Questions tagged [elementary-functions]

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

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2
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1answer
49 views

Is it decidable whether the value of a given definite integral has a closed-form expression?

There are many elementary functions such as $e^{-x^2}$ which don't have an elementary antiderivative, but a definite integral of the same integrand has a closed-form value, e.g. $$\int_{-\infty}^\...
-2
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1answer
130 views

Can we derive Euler's Formula $e^{ix}=\cos x +i\sin x$ in an elementary way? [closed]

If we generalize de Moivre's formula $(\cos x+i\sin x)^n=\cos nx+i\sin nx$ to non-integer powers, we can get $(\cos 1+i\sin 1)^x=\cos x+i\sin x$. But I have no idea to find a real number $e$ matching $...
1
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2answers
74 views

Basic: $x^2 - 1$ and $-x^2 + 1$ have the same roots and therefore both could be written as $(x+1)(x-1).$ I guess that’s not right…

I know they aren’t the same function but what’s the issue here? What am I not taking into account? Sorry for such a basic question, it just confuses me.
3
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0answers
33 views

Which ODEs have solutions in terms of elementary functions?

I'm wondering what the most general results are on determining which ODEs have solutions which are expressible in terms of elementary functions and which do not. Is there some kind of ODE equivalent ...
1
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0answers
40 views

Surjective Functions. Beginner in discrete Mathematics

I am solving a problem related to surjective functions and I suspect that the question is wrong. The Question is as follows: Let f : A → B and g : B → A be functions. Then prove that: If f and g are ...
13
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1answer
280 views

An elementary function with asymptotic $f'(x)\sim2f(2x)$ for $x\to0^+$

We want to find an elementary function $f(x)$ that is smooth and strictly increasing on some interval $x\in(0,\epsilon)$, satisfying $\lim\limits_{\,x\to0^+}f(x)=0$, whose asymptotic for $x\to0^+$ is $...
1
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0answers
48 views

Given that 𝑓 ◦ 𝑓 is a bijection, prove that 𝑓 itself must be a bijection [duplicate]

Does this proof work to show that 𝑓 is a bijection if the composite 𝑓 ◦ 𝑓 is a bijection? We are given that 𝑓 is a function from A → A. Injective: Let x, y ∈ A. Assume 𝑓(x) = 𝑓(y). By the ...
2
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1answer
69 views

Which functions $f$ satisfy $f(x-a-c)f(x-b)=\alpha f(x-c-\tfrac{a+b}{2})f(x-\tfrac{a+b}{2})$ for every $x$ where $a,b,c,\alpha$ are constants?

Let $a,b,c \in \mathbb R$ be real constants. I'm searching for functions $f : \mathbb R \to \mathbb C$ with the following property: For every such $a,b,c$ there exists a constant $\alpha = \alpha(a,b,...
2
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1answer
118 views

Need a function that measure the proximity of a given value to a target value

Like the title says, I'm looking for a function that take a given value, and returns a value between 0 and 1 which measures how much the given value is near to a target value. Ideally, the returned ...
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0answers
78 views

Is there a bounded increasing elementary function with unbounded derivative?

Is there an elementary function function $f(x)$ with the following properties: $f(x)$ is defined and infinitely differentiable on all of $\mathbb{R}$. $f(x)$ is bounded. $f^\prime(x)$ is nonnegative ...
1
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3answers
97 views

Constructing Very “Flat” Functions

Consider the function $f(x)=\tan(\sin(x))-\sin(\tan(x))$. It has (what I think is) a remarkable property. The Taylor series expansion of $f(x)$ around $x=0$ is $$\frac{1}{30}x^7+\frac{29}{756}x^9+\...
2
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0answers
82 views

Elementary function producing primes

I know there is no known elementary function $f:\mathbb{R}\to\mathbb{R}$ such that $f(n)$ is the $n$-th prime number, but is there a proof for the non-existance of one? How about one which only takes ...
1
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1answer
35 views

Which kinds of rational functions of one variable have an inverse relation that contains a branch that is a rational function?

