n-k}(c^k-b^k)}$ Show that if $P_3(a) \neq 0$ then for all $n \geq 3, P_n(a)\neq 0$ Using mathematical ...
2
votes
1answer
47 views

Is this expression bounded?

I wonder: is $$ \left( 1 + \frac{n}{a} \right)^{-a} \prod_{k = 1}^n \left( 1 + \frac{a}{k} \right) $$ uniformly bounded in $n \in \mathbb{N}$ and $0 < a \leq n$? Following Jack's answer, I have ...
0
votes
2answers
62 views

Help to factorize $(x+y)^n-(x-y)^n$. [duplicate]

let $x,y,n>0$. I have been struggling to factorize $(x+y)^n-(x-y)^n$. I have found a product of $y$ and a double summation. Each time I expand it for small integers, my result is wrong. Please ...
1
vote
3answers
58 views

What's the domain of $(\log_{\frac{1}{2}}{x})^x$?

What's the domain of $$(\log_{\frac{1}{2}}{x})^x$$ I know that for defining $\log_{a}{x}$, $x$ must be grater than zero ($x\gt 0$) and ($a\gt$ and $a\neq 1$) but I asked this question in Quora and ...
0
votes
0answers
12 views

How to get symmetry function of a function in proportion to any other functions generally?

How can I get symmetry equation of a function in proportion to an other one? for instance we know that: $y=\log_{10}{x}$ and $y=\log_{\frac{1}{10}}{x}$ are symmetric in proportion of $y=0$. now, I ...
4
votes
3answers
65 views

How to prove this floor function equation?

How can I prove the following equation? $$ \lfloor nx \rfloor = \lfloor x \rfloor + \Big\lfloor x + \frac{1}{n} \Big\rfloor + \Big\lfloor x + \frac{2}{n} \Big\rfloor + \Big\lfloor x + ...
0
votes
0answers
34 views

Elementary form of $\arg(\Gamma(\alpha+i\beta))$?

It is claimed that $$\arg(\Gamma(\alpha+i\beta))=\lambda(\alpha,\beta)−\sigma(\alpha,\beta)$$ with $\lambda(\alpha,\beta)$ and $\sigma(\alpha,\beta)$ given in term of $\cos$, $\sinh$, $\sin$, $\cosh$ ...
0
votes
4answers
39 views

What is the meaning of $n\log^2(n)$

I know this seems ridiculously an obvious thing but: What does $n\log^2(n)$ mean? Does it mean $n\log (\log(n))$ or $n\log(n)^2$? I am trying to compare it with $n\log(n)$ to check which one of the ...
1
vote
0answers
28 views

Liouville–Hardy theorem: when is $\int f(x) \log(x) dx$ elementary?

I am currently writing a report on Liouville's theorems on integration in finite terms, and I am in the process of proving the Liouville–Hardy theorem. This is what I understand so far. Theorem ...
1
vote
1answer
44 views

What is the smallest fraction produced by a sum of fractions with bounded denominator?

For $x$ a sum of fractions: $$ x = \sum_{i=1}^{N}\frac{a_i}{b_i} $$ for all $a_i, b_i \in \mathbb{Z}$ with $ 0 < b_i \leq D$ and $N$ are non-zero positive integers, I know that the denominator of ...
0
votes
1answer
31 views

What are equivalent parametric equations?

What are equivalent parametric equations? Is there a fast method to prove that 2 parametric equations are non-equivalent?
0
votes
1answer
33 views

Minimum of a+b+c given an exponential equation

I am reviewing for my exam next week and in my notes we have this problem: Given x,y,z as real numbers and $4^{\sqrt{5x+9y+4z}}-68\cdot 2^{\sqrt{5x+9y+4z}}+256=0$. Find the product of the minimum and ...
0
votes
3answers
43 views

Find the values of $\sin z=0$

Hello can someone check my work and tell me what to do next. Find the values of $\sin z = 0$ we know that; $$\sin z = {{{e^{zi}} - {e^{ - zi}}} \over {2i}}$$ So; $${{{e^{zi}} - {e^{ - zi}}} ...
3
votes
2answers
79 views

How do I find the value of $\tan\bigl(\frac{\pi}{4}-3i\bigr)$ in terms of elementary functions?

The question is to compute $t=\tan \left( {{\pi \over 4} - 3i} \right)$. So I change it into $$t={{\sin ({\pi \over 4} - 3i)} \over {\cos ({\pi \over 4} - 3i)}}$$ Then $$\eqalign{t= & ...
1
vote
1answer
45 views

Finding the inverse of a non linear function

Let $F:\mathbb{R}^2\to \mathbb{R}^2$ be the diffeomorphism given by $$F(x,y)=(y+\sin x, x) $$ Find $F^{-1}$. I know that the answer is $F^{-1}(x,y)=(y,x-\sin y$), this can be shown to be true by ...
1
vote
1answer
27 views

Examples of functions with properties

I'm looking for example of functions defined on $[0,+\infty)$ with the following properties: 1) continuous, twice differentiable 2) $f(0)=0$, $\lim_{x\to+\infty} f(x)=1/3$ 3) $f^{\prime}>0$, ...
2
votes
0answers
22 views

Number of terms in $f'$ as a polynomial of the number of terms in $f$

Take an elementary function $f$. The derivative $f'$ is an elementary function as well. Let $G(f)$ be the minimum number of terms required to express $f$ as a formula comprising a finite number of ...
0
votes
0answers
73 views

Nice approximations of sums by integrals.

