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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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37 views

Prove: $f$ is injective if and only if for any non-empty set $C$ and for all functions $g, h : C \to A$ such that $f ◦ g = f ◦ h$ we have $g = h$.

I’m attempting to prove this exercise for my elementary analysis class, however, I am having difficulty deconstructing the iff statement. Any help in understanding how the implications are working ...
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2answers
74 views

Are Elementary Functions chosen by convention?

Are the elementary functions chosen by convention? What leads me to this question is that many non-elementary functions have power series representations, allowing them to be computed numerically, ...
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3answers
48 views

$Log(z^2)$ analytic for all of the complex plane except origin.

The question was show $\ln{x^2 + y^2}$ is harmonic in two ways. It was very easy to show by LaPlace's equation, but next I have to show it by showing it is the real part of an analytic function. I am ...
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1answer
28 views

How do I show $\arctan x$ is real analytic on $\mathbb{R}$?

How do I show $\arctan x$ is real analytic on $\mathbb{R}$? First of all we have $$ \arctan x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} \, \forall\, x\in[-1,1] $$ because $$ \arctan x = \int_0^...
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0answers
25 views

Between $\ell^p$ and $\ell^\infty$ balls?

Take the $\ell^p$ ball in $\mathbb R^d$ of radius $r$ centered at the origin is $$\{x \in \mathbb{R}^d : \sum_{i=1}^d |x_i|^p \le r^p\}.$$ Enclose it by the smallest square possible which is the $\...
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1answer
29 views

What is the name of this convex figure?

The figure is given by $x^4+y^4\leq\frac1{2^4}$ https://www.wolframalpha.com/input/?i=x%5E4+%2B+y%5E4+%3C1%2F16. In general is there a name for the figure $x^{2t}+y^{2t}\leq\frac1{2^{2t}}$ or ...
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4answers
105 views

Should we distinguish the minus sign from the negative sign?

In the set $\mathbb{C}$ of complex numbers, the minus sign "-" may be used for following: As a unary operator $-_u$, given a complex number $a$, $-_ua$ is the unique number (called the negative of a) ...
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1answer
37 views

Which kinds of compositions of invertible elementary and nonelementary functions are elementary?

Let $f$ be a bijective elementary function, elementary invertible or not. Let $h$ be a bijective nonelementary function, elementary invertible or not. Which of the compositions $h(f(x))$ and $f(h(x))$ ...
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0answers
12 views

Which kinds of simply composited elementary functions have elementary antiderivatives?

The elementary functions are defined in differential algebra. That are the functions $X\in\mathbb{C}\to Y\in\mathbb{C}$ that are composed of $\exp$, $\ln$ and/or unary or multiary univalued algebraic ...
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0answers
17 views

Algebraic dependence of the transcendental elementary standard functions?

The transcendental elementary standard functions are the trigonometric functions, the hyperbolic functions, the arcus functions and the area functions. Which of the transcendental elementary standard ...
6
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1answer
180 views

Can all trigonometric expressions be written in terms of sine and cosine?

I know that sine and cosine can be rewritten in terms of the real and complex parts of the exponential function as a result of Euler's formula. My question is, can every trigonometric expression be ...
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1answer
80 views

Homeomorphism between the unit disc and the unit square

I know that the function that describes the homeomorphism is: $f(x,y) = \begin{cases} \frac{x^{2}+y^{2}}{\max(|x|,|y|)}(x,y) &\quad\text{if } (x,y)\ne (0,0) \\ \text{} (0,0) &\...
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1answer
25 views

Elementary schools summation in denominator?

after being lazy for a long time and being away from any fraction and equations, I am confused with a seriously ridiculous math problem, and I want to confirm my answer: the equation is pretty simple ...
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0answers
39 views

is there a special name for a function that generates a binary tree

If you graph t vs i for n = 7*11^i+1 (= 2^k*j + 2^t - 1) with ...
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3answers
90 views

Non zero solution of $3x\cos(x) + (-3 + x^2)\sin(x)=0$

How can I find exact non-zero solution of $3x\cos(x) + (-3 + x^2)\sin(x)=0$. Simple analysis and the below plot show that the equation has an infinite number of non-zero solutions.
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1answer
121 views

Smooth Elementary Function that Outgrows All Tower Functions?

This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer. Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth. Is ...
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2answers
54 views

Solution for $x = a^x$

We know $e^x = x$ has no real solution, and $1^x = x$ has a solution $x=1$. For what value $1 < a < e$, $a^x = x$ has a real solution? (and no $b>a$ yields a solution.) I do not have any ...
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2answers
42 views

Prove that $\sqrt{(p-1)^2 p^2 + 4} = (p - 1)p + \epsilon$ for $p > 1$ and $0 < \epsilon < 1$.

