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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12 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.
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1answer
15 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
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1answer
53 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 $e^{t^2}$ ...
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1answer
14 views

Differentiation of Rodrigues' formula

Iam trying to differentiate Rodrigues' formula m times with respect to x. to attain the form Any help ?
11
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1answer
147 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
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3answers
81 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
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1answer
155 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
76 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 ...
6
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0answers
75 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
57 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, ...
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0answers
16 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
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0answers
49 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.
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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 ...
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1answer
36 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
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5answers
104 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 $f(0)=1$ ...
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2answers
34 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
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2answers
59 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
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2answers
75 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
360 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
53 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
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1answer
64 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 ...
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0answers
22 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
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3answers
102 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
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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
24 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 ...
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0answers
21 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
41 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
44 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
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1answer
49 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
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2answers
66 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 ...
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3answers
60 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 ...
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0answers
16 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
68 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 + ...
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0answers
36 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 $\cos$, $\sinh$, $\sin$, $\cosh$ ...
3
votes
4answers
59 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
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0answers
47 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 ...
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1answer
47 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
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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
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1answer
34 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 ...
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3answers
48 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
80 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
51 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
28 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
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0answers
79 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
67 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
53 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 ...
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4answers
68 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
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1answer
29 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 ...