Let $f(x):\Bbb Z^+\rightarrow \Bbb R^+$ be a non-monotone function. If for every $m\in\Bbb N$, $$S(m) =\sum_{n=1}^N\frac{1}{(1+f(n))^m}$$ be sum of interest, then is there a way to approximate this ...
3
votes
2answers
60 views

Proving that a function is absolutely monotonic on a given interval

According to the following Wolfram Alpha article: http://mathworld.wolfram.com/AbsolutelyMonotonicFunction.html, the function $f(x) = -\ln(-x)$ is absolutely monotonic on the interval $[-1,0)$. My ...
3
votes
1answer
52 views

Asymptotic elementary expression for the antiderivative of $x^x$

It is well known that there exists no elementary function $f$ with $$\int x^x\,dx \quad = \quad f$$ Is there an elementary function $g$ such that $$\int x^x\,dx \quad \tilde{} \quad g$$ in the ...
-1
votes
4answers
66 views

How do I figure out if a 'function is odd or even'?

I am currently doing an highschool math problem and I do not know what the question is asking for when it asks 'state which functions are odd and which are even for the below'. $f(x)= x^2+1 $ ...
0
votes
1answer
28 views

Is there an intersection point for $f_1 = 100n²$ and $f_2 = 2ⁿ$?

As far as i know, there is, and it can be answered by resolving the equation $100n^2 = 2^n$. But how can it be resolved? I already tried to transform the equation using $\log 2$, but i just don't get ...
0
votes
4answers
69 views

How to produce 100 with only four 3s and mathematical symbols

An approach like this doesn't count!        $\underset{k \in \mathbb N, k \leq (3+3/3)} {\sum k^3}$ Thanks for your answers but I need something more strong... You are ...
0
votes
2answers
66 views

Help with parametric quadratic equation

We have the equation $x^2+ax+2a+1=0$ which has real roots $x_1$ and $x_2$ and a is a parameter. I need to answer to the following questions: Find all values of a for which $x_1=(a-1)x_2$. For which ...
1
vote
4answers
39 views

Simplfiying $x(5xy+2x-1)=y(5xy+2y-1)$

I want to simplify: $x(5xy+2x-1)=y(5xy+2y-1)$ to $(x-y)(5xy+something-1)=0$ but I can't figure out what to do with the $2x$ and $2y$ on both sides.
0
votes
2answers
40 views

Inverses / Bijections

Let $f:A\to B$, and $g:B\to A$ such that $$ g(f(a))=a \ ,\ \forall\ a \in A, $$ and $$ f(g(b))=b\ ,\forall\ b \in B. $$ Does this mean that $f,g$ are inverses and bijections? Bests
0
votes
0answers
35 views

Solve this Simple quadratic equation $cU^2-2(a+ b)U+2(a-b)V- cV^2=0$

I need help solving this symmetrical quadratic equation where $\gcd(a,b,c)=1$$cU^2-2(a+ b)U+2(a-b)V- cV^2=0$ Is there an easier method than the quadratic formula? Any hint?
2
votes
2answers
34 views

If $\sin \theta = \frac{3}{5}$, what are the possible values of $\cos\theta$?

If $\sin \theta = \frac{3}{5}$, what are the possible values of $\cos\theta$ ? I do know that .. $\sin^2\theta + \cos^2\theta = 1$ .. and you can solve that equation, resulting in $\cos \theta = ...
2
votes
1answer
41 views

Finding an upper bound for this expression

Given two non-negative numbers $a$ and $b$, I'm trying to found an upper bound for $-\sqrt{a} + \sqrt{a+b}$ But I'd like the bound not to be dependable on any square roots and have only one term. ...
4
votes
5answers
47 views

Tool for expressing $x=f^{-1}(y)$ if $y=f(x)$ is given

I have many equations of nature - $y=ax^{12}+bx^5+5x^4+1$ For all these equations, I need to express x in terms of y. What tool or software would you recommend for this? I would much prefer to ...
4
votes
0answers
56 views

Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
0
votes
2answers
51 views

Asymptotic approximation of the arctangent?

That is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x)) = ...
1
vote
0answers
43 views

Are there any functions which were proposed as elementary by mathematicians but not considered elementary now?

Are there any functions which were proposed by various mathematicians to be included in the set of elementary functions because of their properties but not considered elementary as of now?
0
votes
0answers
58 views

Monotonically decreasing function for multiplication product?

I have a set of numbers $S = [100,999]$ for which I want the maximum product $p$ such that $p = a \times b$ for all $a,b \in S$ also fulfilling some condition $C$. I would like $p$ to be monotonically ...
3
votes
1answer
524 views

Need to prove that the equation has only 1 solution.

I have been trying to solve the following equation: $5^x+7^x=12^x.$ Obviously, x=1 is a solution but how do I prove that there are no other solutions.
3
votes
2answers
81 views

prove a sequence without knowing its convergence

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent? IF given convergence, I know that its limit is $6$, but the question ...