Here $p$ is prime but that does not affect the calculation. Assume that $p > 1$ is any such integer. Numerically at $p = 2$ we have $\epsilon = 0.8...$ which is the maximum value of $\epsilon$.
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1answer
23 views

Is this pair of equations impossible to solve for x? $y_1 = x_2 - v^{\pm 1}e^{-x_1}$, or equivalently $(x - y)c^{\exp(-x)} = z$

Original: I'm trying to solve the following for $x_1$ and $x_2$, $$ y_1 = x_2 - v\, e^{-x_1} $$ $$ y_2 = x_1 - \frac{1}{v}\, e^{-x_2} $$ in terms of $y_1$, $y_2$, and $v$, which are known and real, ...
8
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0answers
98 views

integrate $F(x)$: NO complex analysis, NO multivariable calculus

Suppose I have an elementary function $F(x)$ for which $\int_{-\infty}^\infty F(x) \, \text{d}x $ has an elementary value. Here 'elementary value' means anything generated by $0,1,+,-,\div,\times,\exp,...
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1answer
23 views

Set of functions satisfying a given condition

What is the set of functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $(f(x+y))^2 = f(x^2) + f(y^2)$ for all $x, y, \in \mathbb{Z}$. I've computed that $f(z) = 0$ and $f(z) = 1$ are the only ...
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0answers
10 views

Performing additional calculation of the result of a function

Say I want to perform a division on result of a function and I want that to be a part of the function. How can I do it, here is an example: $f(n)=\begin{cases} n+1 & n\equiv 1\pmod2 \\ 7n & n\...
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2answers
114 views

“Every elementary function is differentiable.”

Edwards and Larson (Calculus, 2018) claim that: You can differentiate any elementary function. It seems though that this claim is false. How then can we modify the above claim so that it becomes ...
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4answers
82 views

The absolute value function $|\cdot|$ is elementary, but not differentiable?

As usual, define the absolute value function $|\cdot|:\mathbb R \rightarrow \mathbb R$ by $$|x| = \left\{ \begin{array}{ll} x & \text{for } x \geq 0,\\ -x & \text{for } x < 0.\\...
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3answers
94 views

Representing integer as product of $2^n$ and an odd number

I have a non-zero integer $x \in \mathbb{N}$, and I want to represent it as $$ x = 2^N \cdot Q $$ where $N,Q \in \mathbb{N_0}$, and Q is an odd number. That means, $x$ is separated into an part that ...
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0answers
31 views

Terminology of function; $f(n)=(4^n-2) / 7$

Let $n\in\mathbb{N}$, and I define the function: $$f(n) = \frac{4^n - 2}{7}$$ Is this $f(n)$ called a quartic function? Is $f(n)$ a polynomial function of degree four? Can a polynomial be restricted ...
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2answers
127 views

Proving that the absolute value function is elementary.

My proof is: The absolute value function f is defined as x when $x\geq 0 $, hence it is a polynomail hence elementary function and it is defined as -x when $x < 0$ hence it is also a polynomial ...
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5answers
65 views

Why isn't $y=a(x^2-Sx+P)$ same with $x^2-Sx+P$

If we have roots of the function $y=ax^2+bx+c$ we can calculate $S=\frac{-b}{a}$ and also $P=\frac{c}{a}$ . Then we know that we can form the function this way: $$x^2-Sx+P$$ So on the other side we ...
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0answers
18 views

Finding factors s.t. f(x) > g(x)

I am looking forward for an answer to the following question: I have a function $f(x) = 1 - e^{-(x / \lambda)^k}$ . I want to find $\lambda'$ and $k'$ in $g(x)$ such that $g(x) < f(x) \in \mathbb{...
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3answers
59 views

Prove $\int_0^1\frac{1}{x}dx=\infty$ [closed]

How do I prove the following? $$\int_0^1\frac{1}{x}dx=\infty $$
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1answer
57 views

What does “$\forall$” mean?

I am studying cryptography and in the first lesson I saw "$\forall$" element in the following formula: $\forall x\neq x_0:P(X) = 0$ What does $\forall$ mean?
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0answers
37 views

Did I derive a function for finding the angle of a complex number?

