Questions tagged [elementary-functions]

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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31 views

primitive roots of unity to solve equations over $\mathbb Z_p$

So, I have been thinking of the equation $x^n-1 \equiv 0 \in \mathbb{Z}_p $, $p$ prime. So, I noticed something weird and I wonder if there is a theory for that. Let $\omega_n $ be the nth primitive ...
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4answers
166 views

Are there any elementary functions $\beta(x)$ that follows this integral $\int_{y-1}^{y} \beta(x) dx =\cos(y)$

Are there any simple functions $\beta(x)$ that follows this integral $$\int_{y-1}^{y} \beta(x) dx =\cos(y)$$ I think there is an infinite amount of solutions that are continuous everywhere but how can ...
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46 views

Simplify $\mu(x)=\frac{a_x}{b_x}$ where $a_x+b_x=a_{x+1}$ & $\sqrt{a_n^2-b_n^2}+1=b_{n+1}$

can you simplify this function $\mu(x)$ into elementary functions where $\mu(x)=\frac{a_x}{b_x}$ where $a_{x-1}+b_{x-1}=a_x$ & $\sqrt{a_{x-1}^2-b_{n-1}^2}+1=b_n$ $$a_1=b_1=1$$ $$\mu(1)=1,\mu(2)=2,\...
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1answer
50 views

is there a function $\gamma(x)$ where when $a$ & $b$ and $a+1$ & $b+1$ are co-prime, $\gamma(\frac{a}{b})>\gamma(\frac{a+1}{b+1})$

is there a function $\gamma(x)$ where when $a$ & $b$ and $a+1$ & $b+1$ are co-prime, $\gamma(\frac{a}{b})>\gamma(\frac{a+1}{b+1})$ when you start with $\gamma(\frac{1}{2})$ you get an ...
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206 views

Can you write $1-x-\frac{x^2}{2!}+\frac{x^3}{3!}-\dots$ with elementary functions

Can you write $$1-x-\frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}-\frac{x^7}{7!}-\dots$$ with elementary functions, where the function is related to the Thue-Morse ...
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26 views

A conjecture about the solvability of rational equations of transcendental functions by elementary numbers

Is my conjecture below mathematically and linguistically correctly formulated and well formulated? How can the conjecture be improved and shortened/simplified and made more intelligible? The ...
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1answer
71 views

Why are $\sin,\cos,\tan$ continuous

I'm done with two courses in Analysis, but just can't seem to work out how I'll show the base trigonometric functions to be continuous. Any references or indications for a simple, preferably ...
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1answer
21 views

On an equation that involves the number-of-divisors function and the formula for a sequence of figurate numbers

In this post we try to relate a sequence from The On-Line Encyclopedia of Integer Sequences and a sequence that solves an equation that involves the number-of-divisors function $\sigma_0(n)=\sum_{1\...
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How to shift and compress a parameterized log curve

Suppose I have some nice way to create a log curve that I need for a certain task (here in Python). ...
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85 views

Basic polynomial questions

Find a rational polynomial such that $$P(n)=1\cdot 2+ 2\cdot 3+\cdots + n\cdot(n+1).$$ for all positive integers $n$ (edited). Does there exist an integer polynomial of this form? Ive found that $P(X)...
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1answer
29 views

Existence of an algebraic function on a disc, in an elementary way

The meromorphic functions on an open disc $\Delta$ in $\mathbf{C}$ form a field $M(\Delta)$. How to show in as elementary a way as possible that for every polynomial $P(X) = X^n + a_1 X^{n-1} +\dots + ...
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1answer
693 views

Is there an elementary expression for every real sequence?