Let's consider the rational functions whose numerator and denominator of the function term are coprime. Which kinds of rational functions of one variable have an inverse relation that contains a ...
4
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2answers
78 views

Proof of the derivative of an elementary function is also elementary

I heard of the result that the derivative of an elementary function is also elementary long ago. Now I want to prove it rigorously. I found this answer(I didn't comment because it was an old post):...
0
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0answers
21 views

sum of pairwise products

I dont understand why $\left(\sum\limits_{1 \le i \le k} f_i\right)^2$ defined as sum of pairwise products In the proove of formula i readed that $(f_1 + f_2 + f_3)\cdot(f_1 + f_2 + f_3) = f_1(f_1 + ...
1
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0answers
31 views

Beyond the second derivative in the study of a function [duplicate]

I was wondering about the analysis of derivatives in the study of a function. We know that $f'(x) = 0$ can give us the critical points $x_k$ (maximum, minimum and sometimes inflection points), not ...
0
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0answers
27 views

Inequality with exponent from (0,1)

Let $x,y>0$, $a\in(0,1)$. Is true the following inequality $(x+y)^a\le x^a+y^a$?
1
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1answer
29 views

Determining if functions are “Completely Additive”

Let $F([x,x_2,...x_n],...[y,y_2,...y_n])=[z,z_2,....z_n]$ denote a function with $2$ parameters, each is a vector containing $n$ variables. Let $U_k$ be defined by the following recurrence relation: $...
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2answers
76 views

How can I prove that $\frac{a}{2} + \frac{x}{a} \geq \sqrt{2x}$ [closed]

How can I prove the inequality $$ \frac{a}{2} + \frac{x}{a} \geq \sqrt{2x} $$ for positive $a$ and $x$. It seems simple, but I don't see a way to prove it.
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2answers
102 views

Finding the Degree of a Polynomial, as a function of the Polynomial

Recently, I was wondering about the following question: Given a polynomial $P(x)$ with real coefficients, express its degree $d$ as a function of only $P(x)$, i.e. $d(P(x))$. Only elementary ...
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2answers
93 views

Can a sum of an elementary and non-elementary definite integral be elementary or evaluated without special functions?

Hi I'm asking this in context of this: Suppose we have non-elementary integral $\chi(x) = \int K(x)dx$ and we want to find $$\int^b_aK(x)dx$$ Now lets say there is some function $F(x)$ which has ...
4
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2answers
110 views

Does a function from the empty set to the empty set exist?

I know that every function from the empty set to any other set is the empty function. I also know that there is no function to the empty set from any other set. Now, what if both the domain and ...
0
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3answers
89 views

Polynomial Long Division:

My question essentially boils down to this: (it was part of a question about polynomial ring ideals) Find an integer $b$ such that the rational function \begin{equation} \frac{x^5-bx}{x^2-2x}\in \...
2
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1answer
44 views

Elementary function $f:\mathbb{R}\to \mathbb{R}$ so that $f(0)=0, f(n)=2^n$ for $n\in \mathbb{N}$

Is there any elementary function $f:\mathbb{R}\to \mathbb{R}$ such that $f(0)=0$ and $f(n)=2^n$ for every positive interger $n$? By elementary functions, I mean functions that are sum, product, ...
0
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1answer
38 views

Solve for $x$: $a_1\ln(1+b_1x)+a_2\ln(1+b_2x)=0$, maybe with non-elementary function?

Given: $a_i>0$, $b_1>0$, $b_2<0$ Solve for: $a_1\ln(1+b_1x)+a_2\ln(1+b_2x)=0$ My try: The equation is equivalent to $(1+b_1x)^{a_1}=(1+b_2x)^{-a_2}$, Which seems to be solved with only non-...
0
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1answer
32 views

Prove that there's a unique integer n where : $ 2n^2 − 3n − 2 = 0$.

I have to prove that there's a unique integer n where : $2n^2 − 3n − 2 = 0$. I factorised and got $(2n+1)(n-2) = 0$ I can then find both $n$ solutions : $-1/2$ and $2$. We can therefore see there's ...
0
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1answer
101 views

Solving exponential equation with negative base

I have to solve the equation: $(x-2)^{x^2-6x+1}\leq 1 $. If $x-2>1 $ i.e. $x> 3 $ we have $3< x \leq 3+2\sqrt{2}$. If $0< x-2<1 $ i.e. $ 2< x<3$ we get no solutions. If $x -2=1$ ...
0
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2answers
48 views

How to show $f(x) \leq (1-t)f(x)+tg(x) \leq g(x)$

Given $f(x) < g(x)$ and $f(x), g(x), t \in[0,1]$, how can I show $f(x) \leq (1-t)f(x)+tg(x) \leq g(x)$? It is easy to show $f(x) \leq (1-t)f(x)+tg(x)$. Indeed, $(1-t)f(x)+tg(x) = f(x)-tf(x)+tg(x) = ...
0
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0answers
61 views

Numerator and denominator of an FX pair

Lets say we have the following currency pair: EUR/USD @ 1.17 : No need to mention that it is read as the following : 1€ = 1.17$. Mathematically speaking is it correct to say that EUR is the numerator [...
0
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2answers
63 views

Let $f \colon A \rightarrow B$ be a function and $S$ a subset of $A$.

