Questions tagged [elementary-functions]

Question 1

For some functions (like $f(x)=x+e^x$), we know an inverse exists by monotonicity, but that inverse is not expressible in elementary terms. Is there a general mathematical framework or theorem (beyond ...

Question 2

I'm studying symbolic integration. Liouville's theorem (the version I've learned) states that for an elementary function $f$, if $f'=g$ for some $g$ lying in some elementary differential extension $E=...

Question 3

As I understand it, umbral calculus by default only postulates the effect of evaluation operator on umbral expressions. This means, we are free to add more axioms and relations between umbral elements ...

Question 4

In Mihai Prunescu, Lorenzo Sauras-Altuzarra, and Joseph M. Shunia (2025), A Minimal Substitution Basis for the Kalmar Elementary Functions, the authors define a minimal generating set for the Kalmár ...

Question 5

Inspired by this question. The definition of elementary functions includes finite addition, subtraction, multiplication, division and composition of algebraic functions, trigonometric functions and ...

Question 6

We use the definition of elementary functions given in Spivak's calculus (with some changes so this is not exactly the same). An elementary function is one which can be obtained by a finite number of ...

Question 7

"Routine methods" here mean "methods like integration by parts and integration by substitution" (as the original asker referred to) (too long for the title). The original question ...

Question 8

Let $x \in (0, \pi/2]$. I am trying to prove the following inequality: $$ \frac{\sin^4 x}{x^4} + \frac{\sin x}{13} \ge \cos x. $$ I tried to tackle this by considering known approximations and bounds ...

Question 9

Let $n,m$ be integers with $n > m$. Consider the set of monic polynomials $P(x)$ over $\mathbb{C}$ of degree at most $n$ satisfying the linear constraints on coefficients $$P(x_i) = y_i$$ for some $...

Question 10

would anybody be so kind to check my proof? It is done by contradiction. Any adjustements insights welcomed. Problem: Prove that the sum of two decreasing functions is decreasing. Proof: Suppose $f$ ...

Question 11

A binomial differential is definied as $x^m(a+bx^n)^p$ with $a,b\in\mathbb{R}\setminus\{0\}$ and $m,n,p\in\mathbb{Q}$. A Chebichev's theorem (see https://encyclopediaofmath.org/wiki/...

Question 12

I have two functions, $f(x)$ and $g(x)$. $f(x) = ax^2$, where $-1 \le x \le 4$. $g(x)$ is a piecewise function: $x+b$ where $-1 \le x \le 0$, and $c-x$ where $0 \le x \le 1$. The two functions have ...

Question 13

Periods as defined by Kontsevich and Zagier are complex numbers α whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients, over ...

Question 14

I stumbled across a bland question on integrating $\sqrt{x^2+4}$ and I throw it into the integral calculator, https://www.integral-calculator.com/ Normally this can be solved via hyperbolic/trig ...

Question 15

I would like to find an elementary function, $f$, with the following properties: $f$ is continuous on [-1,1] except at zero $f'$ is unbounded near zero $\lim_{x\to 0^{+}} f(x)$ and $\lim_{x\to 0^{-}} ...

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Questions tagged [elementary-functions]

Is there a general criterion for when a function has an elementary inverse? [duplicate]

Constant field is preserved in elementary differential extension?

Adding equality $\frac{1}{2} \left(e^{2 i \pi B_- x}+e^{2 i \pi B_+ x}\right)=x \ln \left(\frac{B_+-x}{B_-+x}\right)$ to umbral calculus

Can superposition alone generate the Kalmár elementary function $x^y$ from $⟨x + y, x \bmod y, 2^x⟩$?

Do we need to include logarithm and exponent directly in the definition of elementary functions if we go from finite to countable function arithmetic?

Is the condition about composition needed to stay with the elementary functions?

If the antiderivative of an elementary function is also elementary, can we always compute its indefinite integral with "routine methods"?

For $x \in \left(0, \frac{\pi}{2}\right]$, prove that $\frac{\sin^4 x}{x^4} + \frac{\sin x}{13} \geq \cos x$.

Interpolating polynomial with simple roots

The sum of two decreasing functions is always decreasing - Please check my proof. [closed]

chebyschev theorem on the integration of binomial differentials [duplicate]

Composite functions [closed]

Does there exist an elementary number which is not a period?

Strange technique in integrating [duplicate]

Elementary function with unbounded derivative near zero and different one-sided limits at zero

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