By elementary expression for the sequence $\{a_n\}_{n=0}^\infty$, I mean an elementary function $f : X \to \mathbb C$, where $\mathbb N \subset X \subset \mathbb R$, such that $f(n)=a_n$ for all $n$. ...
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2answers
387 views

Is there a smooth, preferably analytic function that grows faster than any function in the sequence $e^x, e^{e^x}, e^{e^{e^x}}…$

Is there a smooth, preferably analytic function that grows faster than any function in the sequence $e^x, e^{e^x}, e^{e^{e^x}}$? Note: Here the answer is NOT required to be an elementary function, as ...
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1answer
41 views

Elementary Relations/Functions and the Solvability of their Inverses

Background I've been interested lately in the idea of solving and inverting equations, and a question came to mind. Feel free to correct my notation, I could benefit from cleaner structuring. Going ...
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1answer
33 views

Are the Bernoulli numbers $B_{2n}$ given by an elementary function of $n$?

I found an old question asking for a proof that the factorial function is nonelementary, and the Claim 2 section of the answer there (by Vincenzo Oliva) doesn't quite make sense to me: https://math....
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1answer
18 views

general approach to comment on bijection of these functions whose graph can't be drawn

consider the following class of functions defined as: If $f : \Bbb R \to \Bbb R$ be the function such that $$ f(x)=x|x|-4 :x \in \Bbb Q$$ $$ f(x)=x|x|-\sqrt{3} :x \notin \Bbb Q$$ then $f(x)$ is ...
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14 views

Differential of the sine function

I know this is very basic, but it's a doubt I have since a couple years. What is the differential of a function like $w = q_1 \sin(\frac{x}{L})$ ? Is it the following? \begin{equation} \delta w = ...
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1answer
25 views

Constructing $\sqrt{\sqrt{1+x^2}-1}$ to be smooth (cancelling the square)

$\DeclareMathOperator{\sign}{sign}$ Is there a way to rewrite $f(x)=\sign(x)\sqrt{\sqrt{1+x^2}-1}$ using (smooth) elementary functions? As far as I can see the function seems infinitely ...
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1answer
48 views

Open Problems to do with Polynomials and/or Elementary Function Theory

I was wondering what are some open problems (even if deemed impossible) in basic function theory (stuff you'd learn in high school) and/or open problems to do with polynomials... Thank you in ...
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1answer
24 views

Proving that $\exists x\in (-1,1),\frac{a}{x^3+2x^2-1}+\frac{b}{x^3+x-2}=0$

I'm trying to prove that there exists $x\in (-1,1)$ $$\frac{a}{x^3+2x^2-1}+\frac{b}{x^3+x-2}=0$$ For any $a,b \in \mathbb{R}^+$. I know I have to use the Intermediate Value Theorem here, but I'm not ...
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1answer
60 views

Given an integer of the form: $2^j*3^k$ - is there a way to quickly find j and k? [closed]

Suppose you have a number that you know is of the form: $2^j*3^k$ (j,k are positive integers). Is there a way to quickly (ie constant,linear time) find what j and k are? Basically is there a faster ...
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1answer
26 views

Ito integral definition of a elementary process

I'm a stochastic calculus beginner. In every ito's integral presentation there is the following definition of a simple process: $$H(t,ω)=∑_{i=1}^{n-1}h_{i}(ω)1_{(t_{i−1},t_{i}]}(t)$$ Then, the ...
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4answers
47 views

How to show that $\{x|x= \lfloor x\rfloor \}= \mathbb Z$ ? ( On the floor function).

While trying to answer this question Find the domain and range of the function : $f(x) = \sqrt{\lfloor x\rfloor -x}$, I found myself in the necessity of asserting that : if a real number is of the ...
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2answers
69 views

Find the domain and range of the function : $f(x) = \sqrt{\lfloor x\rfloor -x}$

Find the domain and range of the function : $f(x) = \sqrt{\lfloor x \rfloor -x}$ where $\lfloor x \rfloor $ is the floor function (greatest integer function) This is how I did it : $f(x) = \sqrt{\...
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25 views

Online Math resource

I am a new math teacher and currently looking for some online programs to use with elementary grades. Any suggestions?
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3answers
30 views

Polynomial coefficients of $f(x) = \frac{1}{100} e^{5x} - \frac{1}{4}$

Can anybody tell me what the polynomial coefficients are for this function? $f(x) = \frac{1}{100} e^{5x} - \frac{1}{4}$ I'm just trying to get the function coefficient vector and then use "roots" ...
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1answer
55 views

Graphing a given function

How to graph a function like $f(x)=e^{x^2}+\cos x$ using pen and paper? I mean without using any graphing calculators. What should be the approach for finding whether the function is one one or many ...
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2answers
242 views

Symmetric functions written in terms of the elementary symmetric polynomials.

