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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-7
votes
0answers
21 views

How to check the availability of particular function [closed]

I want to check the availability of particular function that returns the value of results in binary like 0 or 1....How to check these function using mathematics...
1
vote
0answers
24 views

Power series solution for a DE with Frobenius method

The given DE is $(x²-3)y"+2xy'=0$ Since there is a singular point ($x=\pm\sqrt{3}$) I used the Frobenius method. I found two indicial relationships: $-3r(r+1)=0$ and $-3(r+1)(r+2)=0$ because I have ...
1
vote
1answer
10 views

Composition of mappings on finite sets

If I'm working in the realm of finite sets on the form $\underline{n} = \{1, \ldots, n\}, n \in \mathbb{N} $. Consider any two transformations $f :\underline{n} \to \underline{m}$ and $g ...
0
votes
1answer
36 views

Find a polynomial with certain conditions.

Suppose that: $$f(x) = 3\frac{x^4+x^3+x^2+1}{x^2+x-2}.$$ Find a polynomial $h(x)$ such that $f(x) + h(x)$ has horizontal asymptote of 0 as $x$ approaches positive infinity.
1
vote
0answers
34 views

Is it possible to find set of functions so that their antiderivatives are also in this set?

Antiderivatives of elementary functions sometimes are not elementary functions. Is it possible to add finite number of non-elementary functions to elementary function set so that their ...
1
vote
4answers
53 views

Equation , powers of two

I want to find the sum of the roots of the equation $$4(4^x + 4^{-x}) - 23(2^x + 2^{-x}) + 40 = 0 $$ in real numbers. I tried the substitution $ 2^x = t $ but then it turns into a quartic equation ...
23
votes
0answers
398 views

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...
37
votes
4answers
2k views

Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
1
vote
0answers
27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
3
votes
3answers
108 views

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$
1
vote
0answers
54 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
1
vote
0answers
44 views

Repository of functions which do not have elementary integrals [duplicate]

If there is some function and I suspect that the primitive function cannot be expressed using elementary functions, I would like to have some argument that there indeed is no such expression. One ...
4
votes
2answers
69 views

value of $a+b$ of the following function

$f(x)=x^3-3x^2+5x\;$ and $\;f(a)=1,f(b)=5.\;$ Find $a+b$. I know only one real root exist for each equation as derivative of the function is always positive .I do not intend to use the formula of ...
1
vote
0answers
13 views

efficiently solve for values of a coefficient in a function, so for those values, the function intersects another function a specific number of times.

This is my summer assignment for my freshman "Intro to Numerical Methods with Matlab: Unit 2" course. The task: "Write an efficient Matlab code, which will take any closed $f(x)$ and $g(x)$ and ...
2
votes
2answers
75 views

Properties of $f(a)-f(b)=a-b$

Let: $f(a)-f(b)=a-b$ and $f(a)>f(b)$ and $f(a),f(b)>0$ and $a>b$ Based on the above facts is it sufficient enough to say that $f(a)=a$ and $f(b)=b$?
0
votes
1answer
42 views

How to define a continuous function $f:I\times I\longrightarrow I$ such that $f(0, t)=t$ and $f(1, t)=1$ for all $t\in I$?

Let $I=[0, 1]$. I need some help to define a continuous function $f:I\times I\longrightarrow I$ such that $$f(0, t)=t\quad \textrm{and}\quad f(1, t)=1$$ for all $t\in I$. The nearest I got was: $$f(s, ...
1
vote
1answer
78 views

What does “versus” mean in the context of a graph?

Say you have a graph of say $y=mx+b$, with $x$ on the horizontal axis and $y$ on the vertical axis. You need to give the graph a title, would you say: This is a graph of "$y$ versus $x$?" or This ...
1
vote
1answer
38 views

Inverse of Higher logarithms

Th polylogarithm function is defined by $$Li_s(z)=\sum_{k=1}^\infty\frac{z^k}{k^s}.$$ At $s=1$, we have the natural logarithm function. We have the inverse of natural logarithm function as the ...
0
votes
1answer
23 views

How to work out the probability of 2 people having a different birthday

I'm trying to reproduce the 'birthday problem' where you work out the probabilities of n people having the same / different birthday. Theres a good example here: ...
0
votes
1answer
57 views

