# 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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### Variants of geometric sum formula

I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$ and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$. (See https://en.wikipedia.org/wiki/Geometric_series#Sum) From Sum of ...
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### Is there a spelling mistake or am I missing something

Here, $[ \cdot]$ is $\lfloor \cdot \rfloor$ floor function. N $\in N$. Where did $\frac{[Nx]} N + \frac{1}{2N}$ came from and how does $x$ differs by $\frac{1}{2N}$. Shouldn't it be $\frac{1}N$ if ...
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### Is there an "elementary irrational number" without a certain digit in its decimal presentation?

In this question, I define an "elementary irrational number" as an irrational number which is built up of a finite combination of integers, field operations (addition, multiplication, ...
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### College Algebra Function Transformations Unique?

Let $f(x)=x^2$ be the parent function for the transformation $g(x)=(2x+12)^2 -1$. Follow the two points $(0,0)$ and $(2,4)$ from the parent function through the transformation. Let's simplify $g(x)$ ...
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### Are all multivariate elementary functions $C^\infty$ on any open ball for which they are defined?

Are all multivariate elementary functions $C^\infty$ on any open ball for which they are defined? I believe they are, and can find no counterexample, but do not know how to prove this. (All the ...
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### How to determine the monotonicity of this function

Given the definition of the function: $$f(x) = e^x - e^{-x} + \frac{2x}{(x^2+1)^2} , x∈(-∞，0]$$ What's the monotonicity of this function in the given range? I tried to calculate the derivation to ...
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### Book Recommendations for Elementary functions

I have been through school. I have had a treatment of (Pre)Calculus (I,II,III). I still feel like I don't know enough Trigonometry, or enough of Exponentials, or other functions. Even through my ...
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### Intuition about a simple equality involving min and max of some naturals

I am playing for fun with some (natural) numbers and noticing the following: For any $h = 0, ... , \max(a, b)$: $$\min(a, h) - \min(b, h) = \max(0, h - b) - \max(0, h - a)$$ What is this called? Is ...
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### Dirichlet series of an elementary function

Is there an example of an elementary function (different from Dirichlet polynomials, i.e. cutoff Dirichlet series) which has a know Dirichlet expansion (known coefficients)? I am aware of the ...
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### Non-elementary antiderivative of rational function: $f(x)= \frac{1}{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}$

Is there any function of the form $$f(x)= \frac{1}{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}$$ for $n\in\mathbb{N},a_i\in\mathbb{R}$, that has no elementary antiderivative? If there is none, is there a ...
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### Inequalities for the solution of $x = (x-a) e^{x+a}$.

Let $a > 0$. The equation $$(x-a) \, e^{x+a} = x$$ must be solved for $x > 0$. Since the solution does not have a closed form, I would like to obtain bounds for the solution. Until now, I was ...
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### Integrals of (Jacobi) elliptic functions

Jacobi elliptic functions create a new trigonometry that we don't teach in high schools. Taking the derivative of composite functions of these functions with elementary functions is easy thanks to ...
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### The rational function $\tan\left(n\arctan x\right)$

While messing around with the formula $$\arctan x+\arctan y=\arctan\frac{x+y}{1-xy},\tag1$$ I decided to iteratively find expressions for $n\arctan x$ by applying the formula $(1)$ $n$ times. I was ...
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### Can we always know whether an infinite series converges or diverges, if we are summing over an elementary expression?

Just like the question stated, can we always determine the convergence of an infinite sum of an elementary function? For example, given some elementary function f(n) can we determine if the sum will ...
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### Solving the equation of the form $a - \frac{1}{x}\ln{(\frac{x}{1-x})} + \frac{1}{x^2}$ [closed]

I got an equation of the form $a - \frac{1}{x}\ln{\frac{x}{1-x}} + \frac{1}{x^2} = 0$. $a$ is a constant and the values of $x \in [0,1]$. Does a closed-form solution exist for this? If not, is there ...
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### Can $\sin(\frac{n\pi}{m})$ with $n,m \in \mathbb{Z}$ always be represented using only algebraic functions?

Can $\sin(\frac{n\pi}{m})$ with $n,m \in \mathbb{Z}$ always be represented using only algebraic functions? In other words can $sin$ of a rational multiple of $\pi$ always be represented using only ...
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### Are there basic techniques for proving that no elementary antiderivative exists? [duplicate]

Background: It seems there is an algorithm called the full Risch algorithm that can decide whether a given function has an antiderivative that can be expressed in terms of elementary functions. There ...
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### How to approach this tricky log problem

Consider the following equation: $$(7 + 4\sqrt{3})^{t^2 - 5t + 5} + (7 - 4\sqrt{3})^{t^2 - 5t + 5} = 14$$ I have tried and exhausted all the methods I know of and resorted to brute force to solve this ...
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### Coefficients of Chebyshev polynomials

Not long ago, I derived the formula for Chebyshev polynomials $$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$ How to extract the ...
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### Prove that $e^{-ax} \leq (1 - x)^a + ax^2/2$ for $a > 1$, $0 \le x \le 1$

Here we assume that all variables are reals. I would like to show that $$e^{-ax} \leq (1 - x)^{a} + \frac{1}{2} ax^2 \tag{1}$$ for all $a > 1$, $0 \leq x \leq 1$. By Taylor’s theorem, I know ...
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### if $f(x) = |3x - 1|$ so what is the sum of all $x$ such that $f(f(x)) = x$?

Question: if $f(x) = |3x - 1|$ so the sum of all values of x that satisfies $f(f(x)) = x$ is? alternatives: a) $11/10$ b) $21/20$ c) $27/20$ d) $23/20$ e) $5/4$ My try: I've thought that I had ...
2 votes
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