[A recent post reminded me of this.] How can we fill in the blanks here: For any _____ function $f(x,y,z)$ of three variables that is symmetric in the three variables, there is a _____ function $...
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24 views

Is there a formula to determine the domain and the range of a function composition with $n$ components? How to define domain and range here?

I was thinking about this recent question : what is the range of $f(x)= \frac {1} {\sqrt{x^2 - 1}}$. I tried to express this function as a function composition , that is, with $i(x)=\frac 1x$ $s(...
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34 views

Inequalities involving elementary functions

Let $k\gt0$ and consider the real sequence $(u_n)_{n\geqslant1}$. Moreover, let \begin{equation} T_k u_n :=\begin{cases} u_n &\hbox{ if } \vert u_n\vert\leqslant k\\[5pt] k\frac{u_n}{\vert u_n\...
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1answer
29 views

Are there cases where the extremum ( or local extremum) of a function can be found algebraically ( without calculus)?

Suppose I know that the function $f(x)=x²$ has a minimum value on $\mathbb R$. Could I determine this munimum value algebraically ( without using calculus). If I set $x² \geq m$, I only get this : ...
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1answer
43 views

What is the inverse of $ f(x)= \sqrt{\log_{3} (x)}$. Why is $g(x)= 3^{x^2}$ not properly $f$'s inverse?

$ f(x)= \sqrt{\log_{3} (x)}$. Suppose $g(x)= f^{-1} (x) = I $. In that case $(f\circ g)(x)) = f(I) = x$ ( I use I - for "inverse" - to denote the expression in $x$ defining function $g$). ...
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1answer
33 views

Can a rational function ( with linear num. & denom.) always be expressed as a composition of elementary ( invertible) functions?

I would like to apply the following rule in order to find the inverse function of a given function : if function $f(x)= h(i(j(x)))$ and if functions $h(x)$, $i(x)$ and $j(x)$ are invertible, ...
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1answer
22 views

What would be the approach for determining whether these function is one to one or onto both or neither?

1) f3: Z × Z → Z × Z. f3(x, y) = (x + 1, 2y) 2) f4: Z+ × Z+ → Z+. f4(x, y) = 2x + y − 1 I understand the conditions for a function to be onto and one-to-one but in this case I'm having trouble ...
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1answer
66 views

Maximal of $\cos(a)$ [closed]

Suppose that $\cos(a+b)=\cos(a)+\cos(b)$. Q What is the maximal value of $\cos(a)$ for all $(a,b)\in\{(a,b)\in\mathbb R^2|~\cos(a+b)=\cos(a)+\cos(b)\}.$
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3answers
44 views

Can I square both sides while calculating the Range of a function

Let $f(x)=$ $\dfrac{1}{\sqrt{x-5}}$ is a given function. We have to find its range. I have tried two approaches:- $\sqrt{x-5}>0$ ⇒ $\dfrac{1}{\sqrt{x-5}}>0$ ⇒ $y>0$ ⇒ Range = $(0,∞)$ $f(x)=...
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1answer
66 views

Prove that $g''(x) = -g(x)$

Suppose $f$ is a differentiable function in $\mathbb{R}$ and $f''(x) = −f(x), \forall x \in \mathbb{R}$. Set $$g(x) = f(x) − f(0)C(x) − f'(0)S(x)$$ Prove that $$g''(x) = −g(x),\quad\forall x \in ...
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2answers
52 views