When presented with a complex # written in rectangular form (x + j*y), w/ the goal of converting it to polar form, the following 2 relationships are used: r = √ ( x^2 + y^2 ) θ = tan-1(y/x) The ...
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1answer
38 views

Proving rules for real numbers hold for complex numbers

I'd like to prove that $\frac{e^{z_1}}{e^{z_2}}=e^{z_1-z_2}$. Obviously this is true for real numbers, but here, $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$, so it needs to be proven. $$\frac{e^{z_1}}{e^{...
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0answers
52 views

Criteria for monotonicity of one-variable functions

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is called monotone if $f$ satisfies one of the following conditions $$ x_1,x_2\in\mathbb{R}, x_1<x_2\quad\Longrightarrow\quad f(x_1)\leq f(x_2) \quad (...
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2answers
116 views

Are elementary compositions of nonelementary functions also nonelementary?

Say we have a nonelementary function $F(x)$ on the real numbers. Let $E_1,E_2,\ldots,E_n$ be a sequence of finite elementary functions on the reals. Is it always true that $$ R(x)=(E_1\circ E_2\circ \...
4
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2answers
118 views

What's the inverse function of $ f(x) = 0.02(x-1)^3 + 130(x-1)^{1.5} + 130(x - 1) $?

The equation is graphed here: https://www.desmos.com/calculator/kgvnud77dg I've come up with this equation as part of designing a game. This equation is used to map the user level to their cumulative ...
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2answers
82 views

Can I switch the order of taking minimums?

Suppose I have some function $F(x,y):\mathbb{R}^2 \rightarrow \mathbb{R}$. Is it always the case that $$ \min_{1 \leq x \leq N} (\min_{1 \leq y \leq M} |F(x,y)|) = \min_{1 \leq y \leq M} (\min_{1 \...
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3answers
53 views

Finding the inverse of a (challenging) function

I just started reading Differential Topology by Guillemin and Pollak and one of the very first exercises is the following: Let $B_a$ be the open ball $\{x : \vert x \vert ^2 < a \}$ in $\...
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0answers
31 views

Zeroes of a function with logarithms and peculiar symmetry

I encountered a strange function $F \colon ]-1/2,1/2 \;[ \; \to \mathbb{R}$, defined on the open three-dimensional unit cube centered on the origin and taking real values. The function has some ...
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1answer
61 views

Add more weight to components of equation?

I'm not sure how to ask this question, so please bear with me. I'm not even sure if it's possible with just this information. This is the only information that we received. I'm using this equation ...
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3answers
478 views

Convert any positive number to decimal $x$ such that $0\lt x \lt 1$.

This is a very basic question, I understand that. So I have these numbers: 20, 0.75, 1.25. If you sort them, it would look like this: 20 1.25 0.75 I need to ...
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0answers
83 views

Proving $\int x^x dx$ is not elementary

Please verify if my reasoning is correct. Write $x^x$ as $e^{x\ln x}$. Then let $f(x)=1$ and $g(x)=x\ln x$. $R'(x) + g'(x)R(x) = f(x)$ will be $R'(x) + (\ln x+1)R(x) = 1$, which solution is $R(x) = ...
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1answer
45 views

Derivatives and functions and maxima

Let $$f(x)=\left\{% \begin{array}{ll} x^3-x^2+10x-5,& x\leq 1 \\ -2x+\log_2 (b^2-2),& x>1 \\ \end{array}% \right.$$ Find all possible real values of $b$ such that $f(x)$ has the ...
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1answer
649 views

Are ceiling and floor elementary functions?

According to the Wikipedia entry on elementary functions, the trigonometric functions and their inverses are elementary functions. It doesn't seem to me that the floor and ceiling functions should be ...
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1answer
74 views

Continuity and differentiability of elementary functions

Given a single-variable elementary function (not piecewise), I was wondering if it is continuous and differentiable in all of its maximum domain? (Not considering the trivial examples involving ...
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1answer
22 views

Multiplication Simplification in a Composition Function

I am trying to work out $(\Pi \circ f)(L)$. The functions are defined: $ \Pi(y) = -y^4 + 6y^2 - 5 $, and $ f(L) = 5L^\frac23$ I understand the first simplification to: $-(5L^\frac23)^4 + 6(5L^\...
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0answers
34 views

Is the modulo operation an elementary function?

Since mod operation can be expressed using atan and cot, can it be considered as an elementary function? https://en.wikipedia.org/wiki/Elementary_function http://functions.wolfram.com/IntegerFunctions/...
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1answer
52 views

Is finding proof of convergence of a function towards a cycle or a constant any different?

Since I am not educated enough in the field of mathematical analysis of integer sequences, I am wondering if there is a difference between a sequence converging towards zero and a sequence entering an ...
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0answers
80 views

A Collatz function based on parity $(-1)^n$

The Collatz Conjecture asks wether the iterates of the Collatz function: $$f(n)=\begin{cases} n/2 & n\equiv 0\pmod2 \\ (3n+1) & n\equiv 1\pmod2 \end{cases}$$ will always reach $1$ or the $4, 2,...