How to graph a sin & cos wave

I've got a time column of numbers(x) from 0 - 20 in 0.1 intervals and i've got a sin(x) and cos(x) column. I've got to produce some kind of wave like graph from it. Although I'm not sure which numbers ...
0
votes
1answer
56 views

Inverse of $a f(x)$ and inverse of $a f(x) + b$

Is there a general rule for the inverse of the function $ g(x) = a f(x) $, where $a$ is a constant, assuming $f^{-1}(x)$ is defined? Follow up: $g(x) = a f(x) + b$. Is the following correct, given ...
0
votes
2answers
17 views

Distributive property

I have the following question on distributive property. If I multiply $3$ to the given expression $\frac{1}{3}\pi r^2h$. Question: $3\cdot\frac{1}{3}\pi r^2h$ Based on what I understand for ...
8
votes
0answers
55 views

Is there a name for the class of functions which are infinitely integrable in elementary functions?

Is there a name for the class of functions which are infinitely integrable in elementary functions, that is whose consecutive integrals also elementary not depending on how much times we took the ...
0
votes
0answers
41 views

Why this function is elementary and its pair is not?

Why $$f(x)=\frac{2 \zeta (2)}{\pi ^2}+\frac{6 \zeta(4) x^2}{\pi^4}+\frac{10 \zeta(6) x^4}{\pi^6}+\frac{14 \zeta(8) x^6}{\pi^8}+\cdots$$ is elementary while $$g(x)=\frac{4 \zeta (3)x}{\pi ...
2
votes
2answers
73 views

Liouville's Criterion and Integration of $\sin(z)/z$ using elementary functions

Liouville's Criterion: $\int fe^g$ is elementary iff there is rational function $q$ in $\mathbb{C}(z)$ such that $$f=q'+qg'$$ where $f\neq0$ and $g$ is a non constant function in $\mathbb{C}(z)$. Now ...
0
votes
1answer
23 views

$T: [0,1)\to [0,1), x\mapsto 10x-\lfloor 10x\rfloor$

Consider $T: [0,1)\to [0,1), x\mapsto 10x-\lfloor 10x\rfloor$. Is that the same as $$ 10x (mod 1)? $$ or in which sense is that multiplication with 10 mod 1?
8
votes
0answers
164 views

Is each “elementary + finite functions” function “elementary + finite functions”-integrable?

It is known that there exists elementary functions which are not elementary integrable, i. e. there exists no elementary anti derivative. Example: $f(x) = e^{-x^2}$. Let $A$ be the set of elementary ...
3
votes
2answers
302 views

List of functions not integrable in elementary terms

When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...
2
votes
1answer
12 views

Simultaneous Equation (I think)

I am not sure whether I am just not remembering the technique or I don't have enough clues to solve this one: $T_1 - T_2 = 362$ $\frac{T_1}{T_2} = 5.48$ I cannot seem to solve for $T_1$ or $T_2$ I ...
0
votes
0answers
45 views

max min sum product

I am confused with this elementary thing: For all real $x,y$ define $\vee, \wedge$ by $x\vee y=\max\{x,y\}$ and $x\wedge y=\min\{x,y\}$. I am going to know the closed formula of sum and product ...
7
votes
2answers
125 views

What is the mathematical relevance of whether an expression has a closed form?

In the evaluation of mathematical expressions, particularly integrals, I often find a statement that the expression has or does not have a closed form. I looked up the definition, and the important ...
0
votes
1answer
23 views

Would this function be correct?

I want to express a function that for any x input, it outputs the nearest EVEN integer less than or equal to x. Would $g(x) = \{ \lfloor x \rfloor : 2 \mid x \} $ do the job properly? Read: $g(x)$ ...
3
votes
1answer
152 views

How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int ...
0
votes
0answers
27 views

About a specific mathematical series which is a power of the exponential function

My professor wrote the below exponential function just out of the box when he suggested a kernal for a 1D domain. $f(x) = e^{-\Big(a_1x_1+ \dfrac{1}{2} a_2 x_2^2 + \dfrac{1}{3} a_3 x_3^3 + .... ...
1
vote
3answers
108 views

Equation including tangent function

I've been studying one problem and I need to consider the following problem. Let $n\in\mathbb{N}$. Is $\tan\left(nx\right)=n \tan(x)$ solvable on $\left(0,\pi\right)$? If it is, what are solutions?
1
vote
2answers
86 views

Is there anything “special” about elementary functions?

I just found an article on Liouville's integrability criterion, which gave me a thought. What makes functions like $\mathrm{Si}(x)$, $\mathrm{Ei}(x)$, $\mathrm{erfc}(x)$, etc. inherently different ...
8
votes
3answers
133 views

How to demonstrate the equality of these integral representations of $\pi$?

Each of the following definite integrals are well known to have the value $\pi$: $$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}dx=2\int_{-1}^1\sqrt{1-x^2}dx=\int_{-\infty}^{\infty}\frac{1}{1+x^2}dx=\pi.$$ I ...
2
votes
0answers
23 views

probability subspaces that make entropy function equal to a constant value

Given the entropy fucntion: $$ H = -\sum_i^n p(i) \ln(p(i))\,.$$ where $p(i)$ are probabilities and $n=4$, I would like to know all the points in the probability space that make $H = k$, being $k$ a ...
-4
votes
2answers
85 views

How could we define $\cos (x)$ in terms of $x$?

How could we define $\cos (x)$ in terms of $x$? For example we define $\Gamma(n)=(n-1)!$ which is purely defined in terms of $n$. But how about $\cos(x)$, can it be equal to something that is ...
2
votes
7answers
108 views

Solve for $n$ in $ \left(n+1 \right) 0.5^n=0.05$

$$ \left(n+1 \right) \times 0.5^n =0.05 $$ Is there a way to solve this directly for $n$? I know that by taking logs we can simplify it but we still do not get a value as far as I can see. A solution ...
4
votes
1answer
79 views

Finding the maximum number of $x\in\mathbb R$ such that $a^x+b=\lfloor x\rfloor.$

Question : Letting $a\gt 1, b\in\mathbb R$, at most how many real number solutions does the following equation have? $$a^x+b=\lfloor x\rfloor.$$ Here, $\lfloor x\rfloor$ is the largest ...
5
votes
1answer
119 views

bijection from $\mathbb{Q} - \{a\}$ to $\mathbb{Q}$ using elementary functions only?

I was wondering, can you define a bijection from $\mathbb{Q} - \{a\}$ to $\mathbb{Q}$ using elementary functions only ($a \in \mathbb{Q}$)? Of course there are many set theoretic bijections like ...
1
vote
1answer
80 views

Solving Some Transcendental Equations

How do you solve for $a$ in each of the equations $$a^{a^a}=b^c$$ $$a^{a^{a^a}}=b^c$$ $$a^{a^{a^a}}=b^{c^d}?$$
0
votes
3answers
81 views

Is the function $f\colon \mathbb{R} \to \mathbb{R}$ defined by $f(x)=x+\sin(x)$ uniformly continuous on $\mathbb{R}$

I know how to show not uniformly or how to show it is uniformly continuous but not how to differentiate when to know which one to use
2
votes
0answers
127 views

Integrating non-elementary functions

An elementary function is a function that can be represented by a finite number of exponentials, logarithms, nth roots, and constants through composition. Clearly, an non-elementary function that is ...
2
votes
2answers
74 views

Showing Inequality using Gauss Function

If $\alpha, \beta\in \Bbb{R}$ and $m, n\in \Bbb{N}$ show that the inequality $[(m+n)\alpha]+[(m+n)\beta] \ge [m\alpha]+[m\beta]+[n\alpha+n\beta]$ holds iff m=n I thought that we have to ...
5
votes
3answers
325 views

Can any continuous function be represented as an infinite polynomial?

Can any continuous function be represented as an infinite polynomial? Motivation: the antiderivative $ \int^\ e^{-x^2}dx\ $ can be expressed as an infinite polynomial(write Taylor series for ...
1
vote
0answers
143 views

Learning about maximal and minimal elements of a set…struggling a little bit with these questions.

Ok so I'm learning about partial orders, and I think I know the difference between a minimum/minimal & maximum/maximal element of a partially ordered set... but I just was wondering if someone ...
15
votes
2answers
233 views

How much math do we need to prove all simple numeric identities?

Consider real numeric expressions build only from integers, operators $+,-,\times,/$ and taking a positive expression to a power (no variables involved), e.g. $$\frac{2}{7},\ 2^{1/2},\ ...
0
votes
0answers
44 views

calculate indicator/characteristic function with summation, multiplication

I want to transform the indicator function on an interval [a,b] to a function $f:[x_{min},x_{max}] \to \{0,1\}$ using only elementary operations like summation multiplication modulo (continuous ...