Find the number of roots of $f(x)$ in the interval $0\leq x\leq 2\pi$

If $$f(x)=a_{0}+a_{1}\cos(x)+a_{2}\cos(2x)+\cdots+a_{n}\cos(nx)$$ where $a_{0}, a_{1},\ldots,a_{n}$ are non-zero real numbers and $$a_{n}>\lvert a_{0}\rvert+\lvert a_{1}\rvert+\cdots \lvert a_{n-1}\...
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1answer
47 views

Estimates on growth of $^{n}3$

I was dealing with a problem on tetration and am supposed to explain why this problem was challenging to me- obviously, difficulties stemmed from the amazing growth of $^{n}3$. The question now is: Is ...
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0answers
17 views

Integer-Coefficient Polynomial that returns an integer for all inputs of the form $\sqrt{x} - \sqrt[3]{x}$

I recently saw a problem that asked you to define an integer-coefficient polynomial that returned an integer for all inputs of the form $1-\sqrt[3]{x}$, where $x$ is an integer The way I solved it is ...
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1answer
33 views

What would be the most efficient method to programatically implement symbolic integration?

At the moment, I'm a CS student who's main research focus is graphics. For a current project of mine, I need to write an indefinite integral calculator, in the vein of those available of wolfram alpha ...
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1answer
37 views

Domain and preimage of a relation/function

The preimage of a function $f:X \rightarrow Y$ is defined as $Img^{-1} f = \{ x \in X : f(x) \in Y \} $, which is equivalent to $\{ x \in X : \exists y \in Y : f(x) = y \}$. This last definition is ...
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1answer
47 views

Find all $p\in\mathbb R[x]$ s.t. $p' \left(x^2\right)\cdot p'''(x)=3\cdot p''\left(x^2\right)\cdot x^2$

Find all $p\in\mathbb R[x]$ s.t. $$p'\left(x^2\right)\cdot p'''(x)=3\cdot p''\left(x^2\right)\cdot x^2.$$ Attempt: $$p(x)=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$$ $$p'(x)=4a_4x^3+3a_3x^2+2a_2x+a_1$$ $$p''(x)...
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0answers
72 views

Inequality of modulus of quadratic polymnomials in $\mathbb{C}$

I am trying to work out some stability conditions for ODE methods and during the computations one needs to solve the following inequality: Let $\alpha_1, \alpha_2, \beta_1, \beta_2 \in \mathbb{R}$. ...
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2answers
120 views

Does $f(x) = x - \tanh(x)$ have an inverse function that can be expressed in terms of elementary functions?

I find this question relevant in my current study of the tractrix, namely because this expression appears in one parameterization of the curve. I’ve noticed that the plot of the Cartesian equation of ...
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1answer
29 views

Reference request: Source for “Cauchy's Theorem” (?) on integration in elementary functions

Buried deep in my notes from a course I took many years ago, I find a reference to the following, which (in my notes) is called "Cauchy's Theorem": Theorem. The integral $\int x^p (1-x)^q dx $ can ...
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0answers
19 views

Coprimality of a rational term?

Let $P_0,P_1,P_2,P_3\in\overline{\mathbb{Q}}$, $P_3\neq 0$, and $p(x),q(x)\in\overline{\mathbb{Q}}[x]$ so that $p(x)$ and $q(x)$ are coprime over $\overline{\mathbb{Q}}$. I have the term $${\frac{...
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1answer
24 views

Find a function with certain property

I am searching for an example of the function with the following property. For given function $f : \mathbb{R}\rightarrow\mathbb{R}$, both $e^{x}f(x)$ and $e^{-f(x)}$ are monotonically decreasing. I ...
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6answers
125 views

Find number of real roots of the polynomial $x^3+7x^2+6x+5$.

I want to find the number of real roots of the polynomial $x^3+7x^2+6x+5$. Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude ...
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0answers
34 views

mupliplication of cosine of $\pi/2^k$

Let $x,y\in \{\frac{2\pi i}{2^m}\}_{0\leq i\leq 2^m-1 }$, where $m$ is a positive integer . Q If we have that $\cos A=\cos x\cdot \cos y $, can we say that $A=\pi k$ for some $k\in \mathbb Q$. For $...

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