Let $f \colon A \rightarrow B$ be a function and $S$ a subset of $A$. Are the following containments always valid?. $(1) f^{-1} (f(S))⊆ S$ $(2) S⊆f^{-1} (f(S))$ Attempt $S ⊆ A$ $f^{-1}(S) = \{a ...
1
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0answers
47 views

Is there a good closed form for $\Re\left(\psi\left(\frac{1}{6} \left(3+i \sqrt{3}\right)\right)\right)$?

I wonder whether $\Re\left(\psi\left(\frac{1}{6} \left(3+i \sqrt{3}\right)\right)\right)$ can be expressed as an elementary function? For instance, the imaginary part is $\frac\pi2\tanh\left(\frac\pi{...
0
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0answers
24 views

Is the relation $fRg$ iff $\exists k > 0 \mid f(x) + k < g(x + k)$ on real functions transitive?

Is the relation on the set of real functions $fRg$ iff exists $k > 0$ such that for all $x \in \mathbb{R}$, $f(x) + k < g(x + k)$ transitive? I have proved that it's not reflexive, not symmetric,...
0
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0answers
19 views

When is the continuous extension of an elementary function elementary?

Some continuous extensions of elementary functions are elementary. For example $f(x) = \frac{x}{x}$ is elementary in $\mathbb{R}\setminus\{0\}$ and its continuous extension to $\mathbb{R}$ is $g(x)=1$ ...
2
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1answer
22 views

System of modular equation

Given the following graph I gotta find the values of $r$ and $s$, with $r,s\in\mathbb{Z}$ such that they satisfy $$f(x)=|x-r|+|x-s|$$ I know the solutions are $2$ and $4$ but I don't know how to ...
4
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1answer
107 views

What is the currently accepted “correct” definition of a “transcendental function”?

Caveat: this question has already been asked on this site more than once, but the question has not been addressed completely. The question I want to ask is: there are two common definitions of a "...
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1answer
48 views

Show that $ 2^{ab}+1=(2^{a}+1)(2^{ab-b}-2^{ab-2b}+2^{ab-3b}+…+1) $ where b is an odd number and( a, b) are natural numbers

Show that : $$ 2^{ab}+1=(2^{a}+1)(2^{ab-b}-2^{ab-2b}+2^{ab-3b}+...+1) $$ where b is an odd number and( a, b) are natrual numbers. So far I’ve tried to use a similar identity as that used to prove the ...
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0answers
44 views

Why are the elementary functions the ones that they are?

Many antiderivatives exist but cannot be expressed in terms of elementary functions. This got me thinking, well, what if we expanded the set of elementary functions? Would it work then? Let's think of ...
2
votes
1answer
94 views

Conditions for the inverse of a bijection to be “almost explicit”

The following map $$f:\left[-1,+\infty\right[\to\left[-\frac1e,+\infty\right[,\,x\mapsto x\,e^x$$is a bijection and it can be shown that its inverse (known as the Lambert function W) is representable ...
0
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1answer
95 views

Cartesian product — understanding the definition

I just started with Schaum's Outline of General Topology by Seymour Lipschutz a few days ago, and now I am working with chapter 2 about functions. Most of the material I have already seen in other ...
0
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1answer
26 views

How does one go about solving a linear maximization problem using elementary maths?

I was checking a list of elementary math problems, and one of them was of the kind "a factory produces products $A$ and $B$, the profit of each product is $P_A$ and $P_B$ (The values were given ...
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0answers
34 views

How do I understand if an indefinite integral is not solvable in terms of elementary functions? [duplicate]

Many indefinite integrals cant be solved in terms of elementary functions.e.g.-$$\int \frac{\sin(x)}{x}dx$$ But many hard looking ones are still solvable by weird substitutions and other tricks. Is ...
1
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2answers
84 views

Is it possible to express values such as $\sin^{-1}(\pi/12)$ without inverse trig functions?

Pretty self explanatory. If I had for example something like $\sin^{-1}(\pi/12)$ in an expression, is it ever possible to express that expression without inverse trig functions?
0
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0answers
45 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 ...
7
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4answers
193 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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votes
1answer
58 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 ...
6
votes
0answers
240 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 ...
0
votes
0answers
51 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 ...
0
votes
1answer
81 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 ...
1
vote
1answer
28 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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0answers
16 